\magnification=1200 \parskip=6pt plus 2pt minus 1pt \parindent=0cm \font\tenCal=eusm10 \font\sevenCal=eusm7 \font\fiveCal=eusm5 \newfam\Calfam \textfont\Calfam=\tenCal \scriptfont\Calfam=\sevenCal \scriptscriptfont\Calfam=\fiveCal \def\Cal{\fam\Calfam\tenCal} \font\bigbf=cmbx10 at 12pt \font\tenBbb=msbm10 \font\sevenBbb=msbm7 \font\fiveBbb=msbm5 \newfam\Bbbfam \textfont\Bbbfam=\tenBbb \scriptfont\Bbbfam=\sevenBbb \scriptscriptfont\Bbbfam=\fiveBbb \def\Bbb{\fam\Bbbfam\tenBbb} % Usual sets of numbers \def\bC{{\Bbb C}} \def\bN{{\Bbb N}} \def\bP{{\Bbb P}} \def\bQ{{\Bbb Q}} \def\bR{{\Bbb R}} \def\bZ{{\Bbb Z}} \long\def\RGBColor#1#2{{\pdfliteral{#1 rg}#2\pdfliteral{0 g}}} \long\def\red#1{\RGBColor{1 0 0}{#1}} \def\D#1{#1} \def\url#1{\red{#1}} \let\text\rm \let\mathrm\rm \let\mathbb\Bbb \let\tag\leqno \def\mathcal#1{{\Cal #1}} \def\frac#1#2{{#1\over #2}} \def\srelbar{\mathrel{\smash{\hbox{\rm-}}}} \def\dashrightarrow{\srelbar\,\srelbar\,\rightarrow} \def\ssm{\mathop{\hbox{\Bbb r}}} \def\newline{\hfill\break} \newcount\artind \artind=0 \def\Author{\advance\artind by 1\bigskip\line{\hrulefill}\par\bf} \def\MSC{\par{\it MSC$\,$}:\\} \def\Title{\par\rm} \def\Publi{\par\rm} \def\Full_text{} \def\Show_scanned_page{} \def\Keywords{\par{\it Keywords$\,$}:\ } \def\References{\par{\it References$\,$}:\par} \def\Reviewer{\par{\it Reviewer$\,$}:\ } \def\Cited{\par Cited\ } \def\.{$\cdot$} \def\[{\par[} \def\\{\hfil\break} \centerline{\bigbf Publication list \&\ Reviews of Jean-Pierre Demailly} \medskip \centerline{\bigbf as found on Zentralblatt Math} \medskip \centerline{\bigbf November 2021} \medskip \Author Demailly, Jean-Pierre \Title Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles. (English. Russian original) Zbl 1464.32028 \Publi Sb. Math. 212, No. 3, 305-318 (2021); translation from Mat. Sb. 212, No. 3, 39-53 (2021). \MSC 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) \Keywords ample vector bundle; Griffiths positivity; Hermitian-Yang-Mills equation \References \[1] Berndtsson, B., Curvature of vector bundles associated to holomorphic fibrations, Ann. of Math. (2), 169, 2, 531-560 (2009) -- Zbl 1195.32012 \[2] Campana, F.; Flenner, H., A characterization of ample vector bundles on a curve, Math. Ann., 287, 4, 571-575 (1990) -- Zbl 0728.14033 \[3] Demailly, J.-P.; Skoda, H., Relations entre les notions de positivités de P. A. Griffiths et de S. Nakano pour les fibrés vectoriels, Séminaire Pierre Lelong-Henri Skoda (Analyse). Années 1978/79, 822, 304-309 (1980) -- Zbl 0454.55011 \[4] Donaldson, S. K., Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. (3), 50, 1, 1-26 (1985) -- Zbl 0529.53018 \[5] Griffiths, P. A., Hermitian differential geometry, Chern classes and positive vector bundles, Global analysis, 181-251 (1969) -- Zbl 0201.24001 \[6] Kodaira, K., On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties), Ann. of Math. (2), 60, 28-48 (1954) -- Zbl 0057.14102 \[7] Mourougane, C.; Takayama, S., Hodge metrics and positivity of direct images, J. Reine Angew. Math., 2007, 606, 167-178 (2007) -- Zbl 1128.14030 \[8] Nakano, S., On complex analytic vector bundles, J. Math. Soc. Japan, 7, 1, 1-12 (1955) -- Zbl 0068.34403 \[9] Narasimhan, M. S.; Seshadri, C. S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2), 82, 3, 540-567 (1965) -- Zbl 0171.04803 \[10] Naumann, P., An approach to Griffiths conjecture \[11] Pingali, V. P., A vector bundle version of the Monge-Ampère equation, Adv. Math., 360 (2020) -- Zbl 1452.32047 \[12] Pingali, V. P., A note on Demailly’s approach towards a conjecture of Griffiths, C. R. Math. Acad. Sci. Paris (2021) \[13] Uhlenbeck, K.; Yau, S. T., On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math., 39, S1, S257-S293 (1986) -- Zbl 0615.58045 \[14] Umemura, H., Some results in the theory of vector bundles, Nagoya Math. J., 52, 97-128 (1973) -- Zbl 0271.14005 \[15] Yau, Shing-Tung, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math., 31, 3, 339-411 (1978) -- Zbl 0369.53059 \Author Angella, Daniele (ed.); Cirici, Joana (ed.); Demailly, Jean-Pierre (ed.); Wilson, Scott (ed.) \Title Mini-workshop: almost complex geometry. Abstracts from the mini-workshop held October 4–10, 2020 (online meeting). (English) Zbl 07406499 \Publi Oberwolfach Rep. 17, No. 4, 1657-1691 (2020). {\it Summary}\/: The mini-workshop focused on very recent developments of analytic and algebraic techniques for studying almost complex structures which are not necessarily integrable. It provided a forum to discuss and compare techniques from PDE’s, elliptic theory, deep algebraic structures, as well as geometric flows, and new topological invariants, towards attacking several longstanding open problems in the field of complex and almost complex geometry. \MSC 00B05 Collections of abstracts of lectures 00B25 Proceedings of conferences of miscellaneous specific interest 32-06 Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces 32Q60 Almost complex manifolds 53Cxx Global differential geometry 58Jxx Partial differential equations on manifolds; differential operators 32J27 Compact Kähler manifolds: generalizations, classification \References \[1] ABG + 18, I. Agricola, G. Bazzoni, O. Goertsches, P. Konstantis, and S. Rollenske, On the history of the Hopf problem, Differential Geom. Appl. 57 (2018), 1-9. -- Zbl 1381.53003 \[2] D. Angella, A. Otiman, and N. Tardini, Cohomologies of locally conformally symplec-tic manifolds and solvmanifolds, Ann. Global Anal. Geom. 53 (2018), no. 1, 67-96. -- Zbl 1394.32022 \[3] D. Angella and A. Tomassini, On the $\partial\overline\partial$-lemma and Bott-Chern cohomology, Invent. Math. 192 (2013), no. 1, 71-81. -- Zbl 1271.32011 \[4] G. Bazzoni and V. Muñoz, Classification of minimal algebras over any field up to dimension 6, Trans. Amer. Math. Soc. 364 (2012), no. 2, 1007-1028. -- Zbl 1239.55003 \[5] S. Boucksom, J.-P. Demailly, M. P\u{a}un, Th. Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, arXiv:math/ 0405285 and J. Algebraic Geom. 22 (2013), 201-248. -- Zbl 1267.32017 \[6] N. Buchdahl, On compact Kähler surfaces, Ann. Inst. 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Fourier (Grenoble) 49 (1999), no. 1, vii, x, 263-285. -- Zbl 0926.32026 \[20] A. Latorre and R. Ugarte, L.and Villacampa, On the real homotopy type of generalized complex nilmanifolds, arXiv e-prints (2019), arXiv:1905.11111. \[21] H. V. Lê and J. Vanžura, Cohomology theories on locally conformal symplectic man-ifolds, Asian J. Math. 19 (2015), no. 1, 45-82. -- Zbl 1323.53092 \[22] T.-J. Li and W. Zhang, Almost Kähler forms on rational 4-manifolds, Amer. J. Math. 137 (2015), no. 5, 1209-1256. -- Zbl 1331.53103 \[23] D. McDuff and D. Salamon, J-Holomorphic Curves and Symplectic Topology, Collo-quium Publ., Vol. 52, Amer. Math. Soc, 2004, 669 pp. \[24] A. Milivojevic, The rational topology of closed almost complex manifolds, Ph.D thesis, to appear (2020). \[25] L. Ornea and M. Verbitsky, Supersymmetry and Hodge theory on Sasakian and Vais-man manifolds, Preprint arXiv:1910.01621 (2019). \[26] D. Popovici, Transcendental Kähler cohomology classes, Publ. Res. Inst. Math. Sci. 49 (2013), no. 2, 313-360. -- Zbl 1277.53075 \[27] D. Popovici, Degeneration at E 2 of certain spectral sequences, Internat. J. Math. 27 (2016), no. 14, 1650111, 31 pp. -- Zbl 1365.53067 \[28] T. Shin, Almost complex manifolds are (almost) complex, Preprint arXiv:\break 1903.10043 (2019). \[29] N. Tardini and A. Tomassini, Differential operators on almost-Hermitian manifolds and harmonic forms, Preprint arXiv:1909.06569 (2019). \[30] C.-J. Tsai, L.-S. Tseng, and S.-T. Yau, Cohomology and Hodge theory on symplectic manifolds: III, J. Differential Geom. 103 (2016), no. 1, 83-143. -- Zbl 1353.53085 \[31] V. Tosatti and B. Weinkove, The Calabi-Yau equation, symplectic forms and almost complex structures, Geometry and analysis. No. 1, Adv. Lect. Math. (ALM), vol. 17, 2011, pp. 475-493. -- Zbl 1268.53079 \[32] V. Tosatti, B. Weinkove, and S.-T. Yau, Taming symplectic forms and the Calabi-Yau equation, Proc. Lond. Math. Soc. (3) 97 (2008), no. 2, 401-424. -- Zbl 1153.53054 \[33] M. Verbitsky, Hodge theory on nearly Kähler manifolds, Geom. Topol. 15 (2011), no. 4, 2111-2133. -- Zbl 1246.58002 \[34] S. O. Wilson, Harmonic symmetries for Hermitian manifolds, Preprint arXiv:\break 1906.02952, to appear in Proc. AMS. (2019). \[35] S.-T. Yau, Open problems in geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 1-28. \[36] S. Dinew, D. Popovici, A Generalised Volume Invariant for Aeppli Cohomology Classes of Hermitian-Symplectic Metrics, arXiv e-print DG 2007.10647v1. \[37] S. K. Donaldson, Two-forms on Four-manifolds and Elliptic Equations Inspired by S. S. Chern, 153-172, Nankai Tracts Math., 11, World Sci.Publ., Hackensack, NJ, 2006. -- Zbl 1140.58018 \[38] R. Harvey, H. B. Lawson, An intrinsic characterization of Kähler manifolds, Invent. Math. 74, (1983), 169-198. -- Zbl 0553.32008 \[39] J. Streets, G. Tian, A Parabolic Flow of Pluriclosed Metrics, Int. Math. Res. Notices, 16 (2010), 3101-3133. -- Zbl 1198.53077 \[40] D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math. 36 (1976), 225-255. -- Zbl 0335.57015 \[41] J. Armstrong, On four-dimensional almost-Kähler manifolds, Quart. J. Oxford Ser. (2), 48, 1997. \[42] A. Borel, A spectral sequence for complex analytic bundles, Appendix Two in: F. Hirzebruch Topological Methods in Algebraic Geometry, vol. 131 of Grundlehren Der Mathematischen Wissenschaften. Springer, Berlin [et al.], 1978. \[43] R.L. Bryant, On the geometry of almost complex 6-manifolds, Asian J. Math. 10:3, 2006. -- Zbl 1114.53026 \[44] J. Cirici, S. Wilson, Dolbeault cohomology for almost complex manifolds. arXiv:\break 1809.01416 [math], 2018. \[45] J. Stelzig, On the structure of double complexes. arXiv:1812.00865 [math], 2018. References \[46] M. Albanese and A. Milivojević, 2019. On the minimal sum of Betti numbers of an almost complex manifold. Differential Geometry and its Applications, 62, pp.101-108. -- Zbl 1417.32030 \[47] M. Albanese and A. Milivojević, 2019. Connected sums of almost complex manifolds, prod-ucts of rational homology spheres, and the twisted spin c Dirac operator. Topology and its Applications, 267, p.106890. -- Zbl 1469.32022 \[48] A. Borel and J-P. Serre, 1953. Groupes de Lie et puissances réduites de Steenrod. American Journal of Mathematics, 75 (3), pp.409-448. -- Zbl 0050.39603 \[49] M. Gromov, 1986. Partial differential relations. Ergebnisse der Mathematik und ihrer Gren-zgebiete, (3), pp.Vol-9. \[50] A. Haefliger, 1971. Lectures on the theorem of Gromov. In Proceedings of Liverpool Singu-larities Symposium II (pp. 128-141). Springer, Berlin, Heidelberg. -- Zbl 0222.57020 \[51] S.T. Yau, 1993. Open problems in geometry. In Proc. Symp. Pure Math, 54, No. 1, pp. 1-28. References -- Zbl 0801.53001 \[52] D. Sullivan, 1977. Infinitesimal computations in topology. Publications Mathématiques de l’Institut des HautesÉtudes Scientifiques, 47(1), pp.269-331. -- Zbl 0374.57002 \[53] W. Browder, 2012. Surgery on simply-connected manifolds (Vol. 65). Springer Science \&\ Business Media. \[54] R.E. Stong, 1965. Relations among characteristic numbers-I. Topology, 4(3), pp.267-281. -- Zbl 0136.20503 \[55] Su, Z., 2014. Rational analogs of projective planes. Algebraic \&\ Geometric Topology, 14(1), pp.421-438. -- Zbl 1291.57019 \[56] L. Kennard, and Z. Su, 2019. On dimensions supporting a rational projective plane. Journal of Topology and Analysis, 11(03), pp.535-555. -- Zbl 1432.57056 \[57] M. Albanese, and A. Milivojević, 2019. On the minimal sum of Betti numbers of an almost complex manifold. Differential Geometry and its Applications, 62, pp.101-108. -- Zbl 1417.32030 \[58] J. Cirici, S. O. Wilson, Dolbeault cohomology for almost complex manifolds, arXiv:1809.01416 [math.DG], 2020. \[59] J. Cirici, S. O. Wilson, Topological and geometric aspects of almost Kähler manifolds via harmonic theory, Sel. Math. New Ser., 26, no. 35 (2020). -- Zbl 1442.32034 \[60] P. de Bartolomeis, A. Tomassini, On formality of some symplectic manifolds. Internat. Math. Res. Notices 2001, no. 24, (2001) 1287-1314. \[61] F. Hirzebruch, Some problems on differentiable and complex manifolds, Ann. of Math. (2) 60, (1954). 213-236. -- Zbl 0056.16803 \[62] T. Holt, W. Zhang, Harmonic Forms on the Kodaira-Thurston Manifold, arXiv:\break 2001.10962 [math.DG], 2020. \[63] N. Tardini, A. Tomassini, Differential operators on almost-Hermitian manifolds and har-monic forms, Complex Manifolds, 7, no. 1, (2020) 106-128. -- Zbl 1439.32069 \[64] F. Hirzebruch, Some problems on differentiable and complex manifolds, Ann. of Math. (2) 60, (1954). 213-236. -- Zbl 0056.16803 \[65] H. Chen and W. Zhang, Kodaira Dimensions of Almost Complex Manifolds I, arXiv:1808.00885, 2018. \[66] T. Holt and W. Zhang, Harmonic Forms on the Kodaira-Thurston Manifold, arXiv:2001.10962, 2020. \[67] T. Holt and W. Zhang, Almost Kähler Kodaira-Spencer problem, arXiv:2010.12545, 2020. References \[68] E. Abbena, A. Grassi, Hermitian left invariant metrics on complex Lie groups and cosym-plectic hermitian manifolds, Boll. Un. Mat. Ital. A 5 (1986) 371-379. -- Zbl 0607.53040 \[69] L. Alessandrini, G. Bassanelli, Modifications of compact balanced manifolds, C. R. Acad. Sci. Paris Sér. I Math 320 (1995) no. 12, 1517-1522. -- Zbl 0842.32024 \[70] J Chu, L. Huang, X. Zhu, The Fu-Yau equation on compact astheno-Kähler manifolds, Adv. Math. 346 (2019), 908-945. -- Zbl 1411.58008 \[71] J. Chu, L. Huang, X. Zhu, The Fu-Yau equation in higher dimensions, Peking Math. J. 2 (2019), no. 1, 71-97. -- Zbl 1416.58010 \[72] T. Fei, Z.J. Huang, S. Picard, A construction of infinitely many solutions to the Strominger system, arXiv:1703.10067, To appear in J. Diff. Geom. -- Zbl 1456.81334 \[73] T. Fei, Z.J. Huang, S. Picard, The Anomaly flow over Riemann surfaces, arXiv:\break 1711.08186 to appear in Int. Math. Res. Notices. -- Zbl 1470.35075 \[74] T. Fei , S.-T. Yau, Invariant Solutions to the Strominger System on Complex Lie Groups and Their Quotients, Comm. Math. Phys. 338 (2015) No. 3, 1183-1195. -- Zbl 1319.32022 \[75] M. Fernández, S. Ivanov, L. Ugarte, R. Villacampa, Non-Kähler heterotic-string compact-ifications with non-zero fluxes and constant dilaton, Commun. Math. Phys. 288 (2009), 677-697. -- Zbl 1197.83103 \[76] A. Fino, G. Grantcharov, L. Vezzoni, Solutions to the Hull-Strominger system with torus symmetry, , arXiv:1901.10322. \[77] J.X. Fu, L. S. Tseng, S.T. Yau, Local heterotic torsional models, Comm. Math. Phys. 289 (2009) 1151-1169. -- Zbl 1167.83309 \[78] J.X. Fu, S.T. Yau, A Monge-Ampère type equation motivated by string theory, Comm. Anal. Geom. 15 (2007) 29-76. -- Zbl 1122.35145 \[79] J.X. Fu, S.T. Yau, The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation, J. Differential Geom. 78 (2008), no. 3, 369-428. -- Zbl 1141.53036 \[80] M. Garcia-Fernandez, T-dual solutions of the Hull-Strominger system on non-Kähler three-folds, J. Reine Angew. Math. 766 (2020), 137-150. -- Zbl 1447.58033 \[81] C. Hull, Superstring compactifications with torsion and space-time supersymmetry, In Turin 1985 Proceedings “Superunification and Extra Dimensions” (1986), 347-375. \[82] J. Kollár, Seifert Gm-bundles, arXiv: math/0404386. \[83] J. Li, S.T. Yau, The existence of supersymmetric string theory with torsion, J. Differential Geom. 70 (2005) 143-181. -- Zbl 1102.53052 \[84] M.L. Michelsohn, On the existence of special metrics in complex geometry, Acta Math. 143 (1983), pp. 261-295. -- Zbl 0531.53053 \[85] A. Otal, L. Ugarte, R. Villacampa, Invariant solutions to the Strominger system and the heterotic equations of motion, Nuclear Phys. 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Reporter: Michael Albanese -- Zbl 0355.32028 \Author Demailly, Jean-Pierre (ed.); Dinh, Tien-Cuong (ed.); Hai, Le Mau (ed.); Hiep, Pham Hoang (ed.); Khoai, Ha Huy (ed.); Ma, Xiaonan (ed.); Marinescu, George (ed.); Peternell, Thomas (ed.); Sibony, Nessim (ed.) \Title Preface. (English) Zbl 1433.00044 \Publi Acta Math. Vietnam. 45, No. 1, 1-2 (2020). From the text$,$: This issue of Acta Mathematica Vietnamica is dedicated to the proceedings of the Conference “Nevanlinna theory and complex geometry” in Honor of Le Van Thiem’s Centenary (Hanoi, 26/02/2018–02/03/2018). \MSC 00B25 Proceedings of conferences of miscellaneous specific interest\\ 32-06 Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces {\it Preface} Le Van Thiem received his doctorate in 1945 from Göttingen and he got his Doctorat d’État at the École Normale Supérieure de Paris in 1949. In 1949, Le Van Thiem returned to Vietnam in the middle of a fierce resistance war for independence of Vietnam. He had a great contribution in building the University in the headquarters of the Resistance, and became a teacher of the first Vietnamese mathematicians. For the next dozen years, Vietnamese mathematicians were either his students or students of his students. We can say that Le Van Thiem is the founder of Mathematics in Vietnam. In the first stage of his career, Le Van Thiem made a great contribution in the inverse problem of the value distribution theory of meromorphic functions (Nevanlinna theory). Later, he turned to applied mathematics, contributing to solving problems raised in Vietnamese practice such as oriented explosion, ground water under irrigation schemes, and the problem of calculating petroleum reserves. Le Van Thiem himself is a part of the history of Mathematics in Vietnam. His name is given to a street in Hanoi and some schools throughout the country. This issue of Acta Mathematica Vietnamica is dedicated to the proceedings of the Conference “Nevanlinna theory and Complex Geometry” in Honor of Le Van Thiem’s Centenary (Hanoi, 26/02/2018 -- 02/03/2018). We thank the Institute of Mathematics of Vietnam and its staff for their help to make the Conference possible. \medskip {\it Selected Papers of Le Van Thiem} \References \[1] Thiem, L.-V.: Beitrag sum Typenproblem der Riemannschen Flachen. Comment. Math. Helv. 20, 270–287 (1947) \[2] Thiem, L.-V.: Uber das Umkehrproblem der Werterteilungslehre. Comment. Math. Helv. 23, 26–49 (1949) \[3] Thiem, L.-V.: Le degré de ramification d’une surface de Riemann et la croissance de la caractéristique de la fonction uniformisante. C. R. Acad. Sc. Paris 228, 1192–1195 (1949) \[4] Thiem, L.-V.: Un problème de type généralisé. C. R. Acad. Sc. Paris 228, 1270–1272 (1949) \[5] Thiem, L.-V.: Sur un problème d’inversion dans la théorie des fonctions méromorphes. Ann. Sci. Ecole Normale Sup. 67, 51–98 (1950) \[6] Thiem, L.-V.: Sur un problème d’infiltration à travers un sol à deux couchés. Acta Sci. Vietnam., Sectio Sci. Math. et Phys. 1, 3–9 (1964) \Author Demailly, Jean-Pierre \Title Recent results on the Kobayashi and Green-Griffiths-Lang conjectures. (English) Zbl 1436.32086 \Publi Jpn. J. Math. (3) 15, No. 1, 1-120 (2020). {\it Summary}$\,$: The study of entire holomorphic curves contained in projective algebraic varieties is intimately related to fascinating questions of geometry and number theory – especially through the concepts of curvature and positivity which are central themes in Kodaira’s contributions to mathematics. The aim of these lectures is to present recent results concerning the geometric side of the problem. The Green-Griffiths-Lang conjecture stipulates that for every projective variety $X$ of general type over $\bC$, there exists a proper algebraic subvariety $Y$ of $X$ containing all entire curves $f:\bC\to X$. Using the formalism of directed varieties and jet bundles, we show that this assertion holds true in case $X$ satisfies a strong general type condition that is related to a certain jet-semi-stability property of the tangent bundle $T_X$. It is possible to exploit similar techniques to investigate a famous conjecture of Shoshichi Kobayashi (1970), according to which a generic algebraic hypersurface of dimension nn and of sufficiently large degree $d\geq d_n$ in the complex projective space $\bP^{n+1}$ is hyperbolic: in the early 2000’s, Yum-Tong Siu proposed a strategy that led in 2015 to a proof based on a clever use of slanted vector fields on jet spaces, combined with Nevanlinna theory arguments. In 2016, the conjecture has been settled in a different way by Damian Brotbek, making a more direct use of Wronskian differential operators and associated multiplier ideals; shortly afterwards, Ya Deng showed how the proof could be modified to yield an explicit value of dndn. We give here a short proof based on a substantial simplification of their ideas, producing a bound very similar to Deng’s original estimate, namely $d_n=\lfloor{1\over 3}(en)^{2n+2}\rfloor$. \MSC 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds\\ 32H30 Value distribution theory in higher dimensions\\ 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results\\ 53C55 Global differential geometry of Hermitian and Kählerian manifolds\\ 14J40 $n$-folds $(n>4)$ \Keywords Kobayashi hyperbolic variety; directed manifold; genus of a curve; jet bundle; jet differential; jet metric; Chern connection and curvature; negativity of jet curvature; variety of general type; Kobayashi conjecture; Green-Griffiths conjecture; Lang conjecture \References \[1] Acta Soc. Sci. Fennicae. Nova Ser. A.194134 \[2] Arrondo, E.; Sols, I.; Speiser, R., Global moduli for contacts, Ark. 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Math., 202, 1069-1166 (2015) -- Zbl 1333.32020 \[146] Siu, Y-T; Yeung, S-K, Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane, Invent. Math., 124, 573-618 (1996) -- Zbl 0856.32017 \[147] Siu, Y-T; Yeung, S-K, A generalized Bloch’s theorem and the hyperbolicity of the complement of an ample divisor in an abelian variety, Math. Ann., 306, 743-758 (1996) -- Zbl 0882.32009 \[148] Siu, Y-T; Yeung, S-K, Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees, Amer. J. Math., 119, 1139-1172 (1997) -- Zbl 0947.32012 \[149] Trapani, S., Numerical criteria for the positivity of the difference of ample divisors, Math. Z., 219, 387-401 (1995) -- Zbl 0828.14002 \[150] Tsuji, H., Stability of tangent bundles of minimal algebraic varieties, Topology, 27, 429-442 (1988) -- Zbl 0698.14008 \[151] Venturini, S., The Kobayashi metric on complex spaces, Math. Ann., 305, 25-44 (1996) -- Zbl 0859.32011 \[152] Voisin, C., On a conjecture of Clemens on rational curves on hypersurfaces, J. Differential Geom., 44, 200-213 (1996) -- Zbl 0883.14022 \[153] P. Vojta, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Math., 1239, Springer-Verlag, 1987. -- Zbl 0609.14011 \[154] Winkelmann, J., On Brody and entire curves, Bull. Soc. Math. France, 135, 25-46 (2007) -- Zbl 1159.32015 \[155] Xie, S-Y, On the ampleness of the cotangent bundles of complete intersections, Invent. Math., 212, 941-996 (2018) -- Zbl 1405.14119 \[156] Xu, G., Subvarieties of general hypersurfaces in projective space, J. Differential Geom., 39, 139-172 (1994) -- Zbl 0823.14030 \[157] Zaidenberg, Mg, The complement of a general hypersurface of degree 2n in $\bC\bP^n$ is not hyperbolic, Siberian Math. J., 28, 425-432 (1988) \[158] M.G. Zaidenberg, Hyperbolicity in projective spaces, In: International Symposium on Holomorphic Mappings, Diophantine Geometry and Related Topics, Kyoto, 1992, S\=urikaisekikenky\=usho K\=oky\=uroku, 819, Kyoto Univ., 1993, pp. 136-156. \Author Campana, Frédéric; Demailly, Jean-Pierre; Peternell, Thomas \Title The algebraic dimension of compact complex threefolds with vanishing second Betti number. (English) Zbl 1436.32071 \Publi Compos. Math. 156, No. 4, 679-696 (2020). {\it Summary$\,$}: We study compact complex three-dimensional manifolds with vanishing second Betti number. In particular, we show that a compact complex manifold homeomorphic to the six-dimensional sphere does carry any non-constant meromorphic function. \MSC 32J17 Compact complex 33-folds\\ 32J10 Algebraic dependence theorems \Keywords compact complex threefold; algebraic dimension; algebraic reduction \References \[1] Barth, W. P., Hulek, K., Peters, C. A. M. and Van De Ven, A., Compact complex surfaces, , second edition (Springer, Berlin, 2004). -- Zbl 1036.14016 \[2] Campana, F., Demailly, J. P. and Peternell, Th., The algebraic dimension of compact complex threefolds with vanishing second Betti number, Comp. Math.112 (1998), 77-91. -- Zbl 0910.32032 \[3] Etesi, G., Complex structure on the six dimensional sphere from a spontaneous symmetry breaking, J. Math. Phys.56 (2015), 043508; Erratum ibid. 099901. -- Zbl 1322.81079 \[4] Gellhaus, Ch. and Heinzner, P., Komplexe Flächen mit holomorphen Vektorfeldern, Abh. Math. Sem. Univ. Hambg.60 (1990), 37-46. -- Zbl 0734.32017 \[5] Hartshorne, R., Algebraic geometry, (Springer, Berlin, 1977). -- Zbl 0367.14001 \[6] Inoue, M., On surfaces of type $VII_0$, Invent. Math.24 (1974), 269-310. -- Zbl 0283.32019 \[7] Kunz, E. and Waldi, R., Der Führer einer Gorensteinvarietät, J. Reine Angew. Math.388 (1988), 106-115. -- Zbl 0666.14020 \[8] Lehn, C., Rollenske, S. and Schinko, C., The complex geometry of a hypothetical complex structure on $S^6$, Differential Geom. Appl.57 (2018), 121-137. -- Zbl 1381.53008 \[9] Mori, S., Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2)116 (1982), 133-176. -- Zbl 0557.14021 \[10] Ueno, K., Classification theory of algebraic varieties and compact complex spaces, (Springer, Berlin, 1975). -- Zbl 0299.14007 \Author Demailly, Jean-Pierre; Rahmati, Mohammad Reza \Title Morse cohomology estimates for jet differential operators. (English) Zbl 1420.32011 \Publi Boll. Unione Mat. Ital. 12, No. 1-2, 145-164 (2019). {\it Summary$\,$}: We provide detailed holomorphic Morse estimates for the cohomology of sheaves of jet differentials and their dual sheaves. These estimates apply on arbitrary directed varieties, and a special attention has been given to the analysis of the singular situation. As a consequence, we obtain existence results for global jet differentials and global differential operators under positivity conditions for the canonical or anticanonical sheaf of the directed structure. \MSC 32H30 Value distribution theory in higher dimensions\\ 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results\\ 14J17 Singularities of surfaces or higher-dimensional varieties\\ 14J40 $n$-folds ($n>4$)\\ 53C55 Global differential geometry of Hermitian and K\"ahler geometry \Keywords directed variety; jet bundle; jet differential; jet metric; holomorphic Morse inequalities; canonical sheaf \References \[1] Bloch, A., Sur les systèmes de fonctions uniformes satisfaisant à l’équation d’une variété algébrique dont l’irrégularité dépasse la dimension, J.\ de Math., 5, 19-66, (1926) JFM 52.0373.04 \[2] Bloch, A., Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires, Ann. Ecole Normale, 43, 309-362, (1926) \. JFM 52.0326.01 \[3] Bonavero, L., Inégalités de Morse holomorphes singulières, C. R. Acad. Sci. Paris Sér. I Math., 317, 1163-1166, (1993) \. Zbl 0799.32023 \[4] Demailly, J-P, Champs magnétiques et inégalités de Morse pour la $d''$-cohomologie, Ann. Inst. Fourier (Grenoble), 35, 189-229, (1985) \. Zbl 0565.58017 \[5] Demailly, J.-P.: Singular hermitian metrics on positive line bundles. In: Hulek, K., Peternell, T., Schneider, M., Schreyer, F. (eds.) Lecture Notes in Math. Proceedings of the Bayreuth conference “Complex algebraic varieties”, April 2-6, 1990, n${}^\circ$ 1507, pp. 87-104. Springer (1992) \. Zbl 0784.32024 \[6] Demailly, J.-P.: Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. In: Kollár, J., Lazarsfeld, R. (eds.) Algebraic Geometry-Santa Cruz 1995, Proceedings Symposia in Pure Math, vol 62, pp. 285-360. American Mathematical Society Providence, RI (1997) \[7] Demailly, J-P, Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture, Pure Appl. Math. Q., 7, 1165-1208, (2011) \. Zbl 1316.32014 \[8] Demailly, J-P, Hyperbolic algebraic varieties and holomorphic differential equations. Expanded version of the lectures given at the annual meeting of VIASM, Acta Math. Vietnam, 37, 441-512, (2012) \[9] Demailly, J.-P.: Towards the Green-Griffiths-Lang conjecture. In: Baklouti, A., El Kacimi, A., Kallel, S., Mir N. (eds) Conference “Analysis and Geometry”, Tunis, March 2014, in honor of Mohammed Salah Baouendi, pp. 141-159. Springer (2015) \. Zbl 1327.14048 \[10] Demailly, J.-P.: Recent results on the Kobayashi and Green-Griffiths-Lang conjectures. Contribution to the 16th Takagi lectures in celebration of the 100th anniversary of K.Kodaira’s birth, November 2015, to appear in the Japanese Journal of Mathematics. arXiv: Math.AG/1801.04765 \[11] Green, M., Griffiths, P.: Two applications of algebraic geometry to entire holomorphic mappings. In: The Chern symposium, 1979, proc. internal. sympos. Berkeley, CA, 1979, pp. 41-74. Springer, New York, (1980) \[12] Lang, S., Hyperbolic and diophantine analysis, Bull. Am. Math. Soc., 14, 159-205, (1986) \. Zbl 0602.14019 \[13] Merker, J., Low pole order frames on vertical jets of the universal hypersurface, Ann. Inst. Fourier (Grenoble), 59, 1077-1104, (2009) \. Zbl 1172.32005 \[14] Păun, M., Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity, Math. Ann., 340, 875-892, (2008) \. Zbl 1137.32010 \[15] Siu, Y.T.: Hyperbolicity in complex geometry. The legacy of Niels Henrik Abel, pp. 543-566. Springer, Berlin (2004) \Author Demailly, Jean-Pierre \Title Extension of holomorphic functions and cohomology classes from non reduced analytic subvarieties. (English) Zbl 1404.32036 \Publi Byun, Jisoo (ed.) et al., Geometric complex analysis. In honor of Kang-Tae Kim’s 60th birthday, Gyeongju, Korea, 2017. Selected papers based on the presentations at the 11th and 12th Korean conferences on several complex variables, KSCV 11 symposium and KSCV 12 symposium, July 4–8, 2016 and July 3–7, 2017. Singapore: Springer (ISBN 978-981-13-1671-5/hbk; 978-981-13-1672-2/ebook). Springer Proceedings in Mathematics \&\ Statistics 246, 97-113 (2018). {\it Summary$\,$}: The goal of this survey is to describe some recent results concerning the $L^2$ extension of holomorphic sections or cohomology classes with values in vector bundles satisfying weak semi-positivity properties. The results presented here are generalized versions of the Ohsawa-Takegoshi extension theorem, and borrow many techniques from the long series of papers by T. Ohsawa. The recent achievement that we want to point out is that the surjectivity property holds true for restriction morphisms to non necessarily reduced subvarieties, provided these are defined as zero varieties of multiplier ideal sheaves. The new idea involved to approach the existence problem is to make use of $L^2$ approximation in the Bochner-Kodaira technique. The extension results hold under curvature conditions that look pretty optimal. However, a major unsolved problem is to obtain natural (and hopefully best possible) $L^2$ estimates for the extension in the case of non reduced subvarieties – the case when $Y$ has singularities or several irreducible components is also a substantial issue. For the entire collection see [Zbl 1402.30001]. \MSC 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results\\ 32D15 Continuation of analytic objects in several complex variables\\ 32E05 Holomorphically convex complex spaces, reduction theory \Keywords compact Kähler manifold; singular Hermitian metric; coherent sheaf cohomology; Dolbeault cohomology; plurisubharmonic function; $L^2$ estimates; Ohsawa-Takegoshi extension theorem; multiplier ideal sheaf \Author Demailly, Jean-Pierre \Title Fano manifolds with nef tangent bundles are weakly almost Kähler-Einstein. (English) Zbl 06890455 \Publi Asian J. Math. 22, No. 2, 285-290 (2018). {\it Summary$\,$}: The goal of this short note is to point out that every Fano manifold with a nef tangent bundle possesses an almost Kähler-Einstein metric, in a weak sense. The technique relies on a regularization theorem for closed positive $(1,1)$-currents. We also discuss related semistability questions and Chern inequalities. \MSC 14J45 Fano varieties\\ 14M17 Homogeneous spaces and generalizations\\ 32C30 Integration on analytic sets and spaces, currents\\ 32Q10 Positive curvature manifolds\\ 32Q20 Kähler-Einstein manifolds \Keywords Fano manifold; numerically effective vector bundle; rational homogeneous manifold; Campana-Peternell conjecture; Kähler-Einstein metric; closed positive current; regularization of currents; Schauder fixed point theorem \Author Demailly, Jean-Pierre \Title Precise error estimate of the Brent-McMillan algorithm for Euler’s constant. (English) Zbl 07321502 \Publi Mosc. J. Comb. Number Theory 7, No. 4, 3-38 (2017). {\it Summary$\,$}: R. P. Brent and E. M. McMillan [Math. Comput. 34, 305–312 (1980; Zbl 0442.10002)] introduced in 1980 a new algorithm for the computation of Euler’s constant $\gamma$, based on the use of the Bessel functions $I_0(x)$ and $K_0(x)$. It is the fastest known algorithm for the computation of $\gamma$. The time complexity can still be improved by evaluating a certain divergent asymptotic expansion up to its minimal term. Brent-McMillan conjectured in 1980 that the error is of the same magnitude as the last computed term, and R. P. Brent and F. Johansson [Math. Comput. 84, No. 295, 2351–2359 (2015; Zbl 1320.33007)] partially proved it in 2015. They also gave some numerical evidence for a more precise estimate of the error term. We find here an explicit expression of that optimal estimate, along with a complete self-contained formal proof and an even more precise error bound. \MSC 33C10 Bessel and Airy functions, cylinder functions, ${}_0F_1$\\ 11Y60 Evaluation of number-theoretic constants \Keywords Euler’s constant; Bessel functions; elliptic integral; integration by parts; asymptotic expansion; Euler-Maclaurin formula \Author Demailly, Jean-Pierre; Gaussier, Herv\'e \Title Algebraic embeddings of smooth almost complex structures. (English) Zbl 06802929 \Publi J. Eur. Math. Soc. (JEMS) 19, No. 11, 3391-3419 (2017). {\it Summary$\,$}: The goal of this work is to prove an embedding theorem for compact almost complex manifolds into complex algebraic varieties. It is shown that every almost complex structure can be realized by the transverse structure to an algebraic distribution on an affine algebraic variety, namely an algebraic subbundle of the tangent bundle. In fact, there even exist universal embedding spaces for this problem, and their dimensions grow quadratically with respect to the dimension of the almost complex manifold to embed. We give precise variation formulas for the induced almost complex structures and study the related versality conditions. At the end, we discuss the original question raised by F. Bogomolov: can one embed every compact complex manifold as a $\mathcal{C}^\infty$ smooth subvariety that is transverse to an algebraic foliation on a complex projective algebraic variety? \MSC 32Q60 Almost complex manifolds\\ 32Q40 Embedding theorems\\ 32G05 Deformations of complex structures\\ 53C12 Foliations (differential geometry) \Keywords deformation of complex structures; almost complex manifolds; complex projective variety; Nijenhuis tensor; transverse embedding; Nash algebraic map \Author Demailly, Jean-Pierre \Title Variational approach for complex Monge-Amp\`ere equations and geometric applications. (English) Zbl 06784933 \Publi S\'eminaire Bourbaki. Volume 2015/2016. Expos\'es 1104--1119. Avec table par noms d'auteurs de 1948/49 \`a 2015/16. Paris: Soci\'et\'e Math\'ematique de France (SMF) (ISBN 978-2-85629-855-8/pbk). Ast\'erisque 390, 245-275, Exp. No. 1112 (2017). For the entire collection see [Zbl 1370.00002]. \MSC 32W20 Complex Monge-Amp\`ere operators\\ 32Q25 Calabi-Yau theory 53C55 Hermitian and K\"ahlerian manifolds (global differential geometry) \Author Cao, JunYan; Demailly, Jean-Pierre; Matsumura, Shin-ichi \Title A general extension theorem for cohomology classes on non reduced analytic subspaces. (English) Zbl 1379.32017 \Publi Sci. China, Math. 60, No. 6, 949-962 (2017). The authors generalize the Ohsawa-Takegoshi extension theorem with the goal to prove it with the weakest possible hypothesis. The main theorem goes as follows: Let $E$ be a holomorphic line bundle over a holomorphically convex K\"ahler manifold $X$. Let $h$ be a (possibly) singular Hermitian metric on $E$, $\psi$ a quasi-plurisubharmonic function with neat analytic singularities on $X$. If there exists a continuous function $\delta>0$ on $X$ such that $$\Theta_{E,h}+(1+\alpha\delta)i \partial\bar{\partial} \psi \geq 0$$ in the sense of currents for all $\alpha\in[0,1]$, then the morphism induced by the natural inclusion $\mathcal I(he^{-\psi})\to \mathcal I(h)$, namely $$H^q(X,K_{X}\otimes E\otimes \mathcal I(he^{-\psi}))\to H^q(X,K_{X}\otimes E\otimes \mathcal I(h)),$$ is injective for every $q\geq 0$. It is noted that most of the argument carries on to the case when $X$ is only weakly pseudoconvex, yet at one place there is a problem and hence this case remains open. An alternative proof using an idea of the third author is presented. \Reviewer \D{Z}ywomir Dinew (Krak\'ow) \MSC 32L10 Sections of holomorphic vector bundles\\ 32Q15 K\"ahler manifolds\\ 32E05 Holomorphically convex complex spaces, reduction theory \Keywords holomorphically convex K\"ahler manifold; Ohsawa-Takegoshi extension theorem; singular Hermitian metric; multiplier ideal sheaf \References \[1] Dem \. Zbl 0507.32021 \. DOI: 10.24033/asens.1434 \[2] Demailly J-P. Complex Analytic and Differential Geometry.\\ https://www-fourier.ujf-grenoble.fr/~demailly/manu scripts/agbook.pdf, 2009 \[3] Demailly J-P. Analytic Methods in Algebraic Geometry. Somerville: International Press, 2012; Beijing: Higher Education Press, 2012 \[4] Demailly J-P. On the cohomology of pseudoeffective line bundles. In: Fornaess J, Irgens M, Wold E, eds. Complex Geometry and Dynamics. Abel Symposia, vol. 10. Cham: Springer, 2015, 51--99 \. Zbl 1337.32030 \[5] Demailly J-P. Extension of holomorphic functions defined on non reduced analytic subvarieties. In: The Legacy of Bernhard Riemann after One Hundred and Fifty Years. Advanced Lectures in Mathematics, vol. 35.1. ArXiv:1510.05230v1, 2015 \[6] Demailly J-P, Peternell T, Schneider M. Pseudo-effective line bundles on compact K\"ahler manifolds. Internat J Math, 2001, 6: 689--741 \. Zbl 1111.32302 \. DOI: 10.1142/S0129167X01000861 \[7] Do \. Zbl 0532.58027 \. DOI: 10.2307/2006983 \[8] Folland G B, Kohn J J. The Neumann Problem for the Cauchy-Riemann Complex. Princeton: Princeton University Press, 1972; Tokyo: University of Tokyo Press, 1972 \. Zbl 0247.35093 \[9] Fujino O. A transcendental approach to Koll\'ar's injectivity theorem II. J Reine Angew Math, 2013, 681: 149--174 \. Zbl 1285.32009 \[10] Fujino O, Matsumura S. Injectivity theorem for pseudo-effective line bundles and its applications. ArXiv:1605.02284v1, 2016 \[11] Guan Q, Zhou X. A proof of Demailly's strong openness conjecture. Ann of Math, 2015, 182: 605--616 \. Zbl 1329.32016 \. DOI: 10.4007/annals.2015.182.2.5 \[12] Hi\^ep P H. The weighted log canonical threshold. C R Math Acad Sci Paris, 2014, 352: 283--288 \. Zbl 1296.32013 \. DOI: 10.1016/j.crma.2014.02.010 \[13] \. Zbl 0581.32036 \. DOI: 10.2977/prims/1195181609 \[14] Lempert L. Modules of square integrable holomorphic germs. ArXiv:1404.0407v2, 2014 \[15] Matsumura S. An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities. J Algebraic Geom, in press, arXiv:1308.2033v4, 2013 \[16] Matsumura S. An injectivity theorem with multiplier ideal sheaves for higher direct images under K\"ahler morphisms. ArXiv:1607.05554v1, 2016 \[17] Ohsawa T. On a curvature condition that implies a cohomology injectivity theorem of Koll\'ar-Skoda type. Publ Res Inst Math Sci, 2005, 41: 565--577 \. Zbl 1103.32005 \. DOI: 10.2977/prims/1145475223 \[18] Ohsawa T, Takegoshi \. Zbl 0625.32011 \. DOI: 10.1007/BF01166457 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Extension of holomorphic functions defined on non reduced analytic subvarieties. (English) Zbl 1360.14025 \Publi Ji, Lizhen (ed.) et al., The legacy of Bernhard Riemann after one hundred and fifty years. Volume I. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-318-0/pbk; 978-1-57146-316-6/set). Advanced Lectures in Mathematics (ALM) 35, 1, 191-222 (2016). The article generalizes results and methods of {\it T. Ohsawa} and {\it K. Takegoshi} theorem from [Math. Z. 195, 197--204 (1987; Zbl 0625.32011)] to the case when the subvariety $Y$ is not necessarily reduced, by using a multiplier ideal sheaf and jumping numbers. For a holomorphic vector bundle $E$ on a complex manifold $X$ one can discuss the existence of global holomorphic extensions $F\in H^0(X,E)$ of a section $f\in H^0(Y,E|_{Y})$ together with $L^2$ approximations. The article considers this problem when $X$ is a weakly pseudoconvex K\"ahler manifold with K\"ahler metric $\omega$ and when the holomorphic vector bundle $E$ is equipped with a (possibly singular) hermitian metric $h=e^{-\varphi}$. Let $\psi$ denote a quasi-psh function on $X$ with neat analytic singularities and with log canonical singularities along a analytic subvariety $Y=V(\mathcal{I}(\psi))$ (so that $Y$ is reduced). If the Chern curvature tensor $\Theta_{E,h}$ has the property that $i\Theta_{E,h}+\alpha i\partial\overline{\partial}\otimes Id_{E}$ is Nakano semipositive for all $\alpha\in [1,1+\delta]$ and some $\delta>0$, then for every section $f\in H^0(Y^0,(K_X\otimes E)|_{Y^0})$ on $Y^{0}=Y_{\mathrm{reg}}$ such that $$ \int_{Y_{0}}|f|^2_{\omega,h}dV_{Y^0,\omega}[\psi]<+\infty $$ there exists an extension $F\in H^{0}(X,K_{X}\otimes E)$ whose restriction to $Y^{0}$ is equal to $f$, such that $$ \int_{X}\gamma(\delta \psi)|F|^2_{\omega,h}e^{-\psi}dV_{X,\omega}<\frac{34}{\delta}\int_{Y_{0}}|f|^2_{\omega,h}dV_{Y^0,\omega}[\psi] $$ The remark states that if $F$ is a $(n,0)$-form then the product $|F|^2_{\omega,h}dV_{X,\omega}$ does not depend on $\omega$. The author claims that the constant $\frac{34}{\delta}$ in the inequality is not optimal. The concept of the multiplier ideal sheaf used in the proof is parallel yet more general than the one presented by {\it D. Popovici} [Nagoya Math. J. 180, 1--34 (2005; Zbl 1116.32017)]. For the entire collection see [Zbl 1343.01006]. \Reviewer Malgorzata Marciniak (Flushing) \Cited in 1 Document \MSC 14C30 Transcendental methods, Hodge theory, Hodge conjecture\\ 14F05 Sheaves, derived categories of sheaves, etc.\\ 32C35 Analytic sheaves and cohomology groups \Keywords holomorphic function; plurisubharmonic function; multiplier ideal sheaf; $L^2$ extension theorem; Ohsawa-Takegoshi theorem; log canonical singularities; non reduced subvariety K\"ahler metric; multiplier ideal sheaf; jumping numbers \Author Demailly, Jean-Pierre \Title Numerical analysis and differential equations. 4th edition. (Analyse num\'erique et \'equations diff\'erentielles.) (French) Zbl 1362.65001 \Publi Grenoble Sciences. Les Ulis: EDP Sciences (ISBN 978-2-7598-1926-3/pbk; 978-2-7598-2004-7/ebook). vii, 368~p. (2016). Publisher's description: Cet ouvrage est la quatri\`eme \'edition d'un livre devenu aujourd'hui un classique sur la th\'eorie des \'equations diff\'erentielles ordinaires. Le cours th\'eorique de base est accompagn\'e d'un expos\'e d\'etaill\'e des m\'ethodes num\'eriques qui permettent de r\'esoudre ces \'equations en pratique. \par De multiples techniques de l'analyse num\'erique sont pr\'esent\'ees : interpolation polynomiale, int\'egration num\'erique, m\'ethodes it\'eratives pour la r\'esolution d'\'equations. Suit un expos\'e rigoureux des r\'esultats sur l'existence, l'unicit\'e et la r\'egularit\'e des solutions des \'equations diff\'erentielles, avec \'etude d\'etaill\'ee des \'equations du premier et du second ordre, des \'equations et syst\`emes lin\'eaires \`a coefficients constants. Enfin, sont d\'ecrites les m\'ethodes num\'eriques \`a un pas ou multi-pas, avec \'etude comparative de la stabilit\'e et du co\^ut en temps de calcul. De nombreux exemples concrets, des exercices et probl\`emes d'application en fin de chapitre facilitent l'apprentissage. \par Plusieurs am\'eliorations ont \'et\'e apport\'ees dans cette derni\`ere version. De nouveaux probl\`emes ou exercices ont \'et\'e introduits dans presque tous les chapitres. La principale nouveaut\'e est que l'ouvrage est maintenant un pap-ebook : le site compagnon en acc\`es libre propose au lecteur des compl\'ements th\'eoriques et pratiques, ainsi que la correction d'un grand nombre d'exercices. \par Cet ouvrage accessible aux L3, M1 et M2 de math\'ematiques est tr\`es utilis\'e pour la pr\'eparation aux concours de l'enseignement. Il constitue un outil de r\'ef\'erence pour les enseignants, chercheurs et scientifiques d'autres disciplines. For the previous edition see [Zbl 0869.65041]. See the review of the German edition in [Zbl 0869.65042]. \MSC 65-01 Textbooks (numerical analysis) 34-01 Textbooks (ordinary differential equations) 65L05 Initial value problems for ODE (numerical methods) 65L06 Multistep, Runge-Kutta, and extrapolation methods 65D32 Quadrature and cubature formulas (numerical methods) 65H10 Systems of nonlinear equations (numerical methods) \Author Boman, Jan (ed.); Sigurdsson, Ragnar (ed.); Lerner, Nicolas; Demailly, Jean-Pierre; Atiyah, Michael; Treves, Fran{\c c}ois; Helgason, Sigurdur; Grubb, Gerd; Bony, Jean-Michel; Kiselman, Christer O.; Brostr\"om, Sofia \Title To the memory of Lars H\"ormander (1931--2012). (English) Zbl 1338.35005 \Publi Notices Am. Math. Soc. 62, No. 8, 890-907 (2015). \MSC 35-03 Historical (partial differential equations) 46-03 Historical (functional analysis) 47-03 Historical (operator theory) 01A70 Biographies, obituaries, personalia, bibliographies \Author Demailly, Jean-Pierre \Title On the cohomology of pseudoeffective line bundles. (English) Zbl 1337.32030 \Publi Forn\ae{}ss, John Erik (ed.) et al., Complex geometry and dynamics. The Abel symposium 2013, Trondheim, Norway, July 2--5, 2013. Cham: Springer (ISBN 978-3-319-20336-2/hbk; 978-3-319-20337-9/ebook). Abel Symposia 10, 51-99 (2015). {\it Summary$\,$}: The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact K\"ahler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation, one can still obtain interesting information on cohomology, e.g. a Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his PhD thesis defended in Grenoble, obtained a general K\"ahler vanishing theorem that depends on the concept of numerical dimension of a given pseudoeffective line bundle. The proof of these results depends in a crucial way on a general approximation result for closed $(1,1)$-currents, based on the use of Bergman kernels, and the related intersection theory of currents. Another important ingredient is the recent proof by Guan and Zhou of the strong openness conjecture. As an application, we discuss a structure theorem for compact K\"ahler threefolds without nontrivial subvarieties, following a joint work with F. Campana and M.~Verbitsky. We hope that these notes will serve as a useful guide to the more detailed and more technical papers in the literature; in some cases, we provide here substantially simplified proofs and unifying viewpoints. For the entire collection see [Zbl 1336.32001]. \Cited in 1 Review \Cited in 6 Documents \MSC 32J27 Compact K\"ahler manifolds: generalizations, classification \Keywords pseudoeffective line bundles on K\"ahler manifolds; multiplier ideal sheaves \References \[1] \[2] \. Zbl 1213.32025 \. DOI: 10.1007/s11511-010-0054-7 \[3] \[4] \. Zbl 1146.32017 \. DOI: 10.2977/prims/1210167334 \[5] \[6] \. Zbl 0685.32007 \. DOI: 10.1090/S0894-0347-1989-1001853-2 \[7] \[8] \[9] \[10] \. Zbl 0315.31007 \. DOI: 10.1007/BF01418826 \[11] \. Zbl 0445.32021 \. DOI: 10.1007/BF01420277 \[12] \[13] \[14] \[15] \[16] \[17] \[18] \[19] \[20] \[21] \. Zbl 0777.32016 \[22] \. Zbl 0783.32013 \[23] \[24] \[25] \. Zbl 1067.14013 \[26] \. Zbl 1189.14044 \. DOI: 10.1215/00127094-2010-008 \[27] \[28] \[29] \. Zbl 1298.14006 \. DOI: 10.1007/s11511-014-0107-4 \[30] \. Zbl 0827.14027 \[31] \. Zbl 1111.32302 \. DOI: 10.1142/S0129167X01000861 \[32] \[33] \[34] \[35] \[36] \[37] \[38] \[39] \[40] \[41] \[42] \[43] \. Zbl 1314.32047 \. DOI: 10.1017/S1474748013000091 \[44] \. Zbl 1295.32042 \. DOI: 10.1016/j.crma.2013.10.024 \[45] \. Zbl 0827.32016 \[46] \[47] \[48] \[49] \. Zbl 0865.32019 \[50] \[51] \. Zbl 0625.32011 \. DOI: 10.1007/BF01166457 \[52] \. Zbl 0606.32018 \. DOI: 10.1007/BF01459143 \[53] \. Zbl 0945.14020 \[54] \. Zbl 1296.32013 \. DOI: 10.1016/j.crma.2014.02.010 \[55] \[56] \. Zbl 0153.15401 \. DOI: 10.1007/BF02063212 \[57] \. Zbl 0289.32003 \. DOI: 10.1007/BF01389965 \[58] \. Zbl 0577.32031 \[59] \[60] \[61] \. Zbl 0246.32009 \[62] \. Zbl 0895.32008 \[63] \[64] \. Zbl 0369.53059 \. DOI: 10.1002/cpa.3160310304 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre (ed.); van der Geer, Gerard (ed.); Hacon, Christopher (ed.); Kawamata, Yujiro (ed.); Kobayashi, Toshiyuki (ed.); Miyaoka, Yoichi (ed.); Schmid, Wilfried (ed.) \Title Foreword. (English) Zbl 1332.00108 \Publi J. Math. Sci., Tokyo 22, No. 1, iii-iv (2015). From the text: Professor Kunihiko Kodaira is one of the greatest mathematicians of the twentieth century, and this issue is dedicated to him to commemorate his 100th birthday. The authors of the articles included in this issue belong to various generations. \MSC 00B30 Festschriften\\ 14-06 Proceedings of conferences (algebraic geometry)\\ 32-06 Proceedings of conferences (several complex variables) \Author Demailly, Jean-Pierre \Title Towards the Green-Griffiths-Lang conjecture. (English) Zbl 1327.14048 \Publi Baklouti, Ali (ed.) et al., Analysis and geometry. MIMS-GGTM, Tunis, Tunisia, March 24--27, 2014. Proceedings of the international conference. In honour of Mohammed Salah Baouendi. Cham: Springer (ISBN 978-3-319-17442-6/hbk; 978-3-319-17443-3/ebook). Springer Proceedings in Mathematics \&\ Statistics 127, 141-159 (2015). {\it Summary$\,$}: The Green-Griffiths-Lang conjecture stipulates that for every projective variety $X$ of general type over $\mathbb C$, there exists a proper algebraic subvariety of $X$ containing all non constant entire curves $f:\mathbb C\to X$. Using the formalism of directed varieties, we prove here that this assertion holds true in case $X$ satisfies a strong general type condition that is related to a certain jet semistability property of the tangent bundle $T_X$. We then give a sufficient criterion for the Kobayashi hyperbolicity of an arbitrary directed variety $(X,V)$. For the entire collection see [Zbl 1320.00044]. \MSC 14C30 Transcendental methods, Hodge theory, Hodge conjecture\\ 32J25 Transcendental methods of algebraic geometry\\ 14C20 Divisors, linear systems, invertible sheaves \Keywords projective algebraic variety; variety of general type; entire curve; jet bundle; semple tower; Green-Griffiths-Lang conjecture; holomorphic morse inequality; semi\-stable vector bundle; Kobayashi hyperbolic \References \[1] J.-P. Demailly, Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. AMS Summer School on Algebraic Geometry, Santa Cruz 1995, in Proceedings Symposia in Pure Mathematics, ed. by J. Koll\'ar, R. Lazarsfeld, Am. Math. Soc. Providence, RI, 285-360 (1997) \[2] J.-P. Demailly, Vari\'et\'es hyperboliques et \'equations diff\'erentielles alg\'ebriques. Gaz. Math. 73, 3-23 (juillet 1997). \[3] J.-P. Demailly, J. El Goul, Hyperbolicity of generic surfaces of high degree in projective 3-space. Am. J. Math. 122, 515-546 (2000) \. Zbl 0966.32014 \. DOI: 10.1353/ajm.2000.0019 \[4] J.-P. Demailly, Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture. Pure App. Math. Q. 7, 1165-1208 (2011). November 2010, arxiv:math.AG/ 1011.3636, dedicated to the memory of Eckart Viehweg \. Zbl 1316.32014 \[5] S. Diverio, J. Merker, E. Rousseau, Effective algebraic degeneracy. Invent. Math. 180, 161-223 (2010) \. Zbl 1192.32014 \. DOI: 10.1007/s00222-010-0232-4 \[6] S. Diverio, E. Rousseau, The exceptional set and the Green-Griffiths locus do not always coincide. arxiv:math.AG/1302.4756 (v2) \. Zbl 1359.32020 \[7] M. Green, P. Griffiths, Two applications of algebraic geometry to entire holomorphic mappings, in The Chern Symposium, Proceedings of the International Symposium Berkeley, CA, 1979 (Springer, New York, 1980), pp. 41-74 \[8] S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Pure and Applied Mathematics, vol. 2 (Marcel Dekker Inc., New York, 1970) \[9] S. Kobayashi, Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften, vol. 318 (Springer, Berlin, 1998) \[10] S. Lang, Hyperbolic and Diophantine analysis. Bull. Am. Math. Soc. 14, 159-205 (1986) \. Zbl 0602.14019 \. DOI: 10.1090/S0273-0979-1986-15426-1 \[11] M. McQuillan, Diophantine approximation and foliations. Inst. Hautes \'Etudes Sci. Publ. Math. 87, 121-174 (1998) \. Zbl 1006.32020 \. DOI: 10.1007/BF02698862 \[12] M. McQuillan, Holomorphic curves on hyperplane sections of \. Zbl 0951.14014 \. DOI: 10.1007/s000390050091 \[13] M. P\u{a}un, Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity. Math. Ann. 340, 875-892 (2008) \. Zbl 1137.32010 \. DOI: 10.1007/s00208-007-0172-5 \[14] Y.T. Siu, Some recent transcendental techniques in algebraic and complex geometry, in Proceedings of the International Congress of Mathematicians, Vol. I, (Higher Ed. Press, Beijing, 2002), pp. 439-448 \. Zbl 1028.32012 \[15] Y.T. Siu, Hyperbolicity in Complex Geometry, The legacy of Niels Henrik Abel (Springer, Berlin, 2004), pp. 543-566 \. Zbl 1076.32011 \[16] Y.T. Siu, S.K. Yeung, Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane. Invent. Math. 124, 573-618 (1996) \. Zbl 0856.32017 \. DOI: 10.1007/s002220050064 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Structure theorems for compact K\"ahler manifolds with nef anticanonical bundles. (English) Zbl 1326.32007 \Publi Bracci, Filippo (ed.) et al., Complex analysis and geometry. KSCV 10. Proceddings of the 10th symposium, Gyeongju, Korea, August 7--11, 2014. Tokyo: Springer (ISBN 978-4-431-55743-2/hbk; 978-4-431-55744-9/ebook). Springer Proceedings in Mathematics \&\ Statistics 144, 119-133 (2015). {\it Summary$\,$}: This survey presents various results concerning the geometry of compact K\"ahler manifolds with numerically effective first Chern class: structure of the Albanese morphism of such manifolds, relations tying semipositivity of the Ricci curvature with rational connectedness, positivity properties of the Harder-Narasimhan filtration of the tangent bundle. For the entire collection see [Zbl 1328.32001]. \Cited in 2 Documents \MSC 32-02 Research monographs (several complex variables)\\ 32J27 Compact K\"ahler manifolds: generalizations, classification\\ 32L05 Holomorphic fiber bundles and generalizations \Keywords compact K\"ahler manifold; anticanonical bundle; semipositive Ricci curvature; Ricci flat manifold; rationally connected variety; holonomy principle \References \[1] Boucksom, S., Demailly, J.-P., Paun, M., Peternell, T.: The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension. \. Zbl 1267.32017 \[2] Demailly, J.-P., Peternell, T., Schneider, M.: Pseudo-effective line bundles on compact K\"ahler manifolds. Internat. J. Math. 12, 689-741 (2001) \. Zbl 1111.32302 \. DOI: 10.1142/S0129167X01000861 \[3] Graber, T., Harris, J., Starr, J.: Families of rationally connected varieties. J. Amer. Math. Soc. 16, 57-67 (2003) \. Zbl 1092.14063 \. DOI: 10.1090/S0894-0347-02-00402-2 \[4] P\u{a}un, M.: On the Albanese map of compact K\"ahler manifolds with numerically effective Ricci curvature. Comm. Anal. Geom. 9, 35-60 (2001) \. Zbl 0980.53091 \[5] Campana, F., Peternell, Th., Zhang, Q.: On the Albanese maps of compact K\"ahler manifolds. Proc. Amer. Math. Soc. 131, 549-553 (2003) \. Zbl 1017.32018 \[6] Zhang, Q.: On projective varieties with nef anticanonical divisors. Math. Ann. 332, 697-703 (2005) \. Zbl 1083.14010 \. DOI: 10.1007/s00208-005-0649-z \[7] Peternell, Th: Kodaira dimension of subvarieties II. Intl. J. Math. 17, 619-631 (2006) \. Zbl 1126.14019 \. DOI: 10.1142/S0129167X0600362X \[8] Brunella, M.: On K\"ahler surfaces with semipositive Ricci curvature. Riv. Math. Univ. Parma (N.S.), 1, 441-450 (2010) \. Zbl 1226.32011 \[9] P\u{a}un, M.: Relative adjoint transcendental classes and the Albanese map of compact K\"ahler manifolds with nef Ricci classes. \[10] Cao, J.: A remark on compact K\"ahler manifolds with nef anticanonical bundles and its applications. \[11] Cao, J.: Vanishing theorems and structure theorems on compact K\"ahler manifolds. PhD thesis, Universit\'e de Grenoble, defended at institut Fourier on 18 Sep 2013. \[12] Cao, J., H\"oring, A.: Manifolds with nef anticanonical bundle. \. Zbl 06695049 \[13] de Rham, G.: Sur la reductibilit\'e d'un espace de Riemann. Comment. Math. Helv. 26, 328-344 (1952) \. Zbl 0048.15701 \. DOI: 10.1007/BF02564308 \[14] Bochner, S., Yano, K.: Curvature and Betti Numbers. Annals of Mathematics Studies, No. 32, pp. ix \. Zbl 0051.39402 \[15] Kodaira, K.: On K\"ahler varieties of restricted type. Ann. of Math. 60, 28-48 (1954) \. Zbl 0057.14102 \. DOI: 10.2307/1969701 \[16] Berger, M.: Sur les groupes d'holonomie des vari\'et\'es \`a connexion affine des vari\'et\'es riemanniennes. Bull. Soc. Math. Fr. 83, 279-330 (1955) \. Zbl 0068.36002 \[17] Bishop, R.: A relation between volume, mean curvature and diameter. Amer. Math. Soc. Not. 10, 364 (1963) \[18] Lichnerowicz, A.: Vari\'et\'es k\"ahleriennes et premi\`ere classe de Chern. J. Diff. Geom. 1, 195-224 (1967) \. Zbl 0167.20004 \[19] Lichnerowicz, A.: Vari\'et\'es K\"ahl\'eriennes \`a premi\`ere classe de Chern non n\'egative et vari\'et\'es riemanniennes \`a courbure de Ricci g\'en\'eralis\'ee non n\'egative. J. Diff. Geom. 6, 47-94 (1971) \. Zbl 0231.53063 \[20] Cheeger, J., Gromoll, D.: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Diff. Geom. 6, 119-128 (1971) \. Zbl 0223.53033 \[21] Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. Math. 96, 413-443 (1972) \. Zbl 0246.53049 \. DOI: 10.2307/1970819 \[22] Bogomolov, F.A.: On the decomposition of K\"ahler manifolds with trivial canonical class. Math. USSR Sbornik 22, 580-583 (1974) \. Zbl 0304.32016 \. DOI: 10.1070/ SM1974v022n04ABEH001706 \[23] Bogomolov, F.A.: K\"ahler manifolds with trivial canonical class. Izvestija Akad. Nauk 38, 11-21 (1974) \. Zbl 0292.32020 \[24] Arnol'd, V.I.: Bifurcations of invariant manifolds of differential equations, and normal forms of neighborhoods of elliptic curves. Funct. Anal. Appl. 10, 249-259 (1976). English translation 1977 \[25] Aubin, T.: Equations du type Monge-Amp\`ere sur les vari\'et\'es k\"ahleriennes compactes. C. R. Acad. Sci. Paris Ser. A 283, 119-121 (1976); Bull. Sci. Math. 102, 63-95 (1978) \. Zbl 0333.53040 \[26] Gauduchon, P.: Le th\'eor\`eme de l'excentricit\'e nulle. C. R. Acad. Sci. Paris 285, 387-390 (1977) \. Zbl 0362.53024 \[27] Yau, S.T.: On the Ricci curvature of a complex K\"ahler manifold and the complex Monge-Amp\`ere equation I. Comm. Pure Appl. Math. 31, 339-411 (1978) \. Zbl 0369.53059 \. DOI: 10.1002/cpa.3160310304 \[28] Gromov, M.: Structures m\'etriques pour les vari\'et\'es riemanniennes. Cours r\'edig\'e par J. Lafontaine et P. Pansu, Textes Math\'ematiques, 1, vol. VII, p. 152. Paris, Cedic/Fernand Nathan (1981) \[29] Gromov, M.: Groups of polynomial growth and expanding maps, Appendix by J. Tits. Publ. I.H.E.S. 53, 53-78 (1981) \. Zbl 0474.20018 \[30] Kobayashi, S.: Recent results in complex differential geometry. Jber. dt. Math.-Verein. 83, 147-158 (1981) \. Zbl 0467.53030 \[31] Ueda, T.: On the neighborhood of a compact complex curve with topologically trivial normal bundle. J. Math. Kyoto Univ. 22, 583-607 (1982/83) \. Zbl 0519.32019 \[32] Kobayashi, S.: Topics in complex differential geometry. In: DMV Seminar, vol. 3. Birkh\"auser (1983) \. Zbl 0506.53029 \[33] Beauville, A.: Vari\'et\'es k\"ahleriennes dont la premi\`ere classe de Chern est nulle. J. Diff. Geom. 18, 775-782 (1983) \. Zbl 0537.53056 \[34] Koll\'ar, J., Miyaoka, Y., Mori, S.: Rationally connected varieties. J. Alg. 1, 429-448 (1992) \. Zbl 0780.14026 \[35] Campana, F.: Connexit\'e rationnelle des vari\'et\'es de Fano. Ann. Sci. Ec. Norm. Sup. 25, 539-545 (1992) \. Zbl 0783.14022 \[36] Demailly, J.-P., Peternell, T., Schneider, M.: K\"ahler manifolds with numerically effective Ricci class. Compositio Math. 89, 217-240 (1993) \. Zbl 0884.32023 \[37] Bando, S., Siu, Y.-T.: Stable sheaves and Einstein-Hermitian metrics. In: Mabuchi, T., Noguchi, J., Ochiai, T. (eds.) Geometry and Analysis on Complex Manifolds, pp. 39-50. World Scientific, River Edge (1994) \. Zbl 0880.32004 \[38] Demailly, J.-P., Peternell, T., Schneider, M.: Compact complex manifolds with numerically effective tangent bundles. J. Alg. Geom. 3, 295-345 (1994) \. Zbl 0827.14027 \[39] Campana, F.: Fundamental group and positivity of cotangent bundles of compact K\"ahler manifolds. J. Alg. Geom. 4, 487-502 (1995) \. Zbl 0845.32027 \[40] Demailly, J.-P., Peternell, T., Schneider, M.: Compact K\"ahler manifolds with hermitian semipositive anticanonical bundle. Compositio Math. 101, 217-224 (1996) \. Zbl 1008.32008 \[41] Koll\'ar, J.: Rational Curves on Algebraic Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 32, Springer (1996) \. Zbl 0877.14012 \[42] Zhang, Q.: On projective manifolds with nef anticanonical bundles. J. Reine Angew. Math. 478, 57-60 (1996) \. Zbl 0855.14007 \[43] Cheeger, J., Colding, T.H.: Lower bounds on Ricci curvature and almost rigidity of warped products. Ann. Math. 144, 189-237 (1996) \. Zbl 0865.53037 \. DOI: 10.2307/2118589 \[44] P\u{a}un, M.: Sur le groupe fondamental des vari\'et\'es k\"ahl\'eriennes compactes \`a classe de Ricci num\'eriquement effective. C. R. Acad. Sci. Paris S\'er. I Math. 324, 1249-254 (1997) \[45] Cheeger J., Colding T.H.: On the structure of spaces with Ricci curvature bounded below. J. Differ. Geom., part I: 46, 406-480 (1997), part II: 54, 13-35 (2000), part III: 54, 37-74 (2000) \. Zbl 0902.53034 \[46] P\u{a}un, M.: Sur les vari\'et\'es k\"ahl\'eriennes compactes classe de Ricci num\'eriquement effective. Bull. Sci. Math. 122, 83-92 (1998) \. Zbl 0946.53037 \. DOI: 10.1016/S0007-4497(98)80078-X \[47] Peternell, Th, Serrano, F.: Threefolds with anti canonical bundles. Coll. Math. 49, 465-517 (1998) \. Zbl 0980.14028 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Skoda, Henri (ed.); Demailly, Jean-Pierre; Siu, Yum-Tong \Title In memory of Pierre Lelong. (English) Zbl 1338.01059 \Publi Notices Am. Math. Soc. 61, No. 6, 586-595 (2014). \MSC 01A70 Biographies, obituaries, personalia, bibliographies\\ 32-03 Historical (several complex variables and analytic spaces) \Author Huckleberry, Alan (ed.); Peternell, Thomas (ed.); Siu, Yum-Tong; Ohsawa, Takeo; Demailly, Jean-Pierre; Barlet, Daniel; Trautmann, G\"unther; Lieb, Ingo \Title A tribute to Hans Grauert. (English) Zbl 1338.01041 \Publi Notices Am. Math. Soc. 61, No. 5, 472-483 (2014). \MSC 01A70 Biographies, obituaries, personalia, bibliographies \Author Campana, F.; Demailly, J.-P.; Peternell, T. \Title Rationally connected manifolds and semipositivity of the Ricci curvature. (English) Zbl 1369.53052 \Publi Hacon, Christopher D. (ed.) et al., Recent advances in algebraic geometry. A volume in honor of Rob Lazarsfeld's 60th birthday. Based on the conference, Ann Arbor, MI, USA, May 16--19, 2013. Cambridge: Cambridge University Press (ISBN 978-1-107-64755-8/pbk; 978-1-107-41600-0/ebook). London Mathematical Society Lecture Note Series 417, 71-91 (2014). {\it Summary$\,$}: This paper establishes a structure theorem for compact K\"ahler manifolds with semipositive anticanonical bundle. Up to finite \'etale cover, it is proved that such manifolds split holomorphically and isometrically as a product of Ricci flat varieties and of rationally connected manifolds. The proof is based on a characterization of rationally connected manifolds through the nonexistence of certain twisted contravariant tensor products of the tangent bundle, along with a generalized holonomy principle for pseudoeffective line bundles. A crucial ingredient for this is the characterization of uniruledness by the property that the anticanonical bundle is not pseudoeffective. For the entire collection see [Zbl 1318.14002]. \Cited in 1 Review \Cited in 5 Documents \MSC 53C55 Hermitian and K\"ahlerian manifolds (global differential geometry)\\ 14M22 Rationally connected varieties \Author Demailly, Jean-Pierre; Ph\d{a}m, Ho\`ang Hi\d{\^e}p \Title A sharp lower bound for the log canonical threshold. (English) Zbl 1298.14006 \Publi Acta Math. 212, No. 1, 1-9 (2014). The log canonical threshold of a plurisubharmonic function $\varphi$ with an isolated singularity at $0$ in an open subset of ${\Bbb C}^n$ is the the supremum of $c>0$ such that $\exp(-2c\varphi)$ is integrable on a neighborhood of the origin. The main result is the sharp lower bound $c(\phi)\geq \sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)$, where the intersection numbers $e_j(\varphi)$ are the Lelong numbers of $(dd^c\varphi)^j$ at $0$. \Reviewer Jan Stevens (G\"oteborg) \Cited in 2 Reviews \Cited in 6 Documents \MSC 14B05 Singularities (algebraic geometry)\\ 32U05 Plurisubharmonic functions and generalizations\\ 32U25 Lelong numbers\\\\ 14C20 Divisors, linear systems, invertible sheaves \Keywords log canonical threshold; plurisubharmonic functions; Lelong number \References \[1] Bedford E., Taylor B. A.: The Dirichlet problem for a complex Monge--Amp\`ere equation. Invent. Math. 37, 1--44 (1976) \. Zbl 0315.31007 \. DOI: 10.1007/BF01418826 \[2] Bedford E, Taylor B.A: A new capacity for plurisubharmonic functions. Acta Math. 149, 1--40 (1982) \. Zbl 0547.32012 \. DOI: 10.1007/BF02392348 \[3] Cegrell U.: The general definition of the complex Monge--Amp\`ere operator. Ann. Inst. Fourier (Grenoble) 54, 159--179 (2004) \. Zbl 1065.32020 \. DOI: 10.5802/aif.2014 \[4] Chel'tsov, I. A., Birationally rigid Fano varieties. Uspekhi Mat. Nauk, 60:5 (2005), 71--160 (Russian); English translation in Russian Math. Surveys, 60 (2005), 875-- 965. \[5] Corti A.: Factoring birational maps of threefolds after Sarkisov. J. Algebraic Geom. 4, 223--254 (1995) \. Zbl 0866.14007 \[6] Corti, A., Singularities of linear systems and 3-fold birational geometry, in Explicit Birational Geometry of 3-folds, London Math. Soc. Lecture Note Ser., 281, pp. 259--312. Cambridge Univ. Press, Cambridge, 2000. \. Zbl 0960.14017 \[7] Demailly J.-P.: Nombres de Lelong g\'en\'eralis\'es, th\'eor\`emes d'int\'egralit\'e et d'analyti\-cit\'e. Acta Math. 159, 153--169 (1987) \. Zbl 0629.32011 \. DOI: 10.1007/BF02392558 \[8] Demailly J.-P.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1, 361--409 (1992) \. Zbl 0777.32016 \[9] Demailly, J.-P., Monge--Amp\`ere operators, Lelong numbers and intersection theory, in Complex Analysis and Geometry, Univ. Ser. Math., pp. 115--193. Plenum, New York, 1993. \. Zbl 0792.32006 \[10] Demailly, J.-P., Estimates on Monge--Amp\`ere operators derived from a local algebra inequality, in Complex Analysis and Digital Geometry, Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., 86, pp. 131--143. Uppsala Universitet, Uppsala, 2009. \. Zbl 1209.32024 \[11] Demailly J.-P., Koll\'ar J.: Semi-continuity of complex singularity exponents and K\"ahler--Einstein metrics on Fano orbifolds. Ann. Sci. \'Ecole Norm. Sup, 34, 525--556 (2001) \. Zbl 0994.32021 \. DOI: 10.1016/S0012-9593(01)01069-2 \[12] Eisenbud, D., Commutative Algebra. Graduate Texts in Mathematics, 150. Sprin\-ger, New York, 1995. \. Zbl 0819.13001 \[13] de Fernex T., Ein L., Musta\c{t}\u{a} M.: Bounds for log canonical thresholds with applications to birational rigidity. Math. Res. Lett, 10, 219--236 (2003) \. Zbl 1067.14013 \. DOI: 10.4310/MRL.2003.v10.n2.a9 \[14] de Fernex T., Ein L., Musta\c{t}\u{a} M.: Multiplicities and log canonical threshold. J. Algebraic Geom. 13, 603--615 (2004) \. Zbl 1068.14006 \. DOI: 10.1090/S1056-3911-04-00346-7 \[15] Howald J.A.: Multiplier ideals of monomial ideals. Trans. Amer. Math. Soc, 353, 2665--2671 (2001) \. Zbl 0979.13026 \. DOI: 10.1090/S0002-9947-01-02720-9 \[16] Iskovskikh, V.A., Birational rigidity of Fano hypersurfaces in the framework of Mori theory. Uspekhi Mat. Nauk, 56:2 (2001), 3--86 (Russian); English translation in Russian Math. Surveys, 56 (2001), 207--291. \. Zbl 0991.14010 \[17] Iskovskikh, V. A. \& Manin, J. I., Three-dimensional quartics and counterexamples to the L''uroth problem. Mat. Sb., 86(128) (1971), 140--166 (Russian); English translation in Math. USSR--Sb., 15 (1971), 141--166. \. Zbl 0222.14009 \[18] Kiselman, C.O., Un nombre de Lelong raffin\'e, in S\'eminaire d'Analyse Complexe et G\'eom\'etrie 1985--87, pp. 61--70. Facult\'e des Sciences de Tunis \& Facult\'e des Sciences et Techniques de Monastir, Monastir, 1987. \[19] Kiselman C.O.: Attenuating the singularities of plurisubharmonic functions. Ann. Polon. Math, 60, 173--197 (1994) \. Zbl 0827.32016 \[20] Ohsawa T., Takegoshi K.: On the extension of $L^2$ holomorphic functions. Math. Z, 195, 197--204 (1987) \. Zbl 0625.32011 \. DOI: 10.1007/BF01166457 \[21] Pukhlikov A. V.: Birational isomorphisms of four-dimensional quintics. Invent. Math. 87, 303--329 (1987) \. Zbl 0613.14011 \. DOI: 10.1007/BF01389417 \[22] Pukhlikov, A. V., Birationally rigid Fano hypersurfaces. Iz{$\upsilon$}. Ross. Akad. Nauk Ser. Mat., 66:6 (2002), 159--186 (Russian); English translation in Iz{$\upsilon$}. Math., 66 (2002), 1243--1269. \. Zbl 1083.14012 \[23] Skoda H.: Sous-ensembles analytiques d'ordre fini ou infini dans $\bC^n$ . Bull. Soc. Math. France, 100, 353--408 (1972) \. Zbl 0246.32009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre; Dinew, S{\l}awomir; Guedj, Vincent;\\ Hiep, Pham Hoang; Ko{\l}odziej, S{\l}awomir; Zeriahi, Ahmed \Title H\"older continuous solutions to Monge-Amp\`ere equations. (English) Zbl 1296.32012 \Publi J. Eur. Math. Soc. (JEMS) 16, No. 4, 619-647 (2014). Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$. The authors study the following complex Monge-Amp\`{e}re operator $$ \text {MA}(u):=\frac {1}{V_{\omega}}(\omega+dd^cu)^n, \ \ \text { where } \ \ V_{\omega}=\int_X\omega^n, \tag{MA} $$ acting on $\omega$-plurisubharmonic functions, $u\in \mathrm{PSH}(X,\omega)$, which are H\"older continuous, $u\in \text {H\"older}(X,\mathbb R)$. The following theorem gives a better information about the H\"older exponent of the solution to the complex Monge-Amp\`{e}re equation than theorems proved recently by {\it P. Eyssidieux} et al. [J. Am. Math. Soc. 22, No. 3, 607--639 (2009; Zbl 1215.32017)] and {\it S. Dinew} [J. Inst. Math. Jussieu 9, No. 4, 705--718 (2010; Zbl 1207.32034)].\\ {\bf Theorem~A.} Let $\mu=f\omega^n=\text {MA}(u)$ be a probability measure absolutely continuous with respect to the Lebesgue measure with density $f\in L^p$, $p>1$. Then $u$ is H\"older continuous with exponent arbitrary close to $\frac {2}{1+nq}$, where $\frac {1}{p}+\frac {1}{q}=1$. The optimal value of the H\"older exponent in Theorem A is still unknown, but it cannot be better than $\frac {2}{nq}$, see [{\it S. Pli\'s}, Ann. Pol. Math. 86, No. 2, 171--175 (2005; Zbl 1136.32306)] or [{\it V. Guedj} et al., Bull. Lond. Math. Soc. 40, No. 6, 1070--1080 (2008; Zbl 1157.32033)]. Moreover, Theorem A is generalized from the K\"ahler case to the case of big cohomology classes. The rest of the paper is devoted to the study of the range $$ \text {MAH}(X,\omega)=\text {MA}\big(\text {PSH}(X,\omega)\cap \text {H\"older}(X,\mathbb R)\big). $$ A complete characterization of the set $\text {MAH}(X,\omega)$ is unknown, but some of its properties are proved in the following theorem.\\ {\bf Theorem~B.} The set $\text {MAH}(X,\omega)$ has the $L^p$ property, i.e.\ if $\mu \in \text {MAH}(X,\omega)$, $f\geq 0$, $f\in L^p(\mu)$ with $||f||_p=1$, then $f\mu\in \text {MAH}(X,\omega)$. In particular $\text {MAH}(X,\omega)$ is a convex set. {\it T.-C. Dinh} et al. [J. Differ. Geom. 84, No. 3, 465--488 (2010; Zbl 1211.32021)] observed that the measures $\mu\in \text {MAH}(X,\omega)$ have the following property $$ \text {exp}\big(-\epsilon \text {PSH}(X,\omega)\big)\subset L^1(\mu), \ \ \text{for some} \ \ \epsilon >0. \tag {DNS} $$ Condition (DNS) gives a full description of the range $\text {MAH}(X,\omega)$ for $n=1$, see [{\it T.-C. Dinh} and {\it N. Sibony}, Comment. Math. Helv. 81, No. 1, 221--258 (2006; Zbl 1094.32005)]. In the article under review it is proved that, for some special class of measures, this characterization is true in higher dimensions.\\ {\bf Theorem~C.} Let $\mu$ be a probability measure with finitely many isolated singularities which is radial or toric. Then $\mu \in \text {MAH}(X,\omega)$ if and only if condition (DNS) is satisfied. \Reviewer Rafa{\l} Czyz (Krakow) \Cited in 7 Documents \MSC 32U05 Plurisubharmonic functions and generalizations\\ 32U40 Currents 53C55 Hermitian and K\"ahlerian manifolds (global differential geometry) \Keywords Monge-Amp\`{e}re operator; K\"ahler manifold; pluripotential theory; H\"older continuity \Full_text \References \[1] Bedford, E., Taylor, B. A.: The Dirichlet problem for the complex Monge-Amp\`ere oper- ator. Invent. Math. 37, 1-44 (1976) \. Zbl 0315.31007 \. DOI: 10.1007/BF01418826 \. EUDML: 142425 \[2] Bedford, E., Taylor, B. A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1-40 (1982) \. Zbl 0547.32012 \. DOI: 10.1007/BF02392348 \[3] Benelkourchi, S., Guedj, V., Zeriahi, A.: A priori estimates for weak solutions of complex Monge-Amp\`ere equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 7, 81-96 (2008) \. Zbl 1150.32011 \[4] Berman, R.: A thermodynamical formalism for Monge-Amp\`ere equations, Moser- Trudinger inequalities and K\"ahler-Einstein metrics. Adv. Math. 248, 1254-1297 (2013) \. Zbl 1286.58010 \. DOI: 10.1016/j.aim.2013.08.024 \[5] Berman, R., Demailly, J.-P.: Regularity of plurisubharmonic upper envelopes in big coho- mology classes. In: Perspectives in Analysis, Geometry, and Topology in honor of Oleg Viro (Stockholm, 2008), I. Itenberg et al. (eds.), Progr. Math. 296, Birkh\"auser/Springer, Boston, 39-66 (2012) \. Zbl 1258.32010 \. DOI: 10.1007/978-0-8176-8277-4\_3 \. arxiv:0905.1246 \[6] Birkar, C., Cascini, P., Hacon, C., McKernan, J.: Existence of minimal models for va- rieties of log general type. J. Amer. Math. Soc. 23, 405-468 (2010) \. Zbl 1210.14019 \. DOI: 10.1090/S0894-0347-09-00649-3 \. arxiv:math/0610203 \[7] Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A.: Monge-Amp\`ere equations in big cohomology classes. Acta Math. 205, 199-262 (2010) \. Zbl 1213.32025 \. DOI: 10.1007/s11511-010-0054-7 \[8] Demailly, J.-P.: Estimations $L^2$ pour l'op\'erateur $\overline\partial$ d'un fibr\'e vectoriel holomorphe semi- positif au-dessus d'une vari\'et\'e k\"ahl\'erienne compl\`ete. Ann. Sci. \'Ecole Norm. Sup. 15, 457-511 (1982) \. Zbl 0507.32021 \. NUMDAM: ASENS\_1982\_4\_15\_3\_457\_0 \. EUDML: 82103 \[9] Demailly, J.-P.: Regularization of closed positive currents and intersection theory. J. Al- gebraic Geom. 1, 361-409 (1992) \. Zbl 0777.32016 \[10] Demailly, J.-P.: Monge-Amp\`ere operators, Lelong numbers and intersection theory. In: Complex Analysis and Geometry, Univ. Ser. Math., Plenum, New York, 115-193 (1993) \. Zbl 0792.32006 \[11] Demailly, J.-P.: Regularization of closed positive currents of type (1.1) by the flow of a Chern connection. In: Contributions to Complex Analysis and Analytic Geometry (Paris, 1992), Aspects Math. E26, Vieweg, 105-126 (1994) \. Zbl 0824.53064 \[12] Demailly, J.-P., Pali, N.: Degenerate complex Monge-Amp\`ere equations over compact K\"ahler manifolds. Int. J. Math. 21, 357-405 (2010) \. Zbl 1191.53029 \. DOI: 10.1142/S0129167X10006070 \. arxiv:0710.5109 \[13] Dinew, S.: H\"older continuous potentials on manifolds with partially positive curvature. J. Inst. Math. Jussieu 9, 705-718 (2010) \. Zbl 1207.32034 \. DOI: 10.1017/ S1474748010000113 \[14] Dinew, S., Zhang, Z.: On stability and continuity of bounded solutions of degenerate complex Monge-Amp\`ere equations over compact K\"ahler manifolds. Adv. Math. 225, 367-388 (2010) \. Zbl 1210.32020 \. DOI: 10.1016/j.aim.2010.03.001 \[15] Dinh, T. C., Nguyen, V. A., Sibony, N.: Exponential estimates for plurisubhar- monic functions and stochastic dynamics. J. Differential Geom. 84, 465-488 (2010) \. Zbl 1211.32021 \. euclid:jdg/1279114298 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Campana, Fr\'ed\'eric; Demailly, Jean-Pierre; Verbitsky, Misha \Title Compact K\"ahler 3-manifolds without nontrivial subvarieties. (English) Zbl 1293.32028 \Publi Algebr. Geom. 1, No. 2, 131-139 (2014). This is a very interesting paper. The authors prove that compact K\"ahler threefolds without nontrivial subvariety are tori. We describe below the proof in four steps: Step 1 (Corollary 4.3): If $X$ is a compact simple K\"ahler threefold, then $K_X$ is pseudo-effective. That is the Chern class of the canonical line bundle has a positive $(1,1)$-current. $X$ is simple if there is an intersection $A$ of a set of accountable dense Zariski open sets such that each point in $A$ is not contained in any nontrivial subvariety. The description on page 131 of the ``simple'' is a little bit confusing. The authors use the celebrated result of Brunella\\ {\bf Theorem 4.1.} They notice that $h^{2,0} >0$, otherwise $X$ is projective. Step 2 (Corollary 2.6): If $X$ is a compact K\"ahler manifold without nontrivial subvariety, then $K_X$ is nef but not big. This comes from Theorem 2.5. Step 3 (Corollary 3.3): ${\cal X}({\cal O})=0$. This comes from a manipulation of the Riemann-Roch formula with a version of the Hard Lefschetz Theorem 3.1. Step 4 (Lemma 1.4.): Another application of the Riemann-Roch formula noticing that $h^{3,0} \leq 1$. See also the authors' summary. \Reviewer Daniel Guan (Riverside) \MSC 32J17 Compact $3$-folds (analytic spaces)\\ 32J27 Compact K\"ahler manifolds: generalizations, classification\\ 32J25 Transcendental methods of algebraic geometry \Keywords compact K\"ahler threefolds; simple manifolds; holomorphic foliations; complex torus; hyperk\"ahler manifolds \Author Demailly, Jean-Pierre \Title Pierre Lelong: a fundamental work in complex analysis and analytical geometry.\\ (Pierre Lelong: une {\oe}uvre fondatrice en analyse complexe et en g\'eom\'etrie analytique.) (French) Zbl 1296.01027 \Publi Gaz. Math., Soc. Math. Fr. 135, 63-66 (2013). \Cited in 1 Document \MSC 01A70 Biographies, obituaries, personalia, bibliographies\\ 32-03 Historical (several complex variables and analytic spaces) \Author Demailly, Jean-Pierre \Title Episciences: a publishing platform for open archive overlay journals. (English) Zbl 1290.01031 \Publi Eur. Math. Soc. Newsl. 87, 31-32 (2013). \MSC 01A80 Sociology (and profession) of mathematics \Author Demailly, Jean-Pierre; Hacon, Christopher D.; P\u{a}un, Mihai \Title Extension theorems, non-vanishing and the existence of good minimal models. (English) Zbl 1278.14022 \Publi Acta Math. 210, No. 2, 203-259 (2013). Let $X$ be a complex projective manifold (or normal complex projective variety with mild singularities). The aim of the minimal model program is to construct a birational model $X \dashrightarrow X'$ such that either $X'$ admits a fibration with general fibre a Fano variety or $X'$ is a good minimal model, that is some positive multiple of the canonical divisor $K_{X'}$ defines a morphism. If $X$ is covered by rational curves or $X$ is of general type (that is some positive multiple of $K_X$ defines a birational map) the minimal model program is completed in the landmark paper by {\it C. Birkar} et al., [J. Am. Math. Soc. 23, No. 2, 405--468 (2010; Zbl 1210.14019)]. Thus the main challenge is now to study projective manifolds $X$ that are not covered by rational curves and not of general type. By a fundamental result of {\it S. Boucksom} et al. [J. Algebr. Geom. 22, No. 2, 201--248 (2013; Zbl 1267.32017)], the canonical divisor $K_X$ is then pseudoeffective, that is $K_X$ is a limit of effective divisor. However the nonvanishing conjecture claims that some positive multiple of the canonical divisor is actually effective. Once we know that there exists at least one effective pluricanonical divisor $D$ one can hope to establish the existence of good minimal models inductively by proving that the restriction morphism $$ H^0(X, \mathcal O_X(mK_X)) \rightarrow H^0(D, \mathcal O_D(mK_X)) $$ is surjective for $m \gg 0$. A similar extension result played a crucial role in the proof of the existence of flips by {\it C. D. Hacon} and {\it J. McKernan} [J. Am. Math. Soc. 23, No. 2, 469--490 (2010; Zbl 1210.14021)]. In the paper under review the authors realise an important step of this strategy by proving the following ``plt'' extension theorem: \newline Let $X$ be a projective manifold and $S+B$ a $\mathbb Q$-divisor with simple normal crossings such that \newline 1) $(X,S+B)$ is plt (i.e.\ $S$ is a prime divisor with $\mathrm{mult}_S(S+B)=1$ and $\lfloor B \rfloor =0$), and \newline 2) there exists an effective $\mathbb Q$-divisor $D\sim _{\mathbb Q}K_X+S+B$ such that $$S\subset \mathrm{Supp} (D)\subset \mathrm{Supp} (S+B),$$ and \newline 3) for any ample divisor $A$ and any rational number $\epsilon >0$, there is an effective $\mathbb Q$-divisor $D\sim _{\mathbb Q}K_X+S+B+\epsilon A$ whose support does not contain $S$). \newline Consider $\pi: \tilde X\to X$ a log-resolution of $(X, S+B)$, so that we have $$K_{\tilde X}+ \tilde S+ \tilde B= \pi^*(K_X+S+B)+ \tilde E$$ where $\tilde S$ is the strict transform of $S$. Let $m$ be an integer, such that $m(K_X+S+B)$ is Cartier, and let $u$ be a section of $m(K_X+S+B)|_S$, such that $$ Z_{\pi^*(u)}+ m\tilde E|_{\tilde S}\geq m\Xi, $$ where $Z_{\pi^*(u)}$ is the zero divisor of the section $\pi^*(u)$ and $\Xi$ the extension obstruction divisor (cf. [Zbl 1210.14021]). Then $u$ extends to $X$. \newline The main achievement of this theorem compared to earlier extension results is that one does not assume $B$ to be strictly positive (i.e.\ ample or big). The authors conjecture that their statement also holds under the weaker assumption that the pair $(X,S+B)$ is dlt. This stronger extension result would then reduce the minimal model conjecture to the nonvanishing problem. More precisely the authors prove the following theorem: Suppose that the ``dlt'' extension theorem holds in dimension $n$. Suppose also that the non-vanishing conjecture holds for semi-log-canonical pairs of dimension $n$. Then every $n$-dimensional projective manifold that is not covered by rational curves has a good minimal model. \Reviewer Andreas H\"oring (Nice) \Cited in 2 Reviews \Cited in 9 Documents \MSC 14E30 Minimal model program (Mori theory, extremal rays)\\ 14J40 Algebraic $n$-folds ($n>4$)\\ 32J25 Transcendental methods of algebraic geometry \Keywords extension theorem; minimal model; MMP; abundance conjecture; nonvanishing conjecture \References \[1] Ambro F.: Nef dimension of minimal models. Math. Ann., 330, 309--322 (2004) \. Zbl 1081.14074 \. DOI: 10.1007/s00208-004-0550-1 \[2] Ambro F.: The moduli b-divisor of an lc-trivial fibration. Compos. Math., 141, 385--403 (2005) \. Zbl 1094.14025 \. DOI: 10.1112/S0010437X04001071 \[3] Berndtsson, B., The extension theorem of Ohsawa--Takegoshi and the theorem of Donnelly--Fefferman. Ann. Inst. Fourier (Grenoble), 46 (1996), 1083--1094. \. 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Soc., Providence, RI, 2003. \. Zbl 1049.32010 \[41] Ohsawa, T. Generalization of a precise $L^2$ division theorem, in Complex Analysis in Several Variables, Adv. Stud. Pure Math., 42, pp. 249--261. Math. Soc. Japan, Tokyo, 2004. \. Zbl 1078.32004 \[42] Ohsawa, T. \& Takegoshi, K., On the extension of $L^2$ holomorphic functions. Math. Z., 195 (1987), 197--204. \. Zbl 0625.32011 \[43] P\u{a}un, M., Siu's invariance of plurigenera: a one-tower proof. J. Differential Geom., 76 (2007), 485--493. \. Zbl 1122.32014 \[44] P\u{a}un, M. Relative critical exponents, non-vanishing and metrics with minimal singularities. Invent. Math., 187 (2012), 195--258. \. Zbl 1251.32018 \[45] Shokurov, V. V., Letters of a bi-rationalist. VII. Ordered termination. Tr. Mat. Inst. Steklova, 264 (2009), 184--208 (Russian); English translation in Proc. Steklov Inst. Math., 264 (2009), 178--200. \. Zbl 1312.14041 \[46] Siu, Y.-T., Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math., 27 (1974), 53--156. \. Zbl 0289.32003 \[47] Siu, Y.-T. Every Stein subvariety admits a Stein neighborhood. Invent. Math., 38 (1976/77), 89--100. \. Zbl 0343.32014 \[48] Siu, Y.-T. Invariance of plurigenera. Invent. Math., 134 (1998), 661--673. \. Zbl 0955.32017 \[49] Siu, Y.-T. Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type, in Complex Geometry (G\"ottingen, 2000), pp. 223--277. Springer, Berlin--Heidelberg, 2002. \. Zbl 1007.32010 \[50] Siu, Y.-T. Finite generation of canonical ring by analytic method. Sci. China Ser. A, 51 (2008), 481--502. \. Zbl 1153.32021 \[51] Siu, Y.-T. Abundance conjecture, in Geometry and Analysis. No. 2, Adv. Lect. Math., 18, pp. 271--317. Int. Press, Somerville, MA, 2011. \. Zbl 1263.14018 \[52] Skoda, H., Application des techniques $L^2$\`a la th\'eorie des id\'eaux d'une alg\`ebre de fonctions holomorphes avec poids. Ann. Sci. \'Ec. Norm. Super., 5 (1972), 545--579. \. Zbl 0254.32017 \[53] Takayama S.: Pluricanonical systems on algebraic varieties of general type. Invent. Math., 165, 551--587 (2006) \. Zbl 1108.14031 \. DOI: 10.1007/s00222-006-0503-2 \[54] Takayama S.: On the invariance and the lower semi-continuity of plurigenera of algebraic varieties. J. Algebraic Geom., 16, 1--18 (2007) \. Zbl 1113.14028 \. DOI: 10.1090/S1056-3911-06-00455-3 \[55] Tian G.: On K\"ahler--Einstein metrics on certain K\"ahler manifolds with C1(M) \> 0. Invent. Math., 89, 225--246 (1987) \. Zbl 0599.53046 \. DOI: 10.1007/BF01389077 \[56] Tsuji, H., Extension of log pluricanonical forms from subvarieties. Preprint, 2007. arXiv:0709.2710 [math.AG]. \[57] Varolin D., Varolin D.: Takayama-type extension theorem. Compos. Math., 144, 522--540 (2008) \. Zbl 1163.32008 \. DOI: 10.1112/S0010437X07002989 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Applications of pluripotential theory to algebraic geometry. (English) Zbl 1271.32037 \Publi Bracci, Filippo (ed.) et al., Pluripotential theory. Lectures of the CIME course, Cetraro, Italy, 2011. Berlin: Springer; Florence: Fondazione CIME (ISBN 978-3-642-36420-4/pbk; 978-3-642-36421-1/ebook). Lecture Notes in Mathematics 2075. CIME Foundation Subseries, 143-263 (2013). {\it Summary$\,$}: These lectures are devoted to the study of various contemporary problems of algebraic geometry, using fundamental tools from complex potential theory, namely plurisubharmonic functions, positive currents and Monge-Amp\`ere operators. Since their inception by Oka and Lelong in the mid 1940s, plurisubharmonic functions have been used extensively in many areas of algebraic and analytic geometry, as they are the function theoretic counterpart of pseudoconvexity, the complexified version of convexity. One such application is the theory of $L^2$ estimates via the Bochner-Kodaira-H\"ormander technique, which provides very strong existence theorems for sections of holomorphic vector bundles with positive curvature. One can mention here the foundational work achieved by Bochner, Kodaira, Nakano, Morrey, Kohn, Andreotti-Vesentini, Grauert, H\"ormander, Bombieri, Skoda and Ohsawa-Takegoshi in the course of more than four decades. Another development is the theory of holomorphic Morse inequalities (1985), which relate certain curvature integrals with the asymptotic cohomology of large tensor powers of line or vector bundles, and bring a useful complement to the Riemann-Roch formula.We describe here the main techniques involved in the proof of holomorphic Morse inequalities (Sect. 1) and their link with Monge-Amp\`ere operators and intersection theory. Section 2, especially, gives a fundamental approximation theorem for closed (1, 1)-currents, using a Bergman kernel technique in combination with the Ohsawa-Takegoshi theorem. As an application, we study the geometric properties of positives cones of an algebraic variety (nef and pseudo-effective cone), and derive from there some results about asymptotic cohomology functionals in Sect. 3. The last Sect. 4 provides an application to the study of the Green-Griffiths-Lang conjecture. The latter conjecture asserts that every entire curve drawn on a projective variety of general type should satisfy a global algebraic equation; via a probabilistic curvature estimate, holomorphic Morse inequalities imply that entire curves must at least satisfy a global algebraic differential equation. For the entire collection see [Zbl 1266.31001]. \MSC 32U05 Plurisubharmonic functions and generalizations\\ 32W20 Complex Monge-Amp\`ere operators\\ 32U40 Currents \Keywords plurisubharmonic functions; positive currents; Monge-Amp\`ere operator; holomorphic Morse inequalities; Ohsawa-Takegoshi theorem \Author Boucksom, S\'ebastien; Demailly, Jean-Pierre; P\u{a}un, Mihai;\\ Peternell, Thomas \Title The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension. (English) Zbl 1267.32017 \Publi J. Algebr. Geom. 22, No. 2, 201-248 (2013). {\it Summary$\,$}: We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of ``movable curves'', which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive $ (1,1)$-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non-negative Kodaira dimension. \Cited in 7 Reviews \Cited in 71 Documents \MSC 32Q15 K\"ahler manifolds\\ 32J27 Compact K\"ahler manifolds: generalizations, classification \Keywords compact K\"ahler manifold; projective K\"ahler manifold; Kodaira dimension; uniruled manifold \References \[1] Florin Ambro, Nef dimension of minimal models, Math. Ann. 330 (2004), no. 2, 309 -- 322. \. Zbl 1081.14074 \. DOI: 10.1007/s00208-004-0550-1 \.\\ https://doi.org/10.1007\%2Fs00208-004-0550-1 \[2] S\'ebastien Boucksom, Charles Favre, and Mattias Jonsson, Differentiability of volumes of divisors and a problem of Teissier, J. Algebraic Geom. 18 (2009), no. 2, 279 -- 308. \. Zbl 1162.14003 \[3] S\'ebastien Boucksom, On the volume of a line bundle, Internat. J. Math. 13 (2002), no. 10, 1043 -- 1063. \. Zbl 1101.14008 \. 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Zbl 0565.58017 \[15] Jean-Pierre Demailly, Singular Hermitian metrics on positive line bundles, Complex algebraic varieties (Bayreuth, 1990) Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 87 -- 104. \. Zbl 0784.32024 \. DOI: 10.1007/BFb0094512 \.\\ https://doi.org/10.1007\%2FBFb0094512 \[16] Jean-Pierre Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), no. 3, 361 -- 409. \. Zbl 0777.32016 \[17] Demailly, J.-P., Vanishing theorems and effective results in algebraic geometry, School on algebraic Geometry held at ICTP in January 2000, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001. \[18] Jean-Pierre Demailly, K\"ahler manifolds and transcendental techniques in algebraic geometry, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Z\"urich, 2007, pp. 153 -- 186. \. Zbl 1141.14007 \. DOI: 10.4171/022-1/8 \.\\ https://doi.org/10.4171\%2F022-1\%2F8 \[19] Demailly, J.-P., Holomorphic Morse inequalities and asymptotic$\,$cohomology$\,$groups: a tribute to Bernhard Riemann, arXiv:1003.5067, math.CV, to appear in Proceedings of the Riemann International School of Mathematics, Advances in Number Theory and Geometry, held at Verbania (Italy), April 2009. \[20] Jean-Pierre Demailly, Lawrence Ein, and Robert Lazarsfeld, A subadditivity property of multiplier ideals, Michigan Math. J. 48 (2000), 137 -- 156. Dedicated to William Fulton on the occasion of his 60th birthday. \. Zbl 1077.14516 \. DOI: 10.1307/mmj/ 1030132712 \. https://doi.org/10.1307\%2Fmmj\%2F1030132712 \[21] Jean-Pierre Demailly and Mihai Paun, Numerical characterization of the K\"ahler cone of a compact K\"ahler manifold, Ann. of Math. (2) 159 (2004), no. 3, 1247 -- 1274. \. Zbl 1064.32019 \. DOI: 10.4007/annals.2004.159.1247 \.\\ https://doi.org/10.4007\%2Fannals.2004.159.1247 \[22] Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), no. 2, 295 -- 345. \. Zbl 0827.14027 \[23] Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Holomorphic line bundles with partially vanishing cohomology, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 165 -- 198. \. Zbl 0859.14005 \[24] Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Pseudo-effective line bundles on compact K\"ahler manifolds, Internat. J. Math. 12 (2001), no. 6, 689 -- 741. \. Zbl 1111.32302 \. DOI: 10.1142/S0129167X01000861 \.\\ https://doi.org/10.1142\%2FS0129167X01000861 \[25] Thomas Eckl, Tsuji's numerical trivial fibrations, J. Algebraic Geom. 13 (2004), no. 4, 617 -- 639. \. 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France, 1992. \[36] Lazarsfeld, R., Ampleness modulo the branch locus of the bundle associated to a branched covering, Comm. Alg. 28 (2000), 5598-5599. \[37] Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. \[38] Yoichi Miyaoka, The Chern classes and Kodaira dimension of a minimal variety, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 449 -- 476. \[39] Yoichi Miyaoka, On the Kodaira dimension of minimal threefolds, Math. Ann. 281 (1988), no. 2, 325 -- 332. \. Zbl 0625.14023 \. DOI: 10.1007/BF01458437 \.\\ https://doi.org/10.1007\%2FBF01458437 \[40] Yoichi Miyaoka, Abundance conjecture for 3-folds: case $\nu=1$, Compositio Math. 68 (1988), no. 2, 203 -- 220. \. Zbl 0681.14019 \[41] Yoichi Miyaoka and Shigefumi Mori, A numerical criterion for uniruledness, Ann. of Math. (2) 124 (1986), no. 1, 65 -- 69. \. Zbl 0606.14030 \. DOI: 10.2307/1971387 \. https://doi.org/10.2307\%2F1971387 \[42] Shigefumi Mori, Classification of higher-dimensional varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 269 -- 331. \. Zbl 1103.14301 \[43] Shigefumi Mori, Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 (1988), no. 1, 117 -- 253. \. Zbl 0649.14023 \[44] Noboru Nakayama, On Weierstrass models, Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, pp. 405 -- 431. \[45] Noboru Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004. \[46] Mihai Paun, Sur l'effectivit\'e num\'erique des images inverses de fibr\'es en droites, Math. Ann. 310 (1998), no. 3, 411 -- 421 (French). \. Zbl 1023.32014 \. DOI: 10.1007/ s002080050154 \. https://doi.org/10.1007\%2Fs002080050154 \[47] Thomas Peternell, Towards a Mori theory on compact K\"ahler threefolds. III, Bull. Soc. Math. France 129 (2001), no. 3, 339 -- 356 (English, with English and French summaries). \. Zbl 0994.32017 \[48] Thomas Peternell, Kodaira dimension of subvarieties. II, Internat. J. Math. 17 (2006), no. 5, 619 -- 631. \. Zbl 1126.14019 \. DOI: 10.1142/S0129167X0600362X \. https://doi.org/10.1142\%2FS0129167X0600362X \[49] Thomas Peternell, Michael Schneider, and Andrew J. Sommese, Kodaira dimension of subvarieties, Internat. J. Math. 10 (1999), no. 8, 1065 -- 1079. \. Zbl 1077.14515 \. DOI: 10.1142/S0129167X9900046X \. https://doi.org/10.1142\%2FS0129167X9900046X \[50] Thomas Peternell and Andrew J. Sommese, Ample vector bundles and branched coverings. II, The Fano Conference, Univ. Torino, Turin, 2004, pp. 625 -- 645. \. Zbl 1071.14018 \[51] Shepherd-Barron, N., Miyaoka's theorems on the generic semi-negativity of $ T_X$, Ast\'erisque 211 (1992), 103-114. \. Zbl 0809.14034 \[52] Tsuji, H., Numerically trivial fibrations., math.AG/0001023 (2000). \[53] Thomas Bauer, Fr\'ed\'eric Campana, Thomas Eckl, Stefan Kebekus, Thomas Peternell, S{\l}awomir Rams, Tomasz Szemberg, and Lorenz Wotzlaw, A reduction map for nef line bundles, Complex geometry (G\"ottingen, 2000) Springer, Berlin, 2002, pp. 27 -- 36. \[54] Shing Tung Yau, On the Ricci curvature of a compact K\"ahler manifold and the complex Monge-Amp\`ere equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339 -- 411. \. Zbl 0369.53059 \. DOI: 10.1002/cpa.3160310304 \.\\ https://doi.org/10.1002\%2Fcpa.3160310304 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre (ed.); Hulek, Klaus (ed.); Peternell, Thomas (ed.) \Title Complex analysis. Abstracts from the workshop held September 2--8, 2012. (Komplexe Analysis.) (English) Zbl 1349.00135 \Publi Oberwolfach Rep. 9, No. 3, 2597-2656 (2012). {\it Summary$\,$}: The aim of this workshop was to discuss recent developments in several complex variables and complex geometry. Special emphasis was put on the interaction of analytic and algebraic methods. Topics included K\"ahler geometry, Ricci-flat manifolds, moduli theory and themes related to the minimal model program. \MSC 00B05 Collections of abstracts of lectures 00B25 Proceedings of conferences of miscellaneous specific interest\\ 32-06 Proceedings of conferences (several complex variables)\\ 14-06 Proceedings of conferences (algebraic geometry) \References \[1] This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Henri Cartan and multivariate holomorphic functions. (Henri Cartan et les fonctions holomorphes de plusieurs variables.) (French) Zbl 1294.32002 \Publi Harinck, Pascale (ed.) et al., Henri Cartan et Andr\'e Weil. Math\'ematiciens du XX$^{\text e}$ si\`ecle. Journ\'ees math\'ematiques X-UPS, Palaiseau, France, May 3--4, 2012. Palaiseau: Les \'Editions de l'\'Ecole Polytechnique (ISBN 978-2-7302-1610-4/pbk). 99-168 (2012). The paper is a review of some fundamental results on holomorphic functions of several variables, and of the role of Henri Cartan in the development of this theory, in particular in the theory of coherent sheaves which he developed and which stands now as one of the most fundamental tools in complex geometry and in algebraic geometry. The exposition is concise but self-contained and the stress is on the essential facts. A few historical remarks are useful for understanding the motivations behind the ideas. All this makes the text much more attractive than many other texts written on the subject. The bibliographical references organized in sections are also very useful. For the entire collection see [Zbl 1270.01009]. \Reviewer Athanase Papadopoulos (MR3014194) \MSC 32-03 Historical (several complex variables and analytic spaces)\\ 32A10 Holomorphic functions (several variables)\\ 32B10 Germs of analytic sets, local parametrization\\ 32C35 Analytic sheaves and cohomology groups\\ 32H99 Holomorphic mappings on analytic spaces \Keywords holomorphic functions of several variables; coherent sheaves; complex geometry; algebraic geometry \Author Demailly, Jean-Pierre \Title Hyperbolic algebraic varieties and holomorphic differential equations. (English) Zbl 1264.32022 \Publi Acta Math. Vietnam. 37, No. 4, 441-512 (2012). The long and exhaustive article reviews and explains recent methods and results in the theory of hyperbolic algebraic varieties, including several new results of the author which are focussed on some of the most challenging conjectures about hyperbolicity of algebraic manifolds of general type, the conjectures of Green-Griffiths and Lang. The conjectures of Green-Griffiths state that a projective algebraic variety $X$ is of general type if and only if $X$ is measure hyperbolic, and that $X$ then contains a proper subvariety $Y$ such that $f(\mathbb C)\subset Y$ for every non-constant holomorphic map $f:\mathbb C\rightarrow X$. An affirmative solution for these conjectures would imply that every very generic hypersurface in $\mathbb P^{n+1}$, $n\ge 3$, of degree $d\ge 2n+1$ is hyperbolic, see [{\it L. Ein}, Invent. Math. 94, No. 1, 163--169 (1988; Zbl 0701.14002); Math. Ann. 289, No. 3, 465--471 (1991; Zbl 0746.14019)] and [{\it C. Voisin}, J. Differ. Geom. 44, No. 1, 200--213 (1996; Zbl 0883.14022)]. The author studies hyperbolicity problems in a more general setting. He works in the class of directed manifolds and transfers classical hyperbolicity concepts for complex varieties to this category. Special emphasis lies on the application of the Ahlfors-Schwarz lemma, jets of curves, Semple jet bundles, k-jet bundles, holomorphic Morse inequalities, jet differentials and k-jet metrics with negative curvature. A (compact resp.\ projective) directed manifold is understood as a pair $(X,V)$ of a (compact resp.\ projective) connected complex manifold $X$ and an irreducible closed analytic subspace $V$ of the holomorphic tangent bundle $T_X$ such that $V\cap T_{X,x}$ is a linear subspace of $T_{X,x}$ for every $x\in X$. Based on the Brody criterion it turns out that a compact directed manifold $(X,V)$ is hyperbolic iff every holomorphic map $f:\mathbb C\rightarrow X$ with $f'(\mathbb C)\subset V$ is constant. A projective directed manifold $(X,V)$ is by definition algebraic hyperbolic if $X$ admits an Hermitian metric with fundamental form $\omega$ such that for some $\epsilon> 0$ the inequality $-\chi(\overline{C})\ge\epsilon \int_C\omega$ is satisfied for every irreducible closed algebraic curve $C\subset X$ tangent to $V$ and normalized by $\overline{C}$. Hyperbolic projective directed manifolds are algebraic hyperbolic, but the converse is not known. The main results of the paper are partial answers to a generalized version of the conjectures mentioned above. Assume that $(X,V)$ is a projective directed manifold and that the canonical bundle of $V$ is big. The generalized Green-Griffiths-Lang conjecture claims the existence of a proper subvariety $Y$ of $X$ with $f(\mathbb C)\subset Y$ for every non-constant holomorphic $f:\mathbb C\rightarrow X$ with $f'(\mathbb C)\subset V$. Under the assumption that $X$ is of general type the author proves the existence of global algebraic differential operators $P$ on $X$ with $P(f, f',\dots,f^{(k)})=0$ for every such $f$. Another interesting result of the author is related to the hyperbolicity of generic algebraic hypersurfaces in $\mathbb P^{n+1}$ of sufficiently high degree. {\it Y.-T. Siu} [``Hyperbolicity of generic high-degree hypersurfaces in complex projective spaces'', Preprint, \url{arXiv:1209.2723}] has shown that there exists a sequence $(d_n)$ in $\mathbb N$ with the property that a generic algebraic hypersurface in $\mathbb P^{n+1}$ of degree $d\ge d_n$, $n\ge 2$, is hyperbolic. {\it S. Diverio, J. Merker} and {\it E. Rousseau} [Invent. Math. 180, No. 1, 161--223 (2010; Zbl 1192.32014)] proved that $d_n:= 2^{n^5}$ does the job. The paper under review yields a considerable improvement of these estimates. The theorem of Siu is fulfilled for $d_2:=286$, $d_3:=7316$ and $d_n:=\lfloor\frac{n^4}{3}(n\log(n\log(24n)))^n\rfloor$ for $n\ge 4$. \Reviewer Eberhard Oeljeklaus (Bremen) \Cited in 2 Documents \MSC 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds\\ 32L10 Sections of holomorphic vector bundles 53C55 Hermitian and K\"ahlerian manifolds (global differential geometry)\\ 14J40 Algebraic $n$-folds ($n>4$) \Keywords Kobayashi hyperbolic variety; directed manifold; jet bundle; Chern connection; variety of general type; Green-Griffiths conjecture; Lang conjecture \Author Berman, Robert; Demailly, Jean-Pierre \Title Regularity of plurisubharmonic upper envelopes in big cohomology classes. (English) Zbl 1258.32010 \Publi Itenberg, Ilia (ed.) et al., Perspectives in analysis, geometry, and topology. On the occasion of the 60th birthday of Oleg Viro. Based on the Marcus Wallenberg symposium on perspectives in analysis, geometry, and topology, Stockholm, Sweden, May 19--25, 2008. Basel: Birkh\"auser (ISBN 978-0-8176-8276-7/hbk; 978-0-8176-8277-4/ebook). Progress in Mathematics 296, 39-66 (2012). The authors deal with the regularity of certain quasiplurisubharmonic upper envelopes. The main theorem says that if $X$ is a compact complex manifold of the Fujiki class $\mathcal C$ (these are the smooth varieties that are bimeromorphic to compact K\"ahler manifolds, or equivalently, they carry a cohomology class $\lbrace\alpha\rbrace\in H^{1,1}(X,\mathbb R)$ which is big), then let $\lbrace\alpha\rbrace$ be the aforementioned big class and let $T_{0}$ be the current obtained from $\alpha$ by adding the $dd^{c}$ of a quasiplurisubharmonic function $\psi_{0}$ with only analytic singularities, such that it dominates some positive multiple of the fixed Hermitian metric on $X$. Then the following function (called the upper envelope) $$\varphi=\sup\big\{ \psi\leq 0: \psi \text { is } \alpha\text{-plurisubharmonic}\big\}$$ is itself quasiplurisubharmonic and has locally bounded second order derivatives outside the analytic set given by the ``$-\infty$''-set of $\psi_{0}$. The order of blow up near this set is also estimated. The main theorem is then applied to obtain a priori inequalities for the solution of the Dirichlet problem for a degenerate Monge-Amp\`ere operator, to the study of geodesics in the space of K\"ahler potentials and finally to obtain a logarithmic modulus of continuity for Tsuji's supercanonical metrics. For the entire collection see [Zbl 1230.00045]. \Reviewer Zywomir Dinew (Krak\'ow) \Cited in 1 Review \Cited in 16 Documents \MSC 32U05 Plurisubharmonic functions and generalizations \Keywords plurisubharmonic function; quasiplurisubharmonic function; upper envelope; Monge-Amp\`ere operator \Full_text \Author Demailly, Jean-Pierre \Title Analytic methods in algebraic geometry. (English) Zbl 1271.14001 \Publi Surveys of Modern Mathematics 1. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-234-3/pbk). 231~p. (2012). The book under review is based upon a series of lectures given by Jean-Pierre Demailly at the Park City Mathematics Institute in 2008, it was partly published in [IAS/Park City Mathematics Series 17, 295--370 (2010; Zbl 1222.32043)]. The main aim is to give a presentation of analytic techniques especially as it relates to positivity of vector bundles. The book begins by briefly reviewing the concepts of sheaf cohomology, plurisubharmonic functions, currents and other topics. In Chapter 4, it moves on to the Bochner technique and applications to the Akizuki-Nakano-Kodaira vanishing theorem. Chapter 5 covers $L^2$ estimates and multiplier ideal sheaves. Chapter 6 covers pseudo effective and nef line bundles and Kawamata-Viehweg vanishing and applications. The next chapter covers applications of these ideas, results towards Fujita's conjecture, such as Reider's Theorem and work of Siu. Chapter 8 covers Holomorphic Morse inequalities as introduced by Demailly. Chapter 9 covers effective versions of Matsusaka's big theorem. Chapter 11 covers the question of surjectivity of global sections for maps of vector bundles and an application to the Briancon-Skoda Theorem. Chapter 12 covers the Ohsawa-Takegoshi $L^2$ Extension Theorem and Skoda's division theorem. Chapter 13 focuses on approximation of positive currents and plurisubharmoic functions and relations to the Hodge Conjecture. The remainder of the book (chapters 15 through 20), cover various topics in higher dimensional algebraic geometry such as Subadditivity of multiplier ideals, invariance of plurigenera, the K\"ahler cone and Pseudo-efective cone, abundance, and many other topics. \Reviewer Karl Schwede (University Park) \Cited in 1 Review \Cited in 12 Documents \MSC 14-02 Research monographs (algebraic geometry)\\ 14F18 Multiplier ideals\\ 14C20 Divisors, linear systems, invertible sheaves\\ 14J40 Algebraic $n$-folds ($n>4$)\\ 32C30 Integration on analytic sets and spaces, currents\\ 32U40 Currents \Keywords positivity; vanishing theorem; multiplier ideal; current; extension theorem; line bundles \Author Demailly, Jean-Pierre \Title Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture. (English) Zbl 1316.32014 \Publi Pure Appl. Math. Q. 7, No. 4, 1165-1207 (2011). {\it Summary$\,$}: The goal of this work is to study the existence and properties of non constant entire curves f drawn in a complex irreducible n-dimensional variety X, and more specifically to show that they must satisfy certain global algebraic or differential equations as soon as $X$ is projective of general type. By means of holomorphic Morse inequalities and a probabilistic analysis of the cohomology of jet spaces, we are able to reach a significant step towards a generalized version of the Green-Griffiths-Lang conjecture. \Cited in 2 Reviews \Cited in 11 Documents \MSC 32L20 Vanishing theorems (analytic spaces)\\ 14C30 Transcendental methods, Hodge theory, Hodge conjecture\\ 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds \Full_text \Author Demailly, Jean-Pierre \Title A converse to the Andreotti-Grauert theorem.\\ (English. French summary) Zbl 1228.32020 \Publi Ann. Fac. Sci. Toulouse, Math. (6) 20, Spec. Issue, 123-135 (2011). {\it Summary$\,$}: The goal of this paper is to show that there are strong relations between certain Monge-Amp\`ere integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of holomorphic line bundles. Especially, we prove that these relations hold without restriction for projective surfaces, and in the special case of the volume, i.e.\ of asymptotic $0$-cohomology, for all projective manifolds. These results can be seen as a partial converse to the Andreotti-Grauert vanishing theorem. \Reviewer Viorel V\^aj\^aitu (Bucure\c{s}ti) \MSC 32J25 Transcendental methods of algebraic geometry\\ 32L10 Sections of holomorphic vector bundles\\ 14C20 Divisors, linear systems, invertible sheaves\\ 14F99 Homology and cohomology theory (algebraic geometry) \Keywords asymptotic cohomology functions; holomorphic Morse inequalities; volume of a line bundle \Full_text \References \[1] Andreotti (A.), Grauert (H.).-- Th\'eor\`emes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90, p. 193-259 (1962). Zbl0106.05501 MR150342 \. Zbl 0106.05501 \. NUMDAM: BSMF\_1962\_\_90\_\_193\_0 \. EUDML: 87019 \[2] Angelini (F.).-- An algebraic version of Demailly's asymptotic Morse inequalities; arXiv: alg-geom/9503005, Proc. Amer. Math. Soc. 124 p. 3265-3269 (1996). Zbl0860.14019 MR1389502 \. Zbl 0860.14019 \. DOI: 10.1090/S0002-9939-96-03829-4 \. arxiv:alg-geom/9503005 \[3] Boucksom (S.).-- On the volume of a line bundle, Internat. J. Math. 13, p. 1043-1063 (2002). Zbl1101.14008 MR1945706 \. Zbl 1101.14008 \.\\ DOI: 10.1142/S0129167X02001575 \[4] Boucksom (S.), Demailly (J.-P.), P\u{a}un (M.), Peternell (Th.).-- The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension, arXiv: math.AG/0405285, see also Proceedings of the ICM 2006 in Madrid. MR1351504 \[5] Demailly (J.-P.).-- Estimations $L^2$ pour l'op\'erateur $\overline\partial$ d'un fibr\'e vectoriel holomorphe semi-positif au dessus d'une vari\'et\'e k\"ahl\'erienne compl\`ete, Ann. Sci. \'Ecole Norm. Sup. 15, p. 457-511 (1982). Zbl0507.32021 MR690650 \. Zbl 0507.32021 \. NUMDAM: ASENS\_1982\_4\_15\_3\_457\_0 \. EUDML: 82103 \[6] Demailly (J.-P.).-- Champs magn\'etiques et in\'egalit\'es de Morse pour la $d''$-cohomo\-logie, Ann. Inst. Fourier (Grenoble) 35, p. 189-229 (1985). Zbl0565.58017 MR812325 \. Zbl 0565.58017 \. DOI: 10.5802/aif.1034 \. NUMDAM: AIF\_1985\_\_35\_4\_189\_0 \. EUDML: 74695 \[7] Demailly (J.-P.).-- Holomorphic Morse inequalities, Lectures given at the AMS Summer Institute on Complex Analysis held in Santa Cruz, July 1989, Proceedings of Symposia in Pure Mathematics, Vol. 52, Part 2, p. 93-114 (1991). Zbl0755.32008 MR1128538 \. Zbl 0755.32008 \[8] Demailly (J.-P.).-- Regularization of closed positive currents and Intersection Theory, J. Alg. Geom. 1, p. 361-409 (1992). Zbl0777.32016 MR1158622 \. Zbl 0777.32016 \[9] Demailly (J.-P.).-- Holomorphic Morse inequalities and asymptotic cohomology\break groups: a tribute to Bernhard Riemann, Milan Journal of Mathematics 78, p. 265-277 (2010). Zbl1205.32017 MR2684780 \. Zbl 1205.32017 \. DOI: 10.1007/s00032-010-0118-3 \[10] Demailly (J.-P.), Ein (L.) and Lazarsfeld (R.).-- A subadditivity property of multiplier ideals, Michigan Math. J. 48, p. 137-156 (2000). Zbl1077.14516 MR1786484 \. Zbl 1077.14516 \. DOI: 10.1307/mmj/1030132712 \. arxiv:math/0002035 \[11] Demailly (J.-P.), P\u{a}un (M.).-- Numerical characterization of the K\"ahler cone of a compact K\"ahler manifold, arXiv: math.AG/0105176~; Annals of Math. 159, p. 1247-1274 (2004). Zbl1064.32019 MR2113021 \. Zbl 1064.32019 \. DOI: 10.4007/annals.2004.159.1247 \[12] de Fernex (T.), K\"uronya (A.), Lazarsfeld (R.).-- Higher cohomology of divisors on a projective variety, Math. Ann. 337, p. 443-455 (2007). Zbl1127.14012 MR2262793 \. Zbl 1127.14012 \. DOI: 10.1007/s00208-006-0044-4 \[13] Fujita (T.).-- Approximating Zariski decomposition of big line bundles, Kodai Math. J. 17, p. 1-3 (1994). Zbl0814.14006 MR1262949 \. Zbl 0814.14006 \. DOI: 10.2996/kmj/1138039894 \[14] Hironaka (H.).-- Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. 79, p. 109-326 (1964). Zbl0122.38603 MR199184 \. Zbl 0122.38603 \. DOI: 10.2307/1970486 \[15] K\"uronya (A.).-- Asymptotic cohomological functions on projective varieties, Amer. J. Math. 128, p. 1475-1519 (2006). Zbl1114.14005 MR2275909 \. Zbl 1114.14005 \. DOI: 10.1353/ajm.2006.0044 \. http://muse.jhu.edu/journals/american\_journal\_of\_mathe\-ma\-tics/v128/128.6kuronya.pdf \[16] Lazarsfeld (R.).-- Positivity in Algebraic Geometry I.-II, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vols. 48-49, Springer Verlag, Berlin, 2004. Zbl1093.14500 MR2095471 \. Zbl 1093.14500 \[17] Totaro (B.).-- Line bundles with partially vanishing cohomology, July 2010, arXiv: math.AG/1007.3955. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Structure theorems for projective and K\"ahler varieties. (English) Zbl 1222.32043 \Publi McNeal, Jeffery (ed.) et al., Analytic and algebraic geometry. Common problems, different methods. Lecture notes from the Park City Mathematics Institute (PCMI) graduate summer school on analytic and algebraic geometry, Park City, UT, USA, Summer 2008. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4908-8/hbk). IAS/Park City Mathematics Series 17, 295-370 (2010). The main purpose of these notes is to describe some basic structure theorems for projective or compact K\"ahler varieties and their cohomology, using recent techniques of complex analysis and potential theory. One central unifying concept is that one of positivity, which can be viewed either in algebraic terms (positivity of divisors and algebraic cycles) or in more analytic terms (plurisubharmonicity, positive currents, Hermitian connections with positive curvature). In the 20th century, powerful $L^2$ techniques have emerged, giving rise to an incredible amount of geometric consequences. Here the author refers to these results and points out several topics: algebro-analytic characterizations of the Kaehler cone, the pseudo-effective cone of divisors, concepts of volume and mobile intersections, super-canonical metrics, non vanishing theorem, finiteness of the canonical ring. For the entire collection see [Zbl 1202.00101]. \Reviewer Gabriela Paola Ovando (Rosario) \Cited in 1 Review \Cited in 1 Document \MSC 32Q15 K\"ahler manifolds \Keywords projective varieties; K\"ahler varieties; positivity \Author Demailly, Jean-Pierre (ed.); Hulek, Klaus (ed.); Peternell, Thomas (ed.) \Title Complex analysis. Abstracts from the workshop held August 29th -- September 4th, 2010. (Komplexe analysis.) (English) Zbl 1209.00048 \Publi Oberwolfach Rep. 7, No. 3, 2283-2333 (2010). {\it Summary$\,$}: The aim of this workshop was to discuss recent developments in several complex variables and complex geometry. Special emphasis was put on the interaction between model theory and the classification theory of complex manifolds. Other topics included K\"ahler geometry, foliations, complex symplectic manifolds and moduli theory. \MSC 00B05 Collections of abstracts of lectures\\ 32-06 Proceedings of conferences (several complex variables)\\ 14-06 Proceedings of conferences (algebraic geometry)\\ 32Qxx Complex manifolds\\ 14Jxx Surfaces and higher-dimensional varieties\\ 14Dxx Families, fibrations \Full_text \Author Demailly, Jean-Pierre \Title Holomorphic Morse inequalities and asymptotic cohomology groups: a tribute to Bernhard Riemann. (English. French summary) Zbl 1205.32017 \Publi Milan J. Math. 78, 265-277 (2010). {\it Summary$\,$}: The goal of this note is to present the potential relationships between certain Monge-Amp\`ere integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of line bundles, as recently introduced by algebraic geometers. The expected most general statements, which are still conjectural, certainly owe a debt to Riemann's pioneering work, which led to the concept of Hilbert polynomials and to the Hirzebruch-Riemann-Roch formula during the XX-th century. \Cited in 2 Documents \MSC 32L10 Sections of holomorphic vector bundles\\ 14B05 Singularities (algebraic geometry)\\ 14C17 Intersection theory, etc. \Keywords holomorphic Morse inequalities; Monge-Amp\`ere integrals; Dolbeault cohomology; asymptotic cohomology groups; Riemann-Roch formula; Hermitian metrics; Chern curvature tensor; plurisuharmonic approximation \References \[1] Andreotti A., Grauert H.: Th\'eor\`emes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. France 90, 193--259 (1962) \. Zbl 0106.05501 \[2] Birkenhage Ch., Lange H.: Complex Abelian Varieties; Second augmented edition, Grundlehren der Math. Wissenschaften, Springer Heidelberg (2004) \[3] Boucksom S.: On the volume of a line bundle. Internat. J. Math. 13, 1043--1063 (2002) \. Zbl 1101.14008 \. DOI: 10.1142/S0129167X02001575 \[4] Berman, R., Demailly, J.-P.: Regularity of plurisubharmonic upper envelopes in big cohomology classes; arXiv: math.CV/0905.1246, to appear in: Perspectives in Analysis, Geometry and Topology, In honor of Oleg Y. Viro, ed. by B. Juhl- J\"oricke, I. Itenberg and M. Passare, Birkha\"user. \[5] Boucksom, S., Demailly, J.-P., P\u{a}un, M., Peternell, Th.: The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension; arXiv: math.AG/0405285, see also Proceedings of the ICM 2006 in Madrid. \. Zbl 1267.32017 \[6] Demailly J.-P.: Champs magn\'etiques et in\'egalit\'es de Morse pour la d''- cohomologie. Ann. Inst. Fourier (Grenoble) 35, 189--229 (1985) \. Zbl 0565.58017 \[7] Demailly, J.-P.: Holomorphic Morse inequalities; Lectures given at the AMS Summer Institute on Complex Analysis held in Santa Cruz, July 1989, Proceedings of Symposia in Pure Mathematics, Vol. 52, Part 2 (1991), 93--114. \[8] Demailly, J.-P.: Singular hermitian metrics on positive line bundles; Proceedings of the Bayreuth conference Complex algebraic varieties, April 2-6, 1990, ed. by K. Hulek, T. Peternell, M. Schneider, F. Schreyer, Lecture Notes in Math. n{$\deg$} 1507, Springer Verlag, 1992. \[9] Demailly J.-P.: Regularization of closed positive currents and Intersection Theory. J. Alg. Geom. 1, 361--409 (1992) \. Zbl 0777.32016 \[10] Demailly J.-P., Ein L., Lazarsfeld R.: A subadditivity property of multiplier ideals. Michigan Math. J. 48, 137--156 (2000) \. Zbl 1077.14516 \. DOI: 10.1307/mmj/ 1030132712 \[11] Demailly J.-P., P\u{a}un M.: Numerical characterization of the K\"ahler cone of a compact K\"ahler manifold; arXiv: math.AG/0105176; Annals of Math. 159, 1247--1274 (2004) \. Zbl 1064.32019 \[12] de Fernex T., K\"uronya A., Lazarsfeld R.: Higher cohomology of divisors on a projective variety. Math. Ann. 337, 443--455 (2007) \. Zbl 1127.14012 \[13] Fujita T.: Approximating Zariski decomposition of big line bundles. Kodai Math. J. 17, 1--3 (1994) \. Zbl 0814.14006 \. DOI: 10.2996/kmj/1138039894 \[14] Hirzebruch F.: Arithmetic genera and the theorem of Riemann-Roch for algebraic varieties. Proc. Nat. Acad. Sci. U.S.A. 40, 110--114 (1954) \. Zbl 0055.38803 \. DOI: 10.1073/pnas.40.2.110 \[15] Hirzebruch, F.: Neue topologische Methoden in der algebraischen Geometrie; Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 9. Springer Verlag, Berlin-G\"ottingen-Heidelberg, 1956. English translation: Topological methods in algebraic geometry; Springer Verlag, Berlin, 1966. \. Zbl 0070.16302 \[16] K\"uronya A.: Asymptotic cohomological functions on projective varieties. Amer. J. Math. 128, 1475--1519 (2006) \. Zbl 1114.14005 \. DOI: 10.1353/ajm.2006.0044 \[17] Laeng, L.: Estimations spectrales asymptotiques en g\'eom\'etrie hermitienne; Th\`ese de Doctorat de l'Universit\'e de Grenoble I, octobre 2002,\\ http://www-fourier.ujf-grenoble.fr/THESE/ps/laeng.ps.gz and\\ http://tel.archives-ouvertes.fr/tel-00002098/en/ \[18] Lazarsfeld R.: Positivity in Algebraic Geometry I, II~; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vols. 48--49. Springer Verlag, Berlin (2004) \[19] Riemann, B.: Theorie der Abel'schen Functionen; J. f\"ur Math. 54 (1857), also in Gesammelte mathematische Werke (1990), 120--144. \. ERAM 054.1427cj \[20] Roch G.: \"Uber die Anzahl der willkurlichen Constanten in algebraischen Functionen. J. f\"ur Math. 64, 372--376 (1865) \. ERAM 064.1685cj This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre; Kobayashi, Shoshichi; Narasimhan, Raghavan; Siu, Yum-Tong \Title Cartan and complex analytic geometry. (English) Zbl 1195.01062 \Publi Notices Am. Math. Soc. 57, No. 8, 952-960 (2010). \MSC 01A70 Biographies, obituaries, personalia, bibliographies\\ 32-03 Historical (several complex variables and analytic spaces) \Author Demailly, Jean-Pierre; Pali, Nefton \Title Degenerate complex Monge-Amp\`ere equations over compact K\"ahler manifolds. (English) Zbl 1191.53029 \Publi Int. J. Math. 21, No. 3, 357-405 (2010). The Calabi conjecture was solved by {\it S.-T. Yau} [Commun. Pure Appl. Math. 31, 339--411 (1978; Zbl 0362.53049)]. {\it A. Bedford} and {\it B. A. Taylor} [Invent. Math. 37, 1--44 (1976; Zbl 0315.31007)] initiated a new method for the study of degenerate complex Monge-Amp\'ere equations. In this paper, the authors prove existence and uniqueness of the solution of some very general type of degenerate complex Monge-Amp\'ere equations, and investigate their regularity. Also the existence and fine regularity properties of the solutions of complex Monge-Amp\'ere equations with respect to a given degenerate metric are proved. \Reviewer Constantin C\u alin (Ia\c si) \Cited in 18 Documents \MSC 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C55 Hermitian and K\"ahlerian manifolds (global differential geometry)\\ 32J15 Compact surfaces (analytic spaces) \Keywords complex Monge-Amp\`ere equations; K\"ahler-Einstein metrics; closed positive currents; plurisubharmonic functions; capacities; Orlicz spaces \Full_text \References \[1] DOI: 10.1007/978-1-4612-5734-9 \. DOI: 10.1007/978-1-4612-5734-9 \[2] DOI: 10.1007/BF01418826 \. DOI: 10.1007/BF01418826 \[3] DOI: 10.1512/iumj.2003.52.2346 \. Zbl 1054.32024 \. DOI: 10.1512/iumj.2003.52.2346 \[4] DOI: 10.1007/s00209-002-0483-x \. Zbl 1076.32036 \. DOI: 10.1007/s00209-002-0483-x \[5] DOI: 10.1090/S0002-9939-07-08858-2 \. Zbl 1116.32024 \. DOI: 10.1090/S0002-9939-07-08858-2 \[6] Demailly J.-P., Ann. Sci. Ecole Norm. Sup. (4) 15 pp 457-- \[7] Demailly J.-P., J. Alg. Geom. 1 pp 361-- \[8] DOI: 10.4007/annals.2004.159.1247 \. Zbl 1064.32019 \. DOI: 10.4007/annals.2004. 159.1247 \[9] DOI: 10.1007/BF02922247 \. Zbl 1087.32020 \. DOI: 10.1007/BF02922247 \[10] Gilbarg D., Elliptic Partial Differential Equations of Second Order (2001) \. Zbl 1042.35002 \[11] DOI: 10.2307/1970486 \. Zbl 0122.38603 \. DOI: 10.2307/1970486 \[12] H\"ormander L., An Introduction to Complex Analysis in Several Variables 7 (1990) \[13] Iwaniec T., Geometric Function Theory and Non-linear Analysis (2001) \. Zbl 1045.30011 \[14] DOI: 10.1007/978-3-642-66282-9 \. DOI: 10.1007/978-3-642-66282-9 \[15] DOI: 10.1007/BF01388524 \. Zbl 0593.14010 \. DOI: 10.1007/BF01388524 \[16] DOI: 10.1007/BF02392879 \. Zbl 0913.35043 \. DOI: 10.1007/BF02392879 \[17] DOI: 10.1512/iumj.2003.52.2220 \. Zbl 1039.32050 \. DOI: 10.1512/iumj.2003.52.2220 \[18] Moishezon B. G., Amer. Math. Soc. Transl. 63 pp 51-- \. Zbl 0186.26204 \. DOI: 10.1090/trans2/063/02 \[19] Pali N., Indiana Univ. Math. J. \[20] DOI: 10.1007/BF01459143 \. Zbl 0606.32018 \. DOI: 10.1007/BF01459143 \[21] Peternell Th., Bayreuth. Math. Schr. 54 pp 1-- \[22] Rao M. M., Pure and Applied Math 146, in: Theory of Orlicz Spaces (1991) \. Zbl 0724.46032 \[23] Siciak J., Extremal Plurisubharmonic Functions and Capacities in $\mathbb{C}$n (1982) \. Zbl 0579.32025 \[24] Skoda H., Bull. Soc. Math. France 100 pp 353-- \[25] DOI: 10.1007/BF01389077 \. Zbl 0599.53046 \. DOI: 10.1007/BF01389077 \[26] Tian G., Acta Math. Sinica (N.S.) 4 pp 250-- \[27] DOI: 10.1007/s11401-005-0533-x \. Zbl 1102.53047 \. DOI: 10.1007/s11401-005-0533-x \[28] DOI: 10.1007/BF02392630 \. Zbl 1036.53052 \. DOI: 10.1007/BF02392630 \[29] DOI: 10.1090/S0894-0347-06-00552-2 \. Zbl 1185.53078 \. DOI: 10.1090/S0894-0347-06-00552-2 \[30] DOI: 10.1007/BF01449219 \. Zbl 0631.53051 \. DOI: 10.1007/BF01449219 \[31] DOI: 10.1002/cpa.3160310304 \. Zbl 0369.53059 \. DOI: 10.1002/cpa.3160310304 \[32] Zhang Z., Int. Math. Res. Not. pp 18-- \[33] Dinh T.-C., Acta Math \[34] DOI: 10.1007/BF01446636 \. Zbl 0832.32010 \. DOI: 10.1007/BF01446636 \[35] DOI: 10.1215/S0012-7094-85-05210-X \. Zbl 0578.32023 \. DOI: 10.1215/S0012-7094-85-05210-X This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Estimates on Monge-Amp\`{e}re operators derived from a local algebra inequality. (English) Zbl 1209.32024 \Publi Passare, Mikael (ed.), Complex analysis and digital geometry. Proceedings from the Kiselmanfest, Uppsala, Sweden, May 2006 on the occasion of Christer Kiselman's retirement. Uppsala: Univ. Uppsala (ISBN 978-91-554-7672-4/pbk). 131-143 (2009). Author's abstract: The goal of this short note is to relate the integrability property of the exponential $e^{-2\varphi}$ of a plurisubharmonic function $\varphi$ with isolated or compactly supported singularities to a priori bounds for the Monge-Amp\`{e}re mass of $(dd^c\varphi)^n$. The inequality is valid locally or globally on an arbitrary open subset $\Omega$ in $\Bbb C^n$. We show that $\int_\Omega (dd\varphi)^n < n^n$ implies $\int_K e^{-2\varphi}<+\infty$ for every compact subset $K$ in $\Omega$, while functions of the form $\varphi(z)=n\log |z-z_0|$, $z_0\in\Omega$, appear as limit cases. The result is derived from an inequality of pure local algebra, which turns out a posteriori to be equivalent to it, proved by A. Corti in dimension $n=2$, and later extended by L. Ein, T. De Fernex and M. Musta\c t\v a to arbitrary dimensions. For the entire collection see [Zbl 1192.00078]. \Reviewer Daniel Barlet (Nancy) \Cited in 2 Reviews \Cited in 2 Documents \MSC 32W20 Complex Monge-Amp\`ere operators\\ 32U10 Plurisubharmonic exhaustion functions\\ 32S05 Local singularities (analytic spaces)\\ 14B05 Singularities (algebraic geometry)\\ 14C17 Intersection theory, etc. \Keywords Monge-Amp\`{e}re operator; local algebra; monomial ideal; Hilbert-Samuel multiplicity; log-canonical threshold; plurisubharmonic function; Ohsawa-Takegoshi $L^2$ extension theorem; approximation of singularities; birational rigidity \Author Demailly, Jean-Pierre (ed.); Hulek, Klaus (ed.); Mok, Ngaiming (ed.); Peternell, Thomas (ed.) \Title Complex analysis. Abstracts from the workshop held August 24--30, 2008. (Komplexe Analysis.) (English) Zbl 1177.14011 \Publi Oberwolfach Rep. 5, No. 3, 2165-2218 (2008). Introduction: The workshop {\it Komplexe Analysis}, organised by Jean-Pierre Demailly (Grenoble), Klaus Hulek (Hannover), Ngaiming Mok (Hong Kong) and Thomas Peternell (Bayreuth) was held August 24th--August 30, 2008. This meeting was well attended with 46 participants from Europe, US, and the Far East. The participants included several leaders in the field as well as many young (non-tenured) researchers. The aim of the meeting was to present recent important results in several complex variables and complex geometry with particular emphasis on topics linking different areas of the field, as well as to discuss new directions and open problems. Altogether there were nineteen talks of $60$ minutes each, a programme which left sufficient time for informal discussions and joint work on research projects. One of the topics at the center of the conference was the classification theory of higher dimensional varieties. Y.~Kawamata lectured on the connections between the minimal model programme and derived categories; A.~Corti discussed an approach to the finite generation of the canonical ring without minimal models, but still in connection with the seminal work which was presented by J.~McKernan in the last Complex Analysis meeting in Oberwolfach 2006, where the finite generation of the canonical ring of varieties of general type was announced. Extension theorems, non vanishing and positivity result for certain direct image sheaves play a role in the global classification of complex manifolds. This was largely discussed by M.~Paun and B.~Berndtsson. In their work analytic methods are central, whereas the talks by Kawamata and Corti were more of an algebraic nature. Also very much on the analytic side and connected to Berndtsson's talk, H.~Tsuji lectured on generalised K\"ahler-Einstein metrics. Families of projective manifolds over higher-dimensional base spaces were considered in the talk by S.~Kebekus. Direct images of coherent sheaves also play a central role in this context. About five years ago, Campana introduced new variations on the concept of ``orbifolds''; they were already the suject of talks in past sessions and have turned out to be of increasing interest -- in the present session, new results on the hyperbolicity of orbifolds were presented in the talk by E.~Rousseau. As to varieties with special geometry, K.~Oguiso spoke on non-algebraic hyperk\"ahler manifolds and, with a rather different flavour, F.~Catanese on complex and real threefolds fibered by rational curves, with a special emphasis on real algebraic geometry. J.~Chen discussed the influence of terminal singularities in three-dimensional geometry, a more algebraic topic. On the analytic side, A.~Teleman reported on recent progress in the classification of non-K\"ahler surfaces in the so called Kodaira class VII, using gauge-theoretical methods, and S.~K.~Yeung lectured on new results on fake projective planes. Group actions and envelopes of holomorphy were the topics of the talk by X.~Zhou. S.~Boucksom discussed equidistribution of Fekete points on complex manifolds, in relation with energy functionals for Monge-Amp\`ere operators. R.~Lazarsfeld presented a very interesting new approach to study properties of linear systems and line bundles via convex geometry. Overall, moduli spaces appeared to be a central theme in the workshop, and were discussed extensively in at least four talks: V.~Gritsenko considered moduli spaces of K3-surfaces; S.~Grushevsky spoke on intersection numbers of divisor on the moduli space of curves, and K.~Ludwig and G.~Farkas lectured on the moduli spaces of spin and Prym curves, their singularities, Kodaira dimension and enumerative geometry. \MSC 14-06 Proceedings of conferences (algebraic geometry)\\ 14Jxx Surfaces and higher-dimensional varieties\\ 32-06 Proceedings of conferences (several complex variables)\\ 32Qxx Complex manifolds 00B05 Collections of abstracts of lectures \Full_text \Author Demailly, Jean-Pierre; Hwang, Jun-Muk; Peternell, Thomas \Title Compact manifolds covered by a torus. (English) Zbl 1144.14035 \Publi J. Geom. Anal. 18, No. 2, 324-340 (2008). Let $X$ be a compact complex manifold that is the image of a complex torus by a surjective holomorphic map $A \to X$. The main theorem of this paper states that $X$ is a K\"ahler manifold and that, up to taking a finite \'etale cover, $X$ is a product of projective spaces and a torus. This very nice statement generalises similar results for projective manifolds by {\it O. Debarre} [C. R. Acad. Sci., Paris, S\'er. I 309, No. 2, 119--122 (1989; Zbl 0699.14050)] and {\it J.-M. Hwang} and {\it N. Mok} [Math. Z. 238, No. 1, 89--100 (2001; Zbl 1076.14021)]. Technically speaking, the proof falls into two independent parts: In the first part, the authors observe that the morphism $A \to X$ is equidimensional. Since a complex torus is K\"ahler, a difficult theorem of {\it J. Varouchas} [Math. Ann. 283, No. 1, 13--52 (1989; Zbl 0632.53059)] then implies that $X$ is also K\"ahler. Note that the authors give a rather short and self-contained proof of Varouchas' theorem in the appendix. For the second part, we observe that the main theorem implies a-posteriori that the tangent bundle of $X$ is nef. The strategy of the proof is now to show that many of the tools used in the study of manifolds with nef tangent bundle by {\it J.-P. Demailly, T. Peternell} and {\it M. Schneider} [J. Algebr. Geom. 3, No. 2, 295--345 (1994; Zbl 0827.14027)] are still available under the hypothesis that $X$ is covered by a torus. For example it is shown that the Albanese map of $X$ is a surjective submersion with connected fibres, and the fundamental group is almost abelian. Furthermore the anticanonical bundle is semi-ample and induces an equidimensional fibration. The main theorem is then established by comparing the Albanese morphism and the anticanonical fibration. \Reviewer Andreas H\"oring (Paris) \Cited in 1 Document \MSC 14J40 Algebraic $n$-folds ($n>4$)\\ 14C30 Transcendental methods, Hodge theory, Hodge conjecture\\ 32J25 Transcendental methods of algebraic geometry \Keywords complex torus; abelian variety; projective space; K\"ahler manifold; Albanese morphism; fundamental group; \'etale cover; nef divisor; nef tangent bundle; anticanonical bundle; numerically flat vector bundle \Full_text \References \[1] Barlet, D.: Espace analytique r\'eduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie. In: S\'eminaire F. Norguet: Fonctions de plusieurs variables complexes, 1974/75. Lecture Notes in Math., vol. 482, pp. 1--158. Springer, Berlin (1975) \[2] Birkenhake, C., Lange, H.: Complex Abelian Varieties, 2nd augmented edn. Grund\-lehren der Mathematischen Wissenschaften, vol. 302. Springer, Berlin (2004) \. Zbl 1056.14063 \[3] Boucksom, S., Demailly, J.-P., P\v{a}un, M., Peternell, Th.: The pseudo-effective cone of a compact K\"ahler manifold and varieties of negative Kodaira dimension. math.AG/ 0405285 (2004) \[4] Campana, F.: Cor\'eduction alg\'ebrique d'un espace analytique faiblement k\"ahl\'erien compact. Invent. Math. 63(2), 187--223 (1981) \. Zbl 0447.32009 \. DOI: 10.1007/ BF01393876 \[5] Campana, F., Peternell, Th.: Cycle spaces. In: Several complex variables, VII, Encyclopaedia Math. Sci., vol. 74, pp. 319--349. Springer, Berlin (1994) \. Zbl 0811.32020 \[6] Debarre, O.: Images lisses d'une vari\'et\'e ab\'elienne simple. C. R. Acad. Sci. Paris 309, 119--122 (1989) \. Zbl 0699.14050 \[7] Demailly, J.-P.: Estimations $L^2$ pour l'op\'erateur $\overline{\partial}$ d'un fibr\'e vectoriel holomorphe semi-positif au-dessus d'une vari\'et\'e k\"ahl\'erienne compl\`ete. Ann. Sci. \'Ecole Norm. Suppl. 4e S\'er. 15, 457--511 (1982) \[8] Demailly, J.-P.: Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1, 361--409 (1992) \. Zbl 0777.32016 \[9] Demailly, J.-P.: Monge-Amp\`ere operators, Lelong numbers and intersection theory. In: Ancona, V., Silva, A. (eds.) Complex Analysis and Geometry, Univ. Series in Math., pp. 115--193. Plenum, New York (1993) \. Zbl 0792.32006 \[10] Demailly, J.-P., P\v{a}un, M.: Numerical characterization of the K\"ahler cone of a compact K\"ahler manifold. Ann. Math. 159(3), 1247--1274 (2004) \. Zbl 1064.32019 \. DOI: 10.4007/annals.2004.159.1247 \[11] Demailly, J.-P., Peternell, Th., Schneider, M.: Compact complex manifolds with numerically effective tangent bundles. J. Algebraic Geom. 3, 295--345 (1994) \. Zbl 0827.14027 \[12] Hwang, J.M., Mok, N.: Projective manifolds dominated by abelian varieties. Math. Z. 238, 89--100 (2001) \. Zbl 1076.14021 \. DOI: 10.1007/PL00004902 \[13] Kawamata, Y.: Pluricanonical systems on minimal algebraic varieties. Invent. Math. 79, 567--588 (1985) \. Zbl 0593.14010 \. DOI: 10.1007/BF01388524 \[14] Moishezon, B.G.: On n-dimensional compact varieties with $n$ algebraically indepedent meromorphic functions. Am. Math. Soc. Transl. II. Ser. 63, 51--177 (1967) \. Zbl 0186.26204 \[15] Nakayama, N.: The lower semi-continuity of the plurigenera of complex varieties. Adv. Stud. Pure Math. 10, 551--590 (1987) \. Zbl 0649.14003 \[16] Okonek, Ch., Schneider, M., Spindler, H.: Vector Bundles on Complex Projective Spaces, Progress in Mathematics, vol. 3. Birkh\"auser, Boston (1980) \. Zbl 0438.32016 \[17] P\v{a}un, M.: Sur l'effectivit\'e num\'erique des images inverses de fibr\'es en droites. Math. Ann. 310, 411--421 (1998) \. Zbl 1023.32014 \. DOI: 10.1007/s002080050154 \[18] Richberg, R.: Stetige Streng pseudokonvexe Funktionen. Math. Ann. 175, 257--286 (1968) \. Zbl 0153.15401 \. DOI: 10.1007/BF02063212 \[19] Ueno, K.: Classification theory of algebraic varieties and compact complex spaces. Lecture Notes in Math., vol. 439. Springer, Berlin (1975) \. Zbl 0299.14007 \[20] Varouchas, J.: Stabilit\'e de la classe des vari\'et\'es k\"ahl\'eriennes par certains morphismes propres. Invent. Math. 77(1), 117--127 (1984) \. Zbl 0529.53049 \. DOI: 10.1007/BF01389138 \[21] Varouchas, J.: K\"ahler spaces and proper open morphisms. Math. Ann. 283, 13--52 (1989) \. Zbl 0632.53059 \. DOI: 10.1007/BF01457500 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre; Kosarew, Siegmund; Malgrange, Bernard \Title Adrien Douady and Banach analytic spaces. (Adrien Douady et les espaces analytiques banachiques.) (French) Zbl 1168.01321 \Publi Gaz. Math., Soc. Math. Fr. 113, 35-38 (2007). \MSC 01A70 Biographies, obituaries, personalia, bibliographies 01A60 Mathematics in the 20th century 46-03 Historical (functional analysis) \Author Demailly, Jean-Pierre \Title K\"ahler manifolds and transcendental techniques in algebraic geometry. (English) Zbl 1141.14007 \Publi Sanz-Sol\'e, Marta (ed.) et al., Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume I: Plenary lectures and ceremonies. Z\"urich: European Mathematical Society (EMS) (ISBN 978-3-03719-022-7/hbk). 153-186 (2007). This paper surveys some of the main recent advances in the study of the geometry of projective or compact K\" ahler manifolds obtained by using local and global complex analytic methods. After a short introduction, section 2 presents well-known definitions and results, necessary for the paper. Section 3 describes the main results obtained by {\it J.-P. Demailly} and {\it M. P\u aun} [Ann. Math. 159, 1247--1274 (2004; Zbl 1064.32019)], namely:\par {\bf Theorem 3.1.} Let $X$ be a compact K\" ahler manifold. Let $\mathcal{P}$ be the set of real $(1, 1)$ cohomology classes $\{\alpha\}$ which are numerically positive on analytic cycles, i.e.\ such that $\int_Y \alpha^p > 0$, for every irreducible analytic set $Y$ in $X$, $p = \dim Y$. Then the K\" ahler cone $\mathcal{K}$ of $X$ is one of the connected components of $\mathcal{P}$.\par {\bf Corollary 3.2.} If $X$ is projective algebraic, then $\mathcal{K} = \mathcal{P}$. These results (which are new even in the projective case) can be seen as a generalization of the well-known Nakai--Moishezon criterion. Sketches of the proofs are given. The section 3 ends with a consequence about the dual of the cone $\mathcal{K}$ in $H_{\mathbb{R}}^{n - 1, n - 1}(X)$. Section 4 is devoted to deformations of compact K\" ahler manifolds. Kodaira showed in the 60s that every K\" ahler surface $X$ is a limit by deformations of algebraic surfaces. The long-standing question whether a similar property holds in higher dimensions was shown in negative by C. Voisin: {\bf Theorem 4.1.} (C. Voisin)\\ (i) In any dimension $\geq 4$, there exist compact K\" ahler manifolds which do not have the homotopy type (or even the homology ring) of a complex projective manifold.\\ (ii) In any dimension $\geq 8$, there exist compact K\" ahler manifolds $X$ such that no compact bimeromorphic model $X'$ of $X$ has the homotopy type of a complex projective manifold. Then, the behaviour of the K\" ahler cone of $X_t$ as $t$ approaches the ``bad strata'' ($X_t$ in a deformation of K\" ahler manifolds) is given (this is a result of the above paper by Demailly - P\u aun). Section 5 presents results on positive cones in $H^{n - 1, n - 1}(X)$ and Serre duality. We shall give only two results: Theorem 5.3. (Demailly - P\u aun) If $X$ is K\" ahler, then the cones $\bar{\mathcal{K}} \subset H^{1,1}(X, \mathbb{R})$ and $\mathcal{N} \subset H^{n - 1, n - 1}_{\mathbb{R}}(X)$ are dual by Poincar\' e duality, and $\mathcal{N}$ is the closed convex cone generated by classes $[Y] \wedge \omega^{p - 1}$, where $Y \subset X$ ranges over $p$-dimensional analytic subsets, $p = 1,2, \ldots, n$, and $\omega$ ranges over K\" ahler forms. The next result is from the paper of {\it S. Boucksom, J.-P. Demailly, M. P\u aun} and {\it Th. Peternell} [The pseudo-effective cone of a compact K\" ahler manifold and varieties of negative Kodaira dimension, \url{arXiv:math/0405285}]: Theorem 5.14. If $X$ is projective, then a class $\alpha \in NS_{\mathbb{R}}(X)$ is pseudo-effective if (and only if) it is in the dual cone of the cone $SME(X)$ of strongly movable curves. The sections ends with some applications and conjectures. The final section 6 presents new results around the invariance of plurigenera. We give only one result which is a special case of a result of P\u aun: Corollary 6.3. (Siu) For any projective family $t \mapsto X_t$ of algebraic varieties, the plurigenera $p_m(X_t) = h^\circ (X_t, m K_{X_t})$ do not depend on $t$. For the entire collection see [Zbl 1111.00009]. \Reviewer Vasile Br\^\i nz\u anescu (Bucure\c{s}ti) \Cited in 4 Documents \MSC 14C30 Transcendental methods, Hodge theory, Hodge conjecture\\ 32C30 Integration on analytic sets and spaces, currents\\ 32L20 Vanishing theorems (analytic spaces) \Keywords projective variety; K\" ahler manifold; Hodge theory; positive current \Author Demailly, Jean-Pierre \Title Towards a revaluation of mathematics and science teaching: GRIP initiatives and SLECC network classes. (Vers une r\'e\'evaluation de l'enseignement des math\'ematiques et des sciences: initiatives du GRIP et r\'eseau de classes SLECC.) (French) Zbl 1343.00016 \Publi Gaz. Math., Soc. Math. Fr. 110, 61-64 (2006). \MSC 00A35 Methodology of mathematics, didactics 97B40 Higher education 97B70 Syllabuses, educational standards \Author Demailly, Jean-Pierre (ed.); Hulek, Klaus (ed.); Peternell, Thomas (ed.) \Title Report 40/2006: Komplexe Analysis (August 27th -- September 2nd, 2006). (English) Zbl 1109.14301 \Publi Oberwolfach Rep. 3, No. 3, 2399-2446 (2006). Abstract: The main aim of this workshop was to discuss recent developments in several complex variables and complex geometry. The topics included: classification of higher dimensional varieties, mirror symmetry, hyperbolicity, K\"ahler geometry and classical geometric questions. Contributions: {\parindent=6mm \item{--}Michel Brion, Log homogeneous varieties (p. 2403) \item{--}Bernd Siebert (joint with Mark Gross), Tropical games and Mirror Symmetry (p. 2405) \item{--}Fedor Bogomolov (joint with Bruno de Oliveira), Symmetric tensors and the geometry of secant varieties in a projective space (p. 2408) \item{--}Philippe Eyssidieux (joint with Vincent Guedj, Ahmed Zeriahi), Singular K\"ahler-Einstein metrics (p. 2409) \item{--}Pelham M.H. Wilson, The geometry of K\"ahler moduli (p. 2411) \item{--}James McKernan (joint with Caucher Birkar, Paolo Cascini, Christopher Hacon), Existence of minimal models for varieties of log general type (p. 2414) \item{--}Jun-Muk Hwang, Rigid targets of surjective holomorphic maps (p. 2419) \item{--}Andreas Gathmann (joint with Hannah Markwig), Tropical enumerative geometry (p. 2421) \item{--}Aleksandr V. Pukhlikov, Birationally rigid Fano fiber spaces (p. 2422) \item{--}Fabrizio Catanese (joint with S\"onke Rollenske), The slope of Kodaira fibrations (p. 2424) \item{--}S\'andor Kov\'acs (joint with Stefan Kebekus), Viehweg's conjecture for two-dimen\-sional bases (p. 2427) \item{--}Erwan Rousseau, Recent developments on hyperbolicity of complex algebraic varieties (p. 2430) \item{--}Priska Jahnke (joint with C. Casagrande, I. Radloff), On the Picard number of almost Fano threefolds (p. 2433) \item{--}Andrew J. Sommese (joint with Charles W. Wampler), Using fiber products to compute exceptional sets (p. 2435) \item{--}Takeo Ohsawa, Levi flat hypersurfaces in complex manifolds (p. 2436) \item{--}Mihai P\u aun, Siu's invariance of plurigenera: a one-tower proof (p. 2437) \item{--}Duco van Straten (joint with K. Jung), Arctic computation of monodromy (p. 2439) \item{--}Michael McQuillan, Residues hyperbolicity and abundance (p. 2442) \par} \MSC 14-XX Algebraic geometry\\ 32-XX Several complex variables and analytic spaces 00B05 Collections of abstracts of lectures \Full_text \Author Demailly, Jean-Pierre; Eckl, Thomas; Peternell, Thomas \Title Line bundles on complex tori and a conjecture of Kodaira. (English) Zbl 1078.32014 \Publi Comment. Math. Helv. 80, No. 2, 229-242 (2005). In 1963 Kodaira proved that every smooth compact K\"ahler surface is almost algebraic in the sense that it can be realized as a deformation of a projective surface. Until recently it was an open problem whether it is true in general that a compact K\"ahler manifold is almost algebraic. An affirmative answer would have implied that every compact K\"ahler manifold has the homotopy type of a projective algebraic manifold and is projective if it is rigid. But in 2004 {\it C. Voisin} [Invent. Math. 157, No. 2, 329--343 (2004; Zbl 1065.32010)] constructed for every dimension greater than three compact K\"ahler manifolds with homotopy type different from the homotopy type of a projective manifold. Her examples are built from compact complex tori by blowing up processes. In the paper under review the authors are equally interested in giving an affirmative answer to the above question for a subclass of compact K\"ahler manifolds and in finding new counterexamples. They discuss different strategies for $\Bbb P(V)$-bundles on complex tori. These considerations are based on the fact that the structure of $\Bbb P(V)$-bundles or $\Bbb P_r$-bundles over a compact complex manifold survives under deformation (Theorem 8). The authors consider the following situation: Let $A$ be a three-dimensional compact complex torus and $L_1, L_2, L_3$ holomorphic line bundles on $A$ representing three linear independent elements in the N\'{e}ron-Severi group NS$(A)$. The manifold $Y=\Bbb P({\cal O}_A\oplus L_1)\times_A\Bbb P({\cal O}_A\oplus L_2)\times_A\Bbb P({\cal O}_A\oplus L_3)$ is a holomorphic $ \Bbb P^3_1$-bundle over $A$ with a natural holomorphic section $Z$ given by the direct summand ${\cal O}_A$ in every factor. The main result of the paper (Theorem 4) asserts that $Y$ is algebraically approximable by projective Albanese bundles $Y_n\rightarrow A_n$ with $Y_n=\Bbb P({\cal O}_{A_n}\oplus L_1)\times_{A_n}\Bbb P({\cal O}_{A_n}\oplus L_2)\times_{A_n}\Bbb P({\cal O}_{A_n}\oplus L_3)$ and $\lim_{n\rightarrow\infty}A_n=A$ in the sense of deformation theory. The proof uses an explicit description of NS$(A)$ in terms of skew-symmetric integer $6\times 6$ matrices and calculations with support of the computer algebra program Macauley 2 [see {\it D. Eisenbud} et al., Algorithms and Computation in Mathematics. 8. (Berlin: Springer) (2002; Zbl 0973.00017)]. Blowing up in the bundle $Y$ every fiber $F$ in the point $F\cap Z$ gives a compact K\"ahler manifold $X$, a holomorphic fiber bundle over $A$ with projective rational fiber. Under the additional assumption that not all of the line bundles $L_1, L_2, L_3$ remain holomorphic under small deformations of $A$, the manifold $X$ is rigid (Proposition 3) and could a priori be a counterexample. But the assumption on the $L_i$ forces $A_n=A$ in Theorem 4, hence $X$ is already projective. The authors explain their ideas how modifications of their construction and more general settings could eventually lead to new counter-examples. \Reviewer Eberhard Oeljeklaus (Bremen) \Cited in 3 Documents \MSC 32J27 Compact K\"ahler manifolds: generalizations, classification\\ 32G05 Deformations of complex structures\\ 32Q15 K\"ahler manifolds \Keywords compact K\"ahler manifold; almost algebraic; algebraically approximable; Kodaira conjecture Software: Macaulay2 \Full_text \Author Demailly, Jean-Pierre (ed.); Hulek, Klaus (ed.); Peternell, Thomas (ed.) \Title Workshop: Complex analysis. (Komplexe Analysis.) (English) Zbl 1078.30501 \Publi Oberwolfach Rep. 1, No. 3, Report 42, 2171-2215 (2004). Contributions:\par-- Daniel Barlet, Application of complex analysis to oscillating integrals p.2171\par-- Ingrid C.Bauer (joint with F.Catanese and F.Grunewald), Beauville surfaces without real structures and group theory p.2171\par-- Michel Brion, Extension of equivariant vector bundles p.2174\par-- Ciprian S.Borcea, Polygon spaces, tangents to quadrics and special Lagrangians p.2177\par-- Bertrand Deroin\par-- Immersed Levi-flat hypersurfaces into non negatively curved complex surfaces p.2179\par-- Wolfgang Ebeling (joint with Sabir M.Gusein-Zade and Jos\'e Seade), Indices of 1-forms on singular varieties p.2182\par-- Akira Fujiki (joint with Massimiliano Pontecorvo), Anti-self-dual hermitian metrics on Inoue surfaces p.2184\par-- Samuel Grushevsky, Addition formulas for theta functions, and linear systems on abelian varieties p.2186\par-- Peter Heinzner (joint with Gerald Schwarz), Cartan decomposition of the moment map p.2188\par-- Jun-Muk Hwang, Bound on the number of curves of a given degree through a general point of a projective variety p.2191\par-- Priska Jahnke (joint with Thomas Peternell and Ivo Radloff), Threefolds with big and nef anticanonical bundles p.2193\par-- Shigeyuki Kondo, A complex ball uniformization for the moduli spaces of del Pezzo surfaces via periods of K3 surfaces p.2195\par-- Michael L\"onne, Braid monodromy of hypersurface singularities p.2197\par-- Laurent Meersseman (joint with Alberto Verjovsky), A foliation of S5 by complex surfaces and its moduli space p.2199\par-- Keiji Oguiso, Automorphisms of hyperk\"ahler manifolds p.2201\par-- Edoardo Sernesi (joint with A.Bruno), The non-Petri locus for pencils p.2203\par-- Andrew J.Sommese (joint with Jan Verschelde and Charles W.Wampler), Numerically decomposing the intersection of algebraic varieties p.2203\par-- Andrei Teleman (joint with Matei Toma), Complex geometric applications of Gauge Theory p.2204 \MSC 30-06 Proceedings of conferences (functions of one complex variable)\\ 32-06 Proceedings of conferences (several complex variables) 00B05 Collections of abstracts of lectures \Author Demailly, Jean-Pierre; Paun, Mihai \Title Numerical characterization of the K\"ahler cone of a compact K\"ahler manifold. (English) Zbl 1064.32019 \Publi Ann. Math. (2) 159, No. 3, 1247-1274 (2004). The K\"ahler cone of a compact K\"ahler manifold is the set of cohomology classes of smooth positive definite clossed $(1,1)$-forms. The authors show that this cone depends only on the intersection product of the cohomology ring, the Hodge structure and the homology classes of analytic cycles: if $X$ is a compact K\"ahler manifold, the K\"ahler cone $\cal{K}$ of $X$ is one of the connected components of the set $\cal{P}$ of real $(1,1)$-cohomology classes $\{\alpha\}$ which are numerically positive on the analytic cycles, i.e.\ such that $\int_Y\alpha^p>0$ for every irreducible analytic set in $X, \ p=\dim Y$. This result can be considered as a generalization of the Nakai-Moishezon criterion, which provide a necessary and sufficient criterion for a line bundle to be ample. If $X$ is projective then $\cal{K}=\cal{P}$. If $X$ is a compact K\"ahler manifold, the $(1,1)$-cohomology class $\alpha $ is nef (numerically effective free) if and only if there exists a K\"ahler metric $\omega$ on $X$ such that $\int _Y\alpha^k\wedge \omega^{p-k}\geq 0$ for all irreducible analytic sets $Y$ and all $k=1,2,\dots, p=\dim Y$. A $(1,1)$-cohomology class $\{\alpha\}$ on $X$ is nef if and only if for every irreducible analytic set $Y$ in $X$, $p=\dim Y$, and for every K\"ahler metric $\omega$ on $X$, one has $\int_Y\alpha\wedge \omega^{p-1}\geq 0$. First, the authors obtain a sufficient condition for a nef class to contain a K\"ahler current. Then the main result is obtained by an induction on the dimension. The obtained result has an important application to the deformation theory of compact K\"ahler manifolds: consider ${\cal{X}}\to S$ a deformation of compact K\"ahler manifolds over an irreducible base $S$. There exists a countable union $S^\prime =\bigcup S_\nu$ of analytic subsets $S_\nu \subset S$, such that the K\"ahler cones ${\cal{K}}_t \subset H^{1,1}(X_t,\Bbb{C})$ are invariant over $S\ssm S^\prime$ under parallel transport with respect to the $(1,1)$-projection $\nabla^{1,1}$ of the Gauss-Manin connection. \Reviewer Vasile Oproiu (Ia\c si) \Cited in 8 Reviews \Cited in 68 Documents \MSC 32Q15 K\"ahler manifolds\\ 32Q25 Calabi-Yau theory 53C55 Hermitian and K\"ahlerian manifolds (global differential geometry)\\ 32J27 Compact K\"ahler manifolds: generalizations, classification \Keywords K\"ahler manifolds; K\"ahler cone; nef cohomology classes; K\"ahler currents \Author Demailly, Jean-Pierre \Title On the geometry of positive cones of projective and K\"ahler varieties. (English) Zbl 1071.14013 \Publi Collino, Alberto (ed.) et al., The Fano conference. Papers of the conference organized to commemorate the 50th anniversary of the death of Gino Fano (1871--1952), Torino, Italy, September 29--October 5, 2002. Torino: Universit\`a di Torino, Dipartimento di Matematica. 395-422 (2004). {\it Summary$\,$}: The goal of these notes is to give a short introduction to several works by S\'ebastien Boucksom, Mihai Paun, Thomas Peternell and myself on the geometry of positive cones of projective or K\"ahler manifolds. Mori theory has shown that the structure of projective algebraic manifolds is -- up to a large extent -- governed by the geometry of its cones of divisors or curves. In the case of divisors, two cones are of primary importance: the cone of ample divisors and the cone of effective divisors (and the closure of these cones as well). We introduce here the analogous transcendental cones for arbitrary compact K\"ahler manifolds, and show that these cones depend only on analytic cycles and on the Hodge structure of the base manifold. Also, we obtain new very precise duality statements connecting the cones of curves and divisors via Serre duality. As a consequence, we are able to prove one of the basic conjectures in the classification of projective algebraic varieties -- a subject which Gino Fano contributed to in many ways: a projective algebraic manifold $X$ is uniruled (i.e.\ covered by rational curves) if and only if its canonical class $c_1(K_X)$ does not lie in the closure of the cone spanned by effective divisors. For the entire collection see [Zbl 1051.00013]. \Cited in 1 Document \MSC 14C30 Transcendental methods, Hodge theory, Hodge conjecture\\ 14C20 Divisors, linear systems, invertible sheaves\\ 32J27 Compact K\"ahler manifolds: generalizations, classification \Keywords transcendental cones; analytic cycles; Hodge structure; duality \Author Demailly, Jean-Pierre; Peternell, Thomas \Title A Kawamata-Viehweg vanishing theorem on compact K\"ahler manifolds. (English) Zbl 1077.32504 \Publi J. Differ. Geom. 63, No. 2, 231-277 (2003). This paper appeared earlier under the same title in the proceedings of a conference. See {\it J.-P. Demailly} and {\it T. Peternell}, Surv. Differ. Geom. 8, 139--169 (2003; Zbl 1053.32011). \Reviewer Imre Patyi (Atlanta) \Cited in 1 Review \Cited in 8 Documents \MSC 32L20 Vanishing theorems (analytic spaces)\\ 32J27 Compact K\"ahler manifolds: generalizations, classification \Keywords Kawamata-Viehweg vanishing theorem; compact K\"ahler spaces; abundance for threefolds \Author Demailly, Jean-Pierre; Peternell, Thomas \Title A Kawamata-Viehweg vanishing theorem on compact K\"ahler manifolds. (English) Zbl 1053.32011 \Publi Yau, S.-T. (ed.), Surveys in differential geometry. Lectures on geometry and topology held in honor of Calabi, Lawson, Siu, and Uhlenbeck at Harvard University, Cambridge, MA, USA, May 3--5, 2002. Somerville, MA: International Press (ISBN 1-57146-114-0/hbk). Surv. Differ. Geom. 8, 139-169 (2003). This article by well-known experts in complex geometry adds results to the ongoing effort to extend some important parts of Mori's theory of complex projective varieties to the case of compact K\"ahler manifolds and spaces. It appeared in the proceedings of a prestigious conference, and as a journal paper in [J. Differ. Geom. 63, No. 2, 231--277 (2003; Zbl 1077.32504)] under the same title. The main results of this long and involved paper are as follows. In claim 0.1 the authors obtain a Kawamata-Viehweg vanishing theorem for the cohomology group $H^q(X,K_X+L)=0$, $q\ge n-1$, where $X$ is a normal compact K\"ahler space of dimension $n$, and $L\to X$ is a nef line bundle with $L^2\not=0$. The proof of claim~0.1 is via demonstrating that the natural coefficient map induces zero in cohomology $H^{n-1}(X,K_X\otimes L\otimes{\Cal J})\to H^{n-1}(X,K_X\otimes L)$, where ${\Cal J}$ is a suitable multiplier ideal sheaf corresponding to a singular metric $h$ on $L$. The latter vanishing is reduced to the study of a divisor $D$ associated to $h$ by Siu decomposition, and consists in showing that $H^0(D,(-L+D)\vert D)=0$, done by working with Hodge index inequalities. Then claim~0.1 is applied to abundance for threefolds given in claim~0.3: If $X$ is a ${\Bbb Q}$-Gorenstein K\"ahler threefold with only terminal singularities and $K_X$ nef, then $\kappa(X)\ge0$ for the Kodaira dimension. The paper is informative and pleasant to read. For the entire collection see [Zbl 1034.53003]. \Reviewer Imre Patyi (Atlanta) \Cited in 1 Review \Cited in 1 Document \MSC 32L20 Vanishing theorems (analytic spaces)\\ 32J27 Compact K\"ahler manifolds: generalizations, classification \Keywords Kawamata-Viehweg vanishing theorem; compact K\"ahler spaces; abundance for threefolds \Author Demailly, Jean-Pierre \Title On the Frobenius integrability of certain holomorphic $p$-forms. (English) Zbl 1011.32019 \Publi Bauer, Ingrid (ed.) et al., Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday. Berlin: Springer. 93-98 (2002). {\it Summary$\,$}: The goal of this note is to exhibit the integrability properties (in the sense of the Frobenius theorem) of holomorphic $p$-forms with values in certain line bundles with semi-negative curvature on a compact K\"ahler manifold. There are in fact very strong restrictions, both on the holomorphic form and on the curvature of the semi-negative line bundle. In particular, these observations provide interesting information on the structure of projective manifolds which admit a contact structure: either they are Fano manifolds or, thanks to results of Kebekus-Peternell-Sommese-Wisniewski, they are biholomorphic to the projectivization of the cotangent bundle of another suitable projective manifold. For the entire collection see [Zbl 0989.00069]. \Cited in 1 Review \Cited in 7 Documents \MSC 32Q15 K\"ahler manifolds\\ 32J25 Transcendental methods of algebraic geometry \Keywords Frobenius integrability; holomorphic $p$-forms \Author Bertin, Jos\'e; Demailly, Jean-Pierre; Illusie, Luc; Peters, Chris \Title Introduction to Hodge theory. Transl. from the French by James Lewis and Chris Peters. (English) Zbl 0996.14003 \Publi SMF/AMS Texts and Monographs. 8. Providence, RI: American Mathematical Society (AMS). ix, 232 p. (2002). This book is the English translation of the French original published in 1996 under the title ``Introduction \`a la th\'eorie de Hodge'' (1996; Zbl 0849.14002). Back then it appeared as volume 3 in the new series ``Panoramas et Synth\`eses'' edited by Soci\'et\'e Math\'ematique de France. Grown out of a set of lectures which the authors had delivered at a conference on the present state of Hodge theory (Grenoble 1994), the aim of the text was to develop a number of fundamental concepts and results of both classical and modern Hodge theory, primarily addressed to graduate students and non-expert researchers in the field. In the English translation of this profound introduction to classical and modern Hodge theory, which discusses the subject in great depth and leads the reader to the forefront of contemporary research in many areas related to Hodge theory, the text has been left entirely unchanged.\par Now as five years before, this book provides a masterly guide through Hodge theory and its various applications. It still maintains its unique up-to-date character, within the textbook and survey literature on the subject, as well as its significant role as an indispensible source for active researchers and teachers in the field, together with the additional advantage that its translation into English makes it now accessible to the entire mathematical and physical community worldwide. Without any doubt, this is exactly what both those communities and this excellent book on Hodge theory needed and deserved. \Reviewer Werner Kleinert (Berlin) \Cited in 5 Documents \MSC 14C30 Transcendental methods, Hodge theory, Hodge conjecture\\ 14-02 Research monographs (algebraic geometry)\\ 14F17 Vanishing theorems\\ 14D07 Variation of Hodge structures 81T30 String and superstring theories 58A14 Hodge theory (global analysis)\\ 14D05 Structure of families\\ 14J32 Calabi-Yau manifolds \Keywords characteristic $p$; De Rham cohomology algebra; Hodge degeneration; variation of Hodge structures; Gauss-Manin connexion; period domains; Picard-Lefschetz theory; Calabi-Yau manifolds; Higgs bundles; Hodge theory; vanishing theorems; mirror symmetry \Author Campana, Fr\'ed\'eric; Demailly, Jean-Pierre \Title $L^2$ -cohomology on the coverings of a compact complex manifold. (French)\\ Zbl 1066.32012 \Publi Ark. Mat. 39, No. 2, 263-282 (2001). The aim of this paper is to define a natural $L^2$-cohomology on any unramified covering of a complex analytic space $X$, with values in the lifting of any coherent analytic sheaf on $X$. This $L^2$ cohomology has been constructed independently by {\it P. Eyssidieux} [Math. Ann. 317, 527--566 (2000; Zbl 0964.32008)]. It is seen that the usual properties of sheaf cohomology such as cohomology exact sequences or spectral sequences hold in this $L^2$-cohomology on $X$. If $X$ is projective and non-singular there are $L^2$ vanishing theorems analogous to those of Kodaira-Serre and Kawamata-Viehweg. When $X$ is compact it is possible to define the $\Gamma$-dimension for Galois coverings. This $\Gamma$-dimension turns out to be finite in this case. An extension of Atiyah's index theorem is given in this context. \Reviewer A. Diaz-Cano (Madrid) \Cited in 2 Documents \MSC 32C35 Analytic sheaves and cohomology groups\\ 32C99 General theory of analytic spaces\\ 32T99 Pseudoconvex domains 58J20 Index theory and related fixed-point theorems (PDE on manifolds) \References \[1] [AG] Andreotti, A. etGrauert, H., Th\'eor\`emes de finitude pour la cohomologie des espaces complexes,Bull. Soc. Math. France 90 (1962), 193--259. \[2] [AV] Andreotti, A. etVesentini, E., Carleman estimates for the Laplace-Beltrami equation in complex manifolds,Inst. Hautes \'Etudes Sci. Publ. Math 25 (1965), 81--130. \. Zbl 0138.06604 \. DOI: 10.1007/BF02684398 \[3] [A] Atiyah, M., Elliptic operators, discrete groups and von Neumann algebras, dans Elliptic Operators, Discrete Groups and von Neumann Algebras. Colloque ''Analyse et Topologie'' en l'honneur de Henri Cartan (Orsay, 1974), Ast\'erisque32--33, p. 43--72, Soc. Math. France, Paris, 1976. \[4] [C] Campana, F., Fundamental group and positivity of cotangent bundles of compact K\"ahler manifolds,J. Algebraic Geom. 4 (1995), 487--502. \. Zbl 0845.32027 \[5] [D] Demailly, J.-P.., Estimations $L^2$ pour l'op\'erateur $\bar \partial $ d'un fibr\'e vectoriel hermitien semi-positif,Ann. Sci. \'Ecole Norm. Sup. 15 (1982), 457--511. \[6] [E1] Eyssidieux, P., Th\'eorie de l'adjonction $L^2$ sur le rev\^etement universel, Preprint, 1997. \[7] [E2] Eyssidieux, P., Syst\`emes lin\'eaires adjoints $L^2$.Ann. Inst. Fourier (Grenoble) 49 (1999),vi, ix-x, 141--176. \[8] [E3] Eyssidieux, P., Invariants de von Neumann des faisceaux analytiques coh\'erents, Math. Ann. 317 (2000), 527--566. \. Zbl 0964.32008 \. DOI: 10.1007/PL00004413 \[9] [G] Gromov, M., K\"ahler hyperbolicity and $L^2$-Hodge theory,J. Differential Geom 33 (1991), 263--291. \. Zbl 0719.53042 \[10] [H]H\"ormander, L.,An Introduction to Complex Analysis in Several Variables. 3rd ed., North-Holland Math. Library7, Elsevier, Amsterdam, 1990. \. Zbl 0685.32001 \[11] [JZ] Jost, J. etZuo, K., Vanishing theorems for $L^2$-cohomology on infinite coverings of compact K\"ahler manifolds and applications in algebraic geometry,Comm. Anal. Geom. 8 (2000), 1--30. \[12] [K] Koll\'ar, J.,Shafarevitch Maps and Automorphic Forms, Princeton Univ. Press, Princeton, N. J., 1995. \[13] [NR] Napier, T. etRamachandran, M.., The $L^2 \bar \partial $ -method, weak Lefschetz theorems, and the topology of K\"ahler manifolds,J. Amer. Math. Soc. 11 (1998), 375--396. \. Zbl 0897.32007 \. DOI: 10.1090/S0894-0347-98-00257-4 \[14] [O] Ohsawa, T., A reduction theorem for cohomology groups of very stronglyq-convex K\"ahler manifolds,Invent. Math. 63 (1981), 335--354; Invent. Math.66 (1982) 391--393. \. Zbl 0457.32007 \. DOI: 10.1007/BF01393882 \[15] [P] Pansu, P., Introduction to $L^2$ Betti numbers, dansRiemannian Geometry (Waterloo, Ont., 1993) (Lovri\'c, M., Maung, M.-O. et Wang, M. Y.-K., \'eds) Fields Inst. Monogr.4, p. 53--86, Amer. Math. Soc., Providence, R. I., 1996. \[16] [W] Weil, A.,Introduction \`a l'\'etude des vari\'et\'es k\"ahl\'eriennes, Hermann, Paris, 1958. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean Pierre \Title Refoundation of mathematics in France. (Italian) Zbl 1106.01306 \Publi Lett. Mat. Pristem 42, 10-14 (2001). The young and (even more) promising author of the present study, Jean-Piere Demailly, approaches the problem of mathematics refoundation from very (highly) French positions, by proudly asserting that the ``nouvelles maths'' have been initiated in France and nowhere else. The role played in France by the ``commission for the long-sum reflexion on the teaching of mathematics'' is emphasized, and the name of its president -- Jean-Pierre Kahane -- is mentioned. Actually, the study is Demailly's contribution to the round table organized in Paris on ``Mathematics and the Teaching of Sciences''. It is quite a revolutionary text. Extremely ``refreshing'' for the reader is the idea of having illustrated the text with coloured, challenging reproductions of Picasso. \Reviewer Cristina Irimia (Ia\c si) \MSC 01A60 Mathematics in the 20th century \Keywords mathematics; France; J. P. Kahane; teaching of science \Author Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael \Title Pseudo-effective line bundles on compact K\"ahler manifolds. (English) Zbl 1111.32302 \Publi Int. J. Math. 12, No. 6, 689-741 (2001). {\it Summary$\,$}: The goal of this work is to pursue the study of pseudo-effective line bundles and vector bundles. Our first result is a generalization of the Hard-Lefschetz theorem for cohomology with values in a pseudo-effective line bundle. The Lefschetz map is shown to be surjective when (and, in general, only when) the pseudo-effective line bundle is twisted by its multiplier ideal sheaf. This result has several geometric applications, e.g., to the study of compact K\"ahler manifolds with pseudo-effective canonical or anti-canonical line bundles. Another concern is to understand pseudo-effectivity in more algebraic terms. In this direction, we introduce the concept of an ``almost'' nef line bundle, and mean by this that the degree of the bundle is nonnegative on sufficiently generic curves. It can be shown that pseudo-effective line bundles are almost nef, and our hope is that the converse also holds. This can be checked in some cases, e.g., for the canonical bundle of a projective 3-fold. From this, we derive some geometric properties of the Albanese map of compact K\"ahler 3-folds. \Cited in 1 Review \Cited in 34 Documents \MSC 32J27 Compact K\"ahler manifolds: generalizations, classification\\ 32Q15 K\"ahler manifolds\\ 32Q57 Classification theorems \References \[1] DOI: 10.1112/plms/s3-7.1.414 \. Zbl 0084.17305 \. DOI: 10.1112/plms/s3-7.1.414 \[2] Beauville A., J. Diff. Geom. 18 pp 745-- (1983) \[3] DOI: 10.1007/BF01393876 \. DOI: 10.1007/BF01393876 \[4] Campana F., Ann. Sci. ENS 25 pp 539-- (1992) \[5] DOI: 10.1002/mana.19971870104$\,.\,$Zbl 0889.32027$\,.\,$DOI: 10.1002/mana.19971870104 \[6] Campana F., Holomorphic 2-forms on complex 3-folds, J. Algebraic Geom. 9 pp 223-- (2000) \[7] DOI: 10.1007/BF02392558 \. Zbl 0629.32011 \. DOI: 10.1007/BF02392558 \[8] DOI: 10.1007/BFb0094512 \. DOI: 10.1007/BFb0094512 \[9] Demailly J.-P., J. Algebraic Geom. 1 pp 361-- (1992) \[10] Demailly J.-P., Compositio Math. 89 pp 217-- (1993) \[11] Demailly J.-P., J. Algebraic Geom. 3 pp 295-- (1994) \[12] Demailly J.-P., Israel Math. Conf. Proc. 9 pp 165-- (1996) \[13] Demailly J.-P., Compositio Math. 101 pp 217-- (1996) \[14] Fischer G., Math. 306 pp 88-- (1979) \[15] Kawamata Y., Adv. Stud. Pure Math. 10 pp 283-- (1987) \[16] DOI: 10.2307/1971351 \. Zbl 0598.14015 \. DOI: 10.2307/1971351 \[17] Koll\'ar J., Algebraic Geom. 1 pp 429-- (1992) \[18] Krasnov V. A., Mat. Zametki 17 pp 119-- (1975) \[19] DOI: 10.5802/aif.65 \. Zbl 0071.09002 \. DOI: 10.5802/aif.65 \[20] Miyaoka Y., Comment. Math. University St. Pauli 42 pp 1-- (1993) \[21] DOI: 10.2307/2007050 \. Zbl 0557.14021 \. DOI: 10.2307/2007050 \[22] Mori S., J. Amer. Math. Soc. 1 pp 177-- (1988) \[23] DOI: 10.2307/1971387 \. Zbl 0606.14030 \. DOI: 10.2307/1971387 \[24] DOI: 10.2977/prims/1195144881 \. Zbl 0926.14004 \. DOI: 10.2977/prims/\break 1195144881 \[25] Bull. Soc. Math. France 127 pp 115-- (1999) \. Zbl 0939.32020 \. DOI: 10.24033/ bsmf.2344 \[26] DOI: 10.1073/pnas.86.19.7299 \. Zbl 0711.53056 \. DOI: 10.1073/pnas.86.19.7299 \[27] DOI: 10.1007/BF01231450 \. Zbl 0861.14036 \. DOI: 10.1007/BF01231450 \[28] DOI: 10.1007/BF01166457 \. Zbl 0625.32011 \. DOI: 10.1007/BF01166457 \[29] DOI: 10.1016/S0764-4442(99)80408-X \. Zbl 0894.53051 \. DOI: 10.1016/S0764-4442 (99)80408-X \[30] DOI: 10.1016/S0007-4497(98)80078-X \. Zbl 0946.53037 \. DOI: 10.1016/S0007-4497 (98)80078-X \[31] DOI: 10.1007/s002080050154 \. Zbl 1023.32014 \. DOI: 10.1007/s002080050154 \[32] DOI: 10.1007/BF02571950 \. Zbl 0815.14009 \. DOI: 10.1007/BF02571950 \[33] DOI: 10.1007/s002080050207 \. Zbl 0919.32016 \. DOI: 10.1007/s002080050207 \[34] Peternell Th., Collectanea Math. 49 pp 465-- (1998) \[35] DOI: 10.1007/BF01406077 \. Zbl 0205.25102 \. DOI: 10.1007/BF01406077 \[36] DOI: 10.1007/BF02063212 \. Zbl 0153.15401 \. DOI: 10.1007/BF02063212 \[37] DOI: 10.1007/BF01389965 \. Zbl 0289.32003 \. DOI: 10.1007/BF01389965 \[38] Skoda H., Bull. Soc. Math. France 100 pp 353-- (1972) \. Zbl 0246.32009 \. DOI: 10.24033/bsmf.1743 \[39] Takegoshi K., Osaka J. Math. 34 pp 783-- (1997) \[40] DOI: 10.1007/BF01458060 \. Zbl 0628.32037 \. DOI: 10.1007/BF01458060 \[41] Zhang Q., J. Reine Angew. Math. 478 pp 57-- (1996) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Multiplier ideal sheaves and analytic methods in algebraic geometry. (English) Zbl 1102.14300 \Publi Demailly, J.-P. (ed.) et al., School on vanishing theorems and effective results in algebraic geometry. Lecture notes of the school held in Trieste, Italy, April 25--May 12, 2000. Trieste: The Abdus Salam International Centre for Theoretical Physics (ISBN 92-95003-09-8/pbk). ICTP Lect. Notes 6, 1-148 (2001). The lecture notes under review, based on the author's course at the 2000 Trieste school on vanishing theorems and effective results in algebraic geometry, are an extended version of the author's CIME lectures [in: Transcendental methods in algebraic geometry. Lect. 3rd sess. CIME , Cetraro, Italy, 1994. Lect. Notes Math. 1646, 1--97 (1996; Zbl 0883.14005)]. They give a comprehensive survey on the application of analytic methods to algebraic geometry, especially to vanishing theorems. Aimed at non-specialist (in the author's words, they are ``written with the idea of serving as an analytic toolbox for algebraic geometers''), they provide a lot of historical and introductory material on the subject, as well as very advanced topics and recent developments. For a detailed account see the review of [loc. cit.]. The main changes are due to the incorporation of {\it Y. T. Siu}'s new result on the deformation invariance of plurigenera of varieties of general type [Invent. Math. 134, No.3, 661-673 (1998; Zbl 0955.32017)]. For the entire collection see [Zbl 0986.00053]. \Reviewer Olaf Teschke (Berlin) \Cited in 3 Reviews \Cited in 30 Documents \MSC 14F17 Vanishing theorems\\ 32J25 Transcendental methods of algebraic geometry\\ 32L10 Sections of holomorphic vector bundles\\ 14C20 Divisors, linear systems, invertible sheaves\\ 32Q15 K\"ahler manifolds\\ 32L20 Vanishing theorems (analytic spaces) \Author Demailly, J.-P. (ed.); G\"ottsche, L. (ed.); Lazarsfeld, R. (ed.) \Title School on vanishing theorems and effective results in algebraic geometry. Lecture notes of the school held in Trieste, Italy, April 25--May 12, 2000. (English) Zbl 0986.00053 \Publi ICTP Lecture Notes. 6. Trieste: The Abdus Salam International Centre for Theoretical Physics. vii, 393 p. (2001). The articles of this volume will be reviewed individually. Indexed articles: {\it Demailly, Jean-Pierre}, Multiplier ideal sheaves and analytic methods in algebraic geometry., 1-148 [Zbl 1102.14300] {\it Smith, Karen E.}, Tight closure and vanishing theorems., 149-213 [Zbl 1079.13500] {\it Helmke, Stefan}, The base point free theorem and the Fujita conjecture., 215-248 [Zbl 1101.14300] {\it Viehweg, Eckart}, Positivity of direct image sheaves and applications to families of higher dimensional manifolds., 249-284 [Zbl 1092.14044] {\it Peternell, Thomas}, Subsheaves in the tangent bundle: Integrability, stability and positivity, 285-334 [Zbl 1027.14009] {\it Hwang, Jun-Muk}, Geometry of minimal rational curves on Fano manifolds., 335-393 [Zbl 1086.14506] \MSC 00B25 Proceedings of conferences of miscellaneous specific interest\\ 14-06 Proceedings of conferences (algebraic geometry) \Keywords School; Vanishing theorems; Proceedings Algebraic geometry; Trieste (Italy) \Full_text \Author Demailly, Jean-Pierre; Koll\'ar, J\'anos \Title Semi-continuity of complex singularity exponents and K\"ahler-Einstein metrics on Fano orbifolds. (English) Zbl 0994.32021 \Publi Ann. Sci. \'Ec. Norm. Sup\'er. (4) 34, No. 4, 525-556 (2001). Let $\varphi$ be a plurisubharmonic function on a complex manifold $X$. The complex singularity exponent $c_K(\varphi)$ of $\varphi$ on a compact set $K\subset X$ is the supremum over $c\ge 0$ such that $\exp(-2c\varphi)$ is integrable on a neighborhood of $K$. The notion plays an important role in complex analysis and algebraic geometry, and several other characteristics of singularities for analytic objects (holomorphic functions, coherent ideal sheaves, divisors, currents) are its particular cases.\par The main results of the paper is lower semicontinuity of the map $\varphi\mapsto c_K (\varphi)$, which means that if $\varphi_j\to \varphi$ in $L^1_{\text {loc}}(X)$ then $\exp(-2c\varphi_j) \to\exp(-2c\varphi)$ in $L^1$-norm over a neighborhood of $K$ for all positive $c4$) 53C55 Hermitian and K\"ahlerian manifolds (global differential geometry) \Keywords Kobayashi hyperbolicity \Author Demailly, Jean-Pierre \Title Hyperbolic varieties and algebraic differential equations. (Vari\'et\'es hyperboliques et \'equations diff\'erentielles alg\'ebriques.) (French) Zbl 0901.32019 \Publi Gaz. Math., Soc. Math. Fr. 73, 3-23 (1997). In this survey article, the author presents the relationship between the existence of entire curves (i.e.\ holomorphic curves $f:\bC \to X)$ on an algebraic variety $X$ and global algebraic differential operators on the variety $X$. We mention that the nonexistence of non constant entire curves is equivalent to the Kobayashi's hyperbolicity.\par The author gives a complete proof of the following vanishing result of {\it M. Green} and {\it Ph. Griffiths}, presented with an incomplete proof in Proc. Int. Chern Symp., Berkely 1979, 41-74 (1980; Zbl 0508.32010)]: ``Let $X$ be a projective algebraic variety and let $f:\bC \to X$ be a non constant entire curve. Then $P(f', \dots, f^{(k)}) \equiv 0$ for any algebraic differential operator $P$ with values in the dual $L^*$ of a holomorphic line bundle $L$ on $X$, with positive curvature''. As an application one obtaines explicit examples of hyperbolic algebraic surfaces of small degree by applying the above vanishing result to wronskian operators. \Reviewer Vasile Br\^{\i}nz\u{a}nescu (Bucure\c{s}ti) \Cited in 4 Documents \MSC 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds\\ 32-02 Research monographs (several complex variables)\\ 32H30 Value distribution theory in higher dimensions\\ 32A22 Nevanlinna theory (local); growth estimates; other inequalities (several complex variables) \Keywords hyperbolic varieties; projective varieties of general type; wronskian \Author Demailly, Jean-Pierre; El Goul, Jawher \Title Meromorphic partial projective connections and hyperbolic projective varieties.\\ (Connexions m\'eromorphes projectives partielles et vari\'et\'es alg\'ebriques hyperboliques.) (French. Abridged English version) Zbl 0898.32016 \Publi C. R. Acad. Sci., Paris, S\'er. I 324, No. 12, 1385-1390 (1997). S. Kobayashi conjectured in [Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, NY (1970; Zbl 0207.37902)] that a generic hypersurface of $\bC\bP^n$ of sufficiently high degree $d$ (where the expected bound is $d\ge 2n-1$) is hyperbolic. The conjecture is true for $\bC\bP^2$, but for $n\ge 3$ a few number of examples are known. For $\bC\bP^3$ (where the expected bound is $5$) the first example of a smooth hyperbolic surface in $\bC\bP^3$ of any degree $d\ge 50$ was obtained by {\it R. Brody} and {\it M. Green} [Duke Math. J. 44, 873-874 (1977; Zbl 0383.32009)] and {\it A. M. Nadel} [Duke Math. J. 58, No. 3, 749-771 (1989; Zbl 0686.32015)] obtained examples of degree $d\ge 21$ and the second author [Manuscr. Math. 90, No. 4, 521-532 (1996)] gave examples of degree $d\ge 14$. In this paper, following some ideas of {\it Y. T. Siu} [Duke Math. J. 55, 213-251 (1987; Zbl 0623.32018)] and A. Nadel, the authors introduce the concept of meromorphic connection and construct Wronskian operators acting on jets of holomorphic curves. Then using some results, the authors give examples of hyperbolic algebraic surfaces in $\bC\bP^3$ with arbitrary degree $d\ge 11$. \Reviewer Raul Iba\~nez (Bilb\~ao) \Cited in 2 Documents \MSC 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds\\ 14H10 Families, algebraic moduli (curves)\\ 32A20 Meromorphic functions (several variables)\\ 32C25 Analytic subsets and submanifolds 53A20 Projective differential geometry 53C55 Hermitian and K\"ahlerian manifolds (global differential geometry) \Keywords meromorphic connection; Wronskian operator; hyperbolic surfaces; complex projective space \Author Demailly, Jean-Pierre \Title Numerical analysis and differential equations. Nouvelle \'ed. (Analyse num\'erique et \'equations diff\'erentielles.) (French) Zbl 0869.65041 \Publi Grenoble: Presses Univ. de Grenoble. 309 p. (1996). See the review of the German translation (1994; Zbl 0869.65042). \Cited in 2 Reviews \Cited in 1 Document \MSC 65L05 Initial value problems for ODE (numerical methods) 65L06 Multistep, Runge-Kutta, and extrapolation methods 65D32 Quadrature and cubature formulas (numerical methods) 65H10 Systems of nonlinear equations (numerical methods) 65-01 Textbooks (numerical analysis) 34-01 Textbooks (ordinary differential equations) \Keywords ordinary differential equation; initial value; problem; roundoff problem; polynomial approximation; quadrature formulas; iterative methods; textbook \Author Demailly, Jean-Pierre \Title $L\sp 2$ vanishing theorems for positive line bundles and adjunction theory. (English) Zbl 0883.14005 \Publi Catanese, F. (ed.) et al., Transcendental methods in algebraic geometry. Lectures given at the 3rd session of the Centro Internazionale Matematico Estivo (CIME), Cetraro, Italy, July 4--12, 1994. Cetraro: Springer. Lect. Notes Math. 1646, 1-97 (1996). The main goal of the paper is to describe a few analytic tools which are useful to study questions such as linear series and vanishing theorems for algebraic vector bundles. Also, algebraic and analytic proofs of some results are compared. One of the first applications of the analytic method in algebraic geometry is Kodaira's use of the Bochner technique (1950-60) to relate cohomology and curvature via harmonic forms. Well known is the Akizuki-Kodaira-Nakano theorem (1954): If $X$ is a nonsingular projective algebraic variety and $L$ is a holomorphic line bundle on $X$ with positive curvature, then $H^q(X,\Omega^p_X\otimes L)=0$ for $p+q>$ dim$X$. H\"ormander (1965) used a refinement of this technique to obtain a fundamental $L^2$ estimate, concerning solutions of the Cauchy-Riemann operator. Except vanishing theorems, more precise quantative information about solutions of $\bar{\partial}$-equations was obtained. Main tools to relate analytic and algebraic geometry are the multiplier ideal sheaf $I(\phi)$ and positive currents. $I(\phi)$ is defined as a sheaf of germs of holomorphic functions $f$ such that $|f|^2e^{-2\phi}$ is locally summable, where $\phi$ is a (locally defined) plurisubharmonic function. Since $I(\phi)$ is a coherent algebraic sheaf over $X$, we have a direct correspondence between analytic and algebraic objects which takes into account singularities efficiently. Currents, introduced by Lelong (1957), play the role of algebraic cycles, and many classical results of intersection theory apply to currents. Also an analytic interpretation of the Seshadri constant of a line bundle is given and it represents a measure of local positivity. One of the motivations for this work was the conjecture of Fujita: If $L$ is an ample (i.e.\ positive) line bundle on a projective $n$-dimensional algebraic variety $X$ then $K_X+(n+2)L$ is very ample. Reider (1988) gave a proof of the Fujita conjecture in the case of surfaces. \par Using an analytic approach, in the paper under review it is shown that $2K_X+L$ is very ample under suitable numerical conditions for $L$. The first part of the proof is to choose an appropriate metric using a complex Monge-Amp\`ere equation and the Aubin-Calabi-Yau theorem. Solution $\phi$ of the equation assumes logarithmic poles and they are controlled using the intersection theory of currents. Detailed relations to the existing algebraic proofs of similar results are given (Ein-Lazarsfeld, Fujita, Siu). In the last section, a proof of the effective Matsusaka big theorem obtained by {\it Y.-T. Siu} [Ann. Inst. Fourier 43, No. 5, 1387-1405; Zbl 0803.32017)] is presented. Siu's proof is based on the very ampleness of $2K_X+mL$ together with the theory of holomorphic Morse inequalities [{\it J.-P. Demailly}, Ann. Inst. Fourier 35, No. 4, 189-229 (1985; Zbl 0565.58017)]. Long and detailed preliminary sections dedicated to the basic facts of complex differential geometry are included which make the main ideas of the paper easier to understand. For the entire collection see [Zbl 0855.00017]. \Reviewer N.Bla\v{z}i\'c (Beograd) \Cited in 1 Review \Cited in 16 Documents \MSC 14F17 Vanishing theorems\\ 32L05 Holomorphic fiber bundles and generalizations\\ 14F43 Other algebro-geometric (co)homologies\\ 14F05 Sheaves, derived categories of sheaves, etc.\\ 32L20 Vanishing theorems (analytic spaces)\\ 32C30 Integration on analytic sets and spaces, currents\\ 32W20 Complex Monge-Amp\`ere operators \Keywords positive line bundle; linear series; vanishing theorems; Lelong number; intersection theory; Bochner technique; $L\sp 2$ estimates; Seshadri constant; numerically effective line bundle; Fujita conjecture; Monge-Ampere equation; very ample line bundle; algebraic vector bundles; effective Matsusaka big theorem \Author Bertin, Jos\'e; Demailly, Jean-Pierre; Illusie, Luc; Peters, Chris \Title Introduction to Hodge theory. (Introduction \`a la th\'eorie de Hodge.) (French. English summary) Zbl 0849.14002 \Publi Panoramas et Synth\`eses. 3. Paris: Soci\'et\'e Math\'ematique de France. vi, 272 p. (1996). The origin of what is currently meant by the notion of Hodge theory can be traced back to {\it W.V.D. Hodge}'s fundamental work accomplished in the 1930s. In modern terminology, Hodge prepared the ground for describing the De Rham cohomology algebra of a Riemannian manifold in terms of its harmonic differential forms. In the following two decades, Hodge's decomposition principle has been extended to the (then) new sheaf-theoretic and cohomological framework of Hermitean differential geometry, complex-analytic geometry, and transcendental algebraic geometry. The names of G. De Rham, A. Weil, K. Kodaira, and many others stand for the tremendous progress achieved during this period, in particular with regard to deformation and classification theory in these areas. The special algebraic structures (Hodge structures) arising from Hodge decompositions and their generalizations have led to a rather independent field of research in geometry, precisely to the so-called Hodge theory, which represents a powerful and indispensible toolkit for contemporary complex geometry, general algebraic geometry, and -- nowadays -- also for mathematical physics. The vast activity in Hodge theory and its related areas, especially during the recent twenty years, is not reflected in the current textbook literature, at least not comprehensively or in an updated form compiling the various recent aspects and applications, so that a panoramic overview of the present state of art must be regarded as a highly welcome (and needed) service to the mathematical community. \par A conference on the present state of Hodge theory, serving exactly that purpose, took place at the University of Grenoble (France) in November 1994. The book under review grew out of the series of lectures which the authors delivered at this meeting. The aim of the text is to develop a number of fundamental concepts and results of classical and modern Hodge theory, and in this the book is prepared for students and non-expert researchers in the field, who wish to get acquainted in depth with the subject, and obtain a profound up-to-date knowledge of its present level of development. -- The material is divided into three main parts, each of which is written by different authors and devoted to various central and complementary aspects of the theory.\par Part I, written by {\it J.-P. Demailly}, is entitled `` $L^2$-Hodge theory and vanishing theorems''. The author discusses in detail two fundamental applications of Hilbert $L^2$-space methods to complex analysis and algebraic geometry, respectively. This part adopts basically the analytic viewpoint and consists, on its side, of two chapters. Chapter 1 provides an introduction to standard complex Hodge theory, including the basics on Hermitean and K\"ahler geometry, differential operators on vector bundles, Hodge decomposition, Hodge degeneration, the spectral sequence of Hodge-Fr\"olicher, Gauss-Manin connexion, and the deformation behavior of the Hodge groups (after Kodaira). Chapter 2 is devoted to $L^2$-estimates for the $\overline \partial $-operator and the resulting vanishing theorems for cohomology groups of K\"ahler manifolds and projective varieties. The main topics here are the classical methods of Oka, Bochner, and H\"ormander in pseudo-convex analysis, their consequences for cohomology vanishing, as well as the more recent but already well-known fundamental contributions by the author himself towards the interpretation of the great vanishing theorems of A. Nadel and of Kawamata-Viehweg. -- The concluding two sections of this chapter deal with the property of very-ampleness of line bundles on projective varieties. The first central result discussed here is the author's analytic approach to the famous conjecture of Fujita, culminating in an improvement of {\it Y.-T. Siu}'s very recent theorem on an effective bound for very-ampleness [cf. ``Effective very ampleness'', Invent. Math. 124, No. 1-3, 563-571 (1996)]. The second central result is an effective version of the classical ``Big embedding theorem of Matsusaka'', whose surprisingly simple proof is due to the author himself (1996), based on some foregoing work of {\it Y.-T. Siu} [Ann. Inst. Fourier 43, No. 5, 1387-1405 (1993; Zbl 0803.32017)], and methodically related to the effective bound for very-ampleness discussed before. These two last sections provide a particularly up-to-data account on the newest developments in analytical Hodge theory and its (algebraic) applications.\par Part II of the text, written by {\it L. Illusie}, is entitled ``Frobenius and Hodge degeneration''. These notes aim at introducing non-specialists to those methods and techniques of algebraic geometry over a field of characteristic $p > 0$, which have been used by P. Deligne and the author to give an algebraic proof of the Hodge degeneration and the Kodaira-Akizuki-Nakano vanishing theorem for smooth projective varieties in characteristic zero. Basically, this part of the book is a careful, detailed introduction to the important work ``Rel\`evements modulo $p^2$ et d\'ecomposition du complexe de De Rham'' [Invent. Math. 89, 247-270 (1987; Zbl 0632.14017)] by {\it P. Deligne} and {\it L. Illusie}. Here the reader is assumed to bring along some basic knowledge of the theory of algebraic schemes and of homological algebra (in categories). After recalling the basics on schemes, differentials and the algebraic De Rham complex in characteristic $p>0$, the author discusses the following topics: smoothness and coverings, the Frobenius morphism and the Cartier isomorphism, derived categories and spectral sequences, decomposition theorems, vanishing theorems in characteristic $p$, degeneration theorems, the standard techniques for passing from characteristic $p$ to characteristic zero, and the proof of the above mentioned degeneration and vanishing theorems. The concluding section of this part points to some recent developments and open problems concerning Hodge theory in characteristic $p$.\par Also this part is essentially self-contained, and most proofs are given in detail. Some proofs are -- quite naturally -- at least outlined, assuming the reader to follow the precise hints to the related textbook literature (mostly EGA) and original papers.\par Part III of the book, written by {\it J. Bertin} and {\it C. Peters}, is entitled ``Variations of Hodge structures, Calabi-Yau manifolds, and mirror symmetry''. It consists again of two main chapters, whose interrelation is beautifully explained in a comprehensive introduction. -- Chapter I is devoted to the comparatively elementary part of the theory of variation of Hodge structures and its applications in complex algebraic geometry. This includes detailed descriptions of the Hodge bundles, the Hodge filtrations, the De Rham cohomology sheaves, the Gauss-Manin connexion in its general setting (after Katz and Oda) and with its transversality property (due to Griffiths), variations and infinitesimal variations of Hodge structures, the Griffiths period domains for polarized Hodge structures, mixed Hodge structures, limits of Hodge structures (after Deligne), the Picard-Lefschetz theory and the local monodromy theorem, Deligne's degeneration criteria for Hodge spectral sequences, and a brief discussion of the method of vanishing cycles. At the end, the authors give a sketch of the use of Higgs bundles for the construction of variations of Hodge structures, mainly by following Simpson's approach [cf.: {\it C. T. Simpson}, Proc. Int. Congr. Math., Kyoto 1990, Vol. I, 747-756 (1991; Zbl 0765.14005)], as well as some comments on M. Saito's work on Hodge modules, intersection cohomology, and ${\cal D}$-modules in algebraic analysis. -- Chapter II reflects the fact that Calabi-Yau manifolds, their Hodge theory, and their mirror symmetry have recently gained enormous significance in both algebraic geometry and theoretical physics, particularly in constructing two-dimensional conformal quantum field theories. The material presented here covers the fundamental facts on Calabi-Yau manifolds, their construction and deformation theory, and their mirror properties. After a digression on the cohomology of hypersurfaces (after Griffiths and Dimca), which is used for the description of the link between the Picard-Fuchs equation and the variation of Calabi-Yau structures, the variation of Hodge structures for families of Calabi-Yau threefolds, their Yukawa couplings, and their mirror symmetries are explained in more depth. The interested reader can find a very complete and comprehensive account on this subject in the recent monography ``Sym\'etrie miroir'' by {\it C. Voisin} [Panoramas et Synth\`eses, No. 2 (1996; see the preceding review)]. In a concluding section, the authors discuss (following an idea of P. Deligne) a possible approach to mirror symmetry via a certain duality between variations of Hodge structures for Calabi-Yau threefolds. A rich bibliography enhances this very systematic and lucid treatise.\par Altogether, the present book, in all its three parts, which consistently refer to each other, may be regarded as a masterly introduction to Hodge theory in its classical and very recent, analytic and algebraic aspects. Aimed to students and non-specialists, it is by far much more than only an introduction to the subject. The material leads the reader to the forefront of research in many areas related to Hodge theory, and that in a detailed and highly self-contained manner. As such, this text is also a valuable source for active researchers and teachers in the field, in particular due to the utmost carefully arranged index at the end of the book. \Reviewer W.Kleinert (Berlin) \Cited in 1 Review \Cited in 3 Documents \MSC 14C30 Transcendental methods, Hodge theory, Hodge conjecture\\ 14F17 Vanishing theorems\\ 14-02 Research monographs (algebraic geometry)\\ 14D07 Variation of Hodge structures 13A35 Characteristic $p$ methods; tight closure 58A14 Hodge theory (global analysis)\\ 14D05 Structure of families 81T30 String and superstring theories\\ 14J32 Calabi-Yau manifolds \Keywords characteristic $p$; De Rham cohomology algebra; Hodge theory; vanishing theorems; very-ampleness of line bundles; Hodge degeneration; De Rham complex; Frobenius morphism; Cartier isomorphism; variation of Hodge structures; Gauss-Manin connexion; period domains; Picard-Lefschetz theory; Higgs bundles; Calabi-Yau manifolds; two-dimensional conformal quantum field theories; Picard-Fuchs equation; Yukawa couplings; mirror symmetries \Author Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael \Title Compact K\"ahler manifolds with Hermitian semipositive anticanonical bundle. (English) Zbl 1008.32008 \Publi Compos. Math. 101, No.2, 217-224 (1996). {\it Summary$\,$}: This note states a structure theorem for compact K\"ahler manifolds with semipositive Ricci curvature: Any such manifold has a finite \'etale covering possessing a de Rham decomposition as a product of irreducible compact K\"ahler manifolds, each one being either Ricci flat (torus, symplectic or Calabi-Yau manifold) or Ricci semipositive without nontrivial holomorphic forms. Related questions and conjectures concerning the latter case are discussed. \Cited in 1 Review \Cited in 9 Documents \MSC 32J27 Compact K\"ahler manifolds: generalizations, classification 53C55 Hermitian and K\"ahlerian manifolds (global differential geometry) \Full_text \References \[1] Aubin, T. : Equations du type Monge-Amp\`ere sur les vari\'et\'es k\"ahleriennes compactes . C. R. Acad. Sci. Paris Ser. A 283 (1976) 119-121; Bull. Sci. Math. 102 (1978) 63-95. \. Zbl 0374.53022 \[2] Beauville, A. : Vari\'et\'es k\"ahleriennes dont la premi\`ere classe de Chern est nulle . J. Diff. Geom. 18 (1983) 775-782. \. Zbl 0537.53056 \. DOI: 10.4310/jdg/1214438181 \[3] Berger, M. : Sur les groupes d'holonomie des vari\'et\'es \`a connexion affine des vari\'et\'es riemanniennes . Bull. Soc. Math. France 83 (1955) 279-330. \. Zbl 0068.36002 \. DOI: 10.24033/bsmf.1464 \. NUMDAM: BSMF\_1955\_\_83\_\_279\_0 \. EUDML: 86895 \[4] Bishop, R. : A relation between volume, mean curvature and diameter . Amer. Math. Soc. Not. 10 (1963) p. 364. \[5] Bogomolov, F.A. : On the decomposition of K\"ahler manifolds with trivial canonical class . Math. USSR Sbornik 22 (1974) 580-583. \. Zbl 0304.32016 \. DOI: 10.1070/ SM1974v022n04ABEH001706 \[6] Bogomolov, F.A. : K\"ahler manifolds with trivial canonical class . Izvestija Akad. Nauk 38 (1974) 11-21. \. Zbl 0299.32022 \. DOI: 10.1070/IM1974v008n01ABEH002093 \[7] Br\"uckmann, P. and Rackwitz, H.-- G.: T-symmetrical tensor forms on complete intersections . Math. Ann. 288 (1990) 627-635. \. Zbl 0724.14032 \. DOI: 10.1007/ BF01444555 \. EUDML: 164757 \[8] Campana, F. : Fundamental group and positivity of cotangent bundles of compact K\"ahler manifolds . Preprint 1993. \. Zbl 0845.32027 \[9] Cheeger, J. and Gromoll, D. : The splitting theorem for manifolds of nonnegative Ricci curvature . J. Diff. Geom. 6 (1971) 119-128. \. Zbl 0223.53033 \. DOI: 10.4310/jdg/1214430220 \[10] Cheeger, J. and Gromoll, D. : On the structure of complete manifolds of nonnegative curvature . Ann. Math. 96 (1972) 413-443. \. Zbl 0246.53049 \. DOI: 10.2307/1970819 \[11] Demailly, J.-P. , Peternell, T. and Schneider, M. : K\"ahler manifolds with numerically effective Ricci class . Compositio Math. 89 (1993) 217-240. \. Zbl 0884.32023 \. NUMDAM: CM\_1993\_\_89\_2\_217\_0 \. EUDML: 90258 \[12] Demailly, J.-P. , Peternell, T. and Schneider, M. : Compact complex manifolds with numerically effective tangent bundles . J. Alg. Geom. 3 (1994) 295-345. \. Zbl 0827.14027 \[13] Kobayashi, S. : Recent results in complex differential geometry . Jber. dt. Math.-Verein. 83 (1981) 147-158. \. Zbl 0467.53030 \[14] Kobayashi, S. : Topics in complex differential geometry . In DMV Seminar , Vol. 3., Birkh\"auser 1983. \. Zbl 0506.53029 \[15] Lichnerowicz, A. : Vari\'et\'es k\"ahleriennes et premi\`ere classe de Chern . J. Diff. Geom. 1 (1967) 195-224. \. Zbl 0167.20004 \. DOI: 10.4310/jdg/1214428089 \[16] Lichnerowicz, A. : Vari\'et\'es K\"ahl\'eriennes \`a premi\`ere classe de Chern non n\'egative et vari\'et\'es riemanniennes \`a courbure de Ricci g\'en\'eralis\'ee non n\'egative . J. Diff. Geom. 6 (1971) 47-94. \. Zbl 0231.53063 \. DOI: 10.4310/jdg/1214430218 \[17] Manivel, L. : Birational invariants of algebraic varieties . Preprint Institut Fourier, no. 257 (1993). \. Zbl 0811.14008 \[18] Ogus, A. : The formal Hodge filtration . Invent. Math. 31 (1976) 193-228. \. Zbl 0339.14004 \. DOI: 10.1007/BF01403145 \. EUDML: 142361 \[19] Yau, S.T. : On the Ricci curvature of a complex K\"ahler manifold and the complex Monge-Amp\`ere equation I . Comm. Pure and Appl. Math. 31 (1978) 339-411. \. Zbl 0369.53059 \. DOI: 10.1002/cpa.3160310304 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Effective bounds for very ample line bundles. (English) Zbl 0862.14004 \Publi Invent. Math. 124, No.1-3, 243-261 (1996). Let $L$ be an ample line bundle on a nonsingular projective $n$-fold $X$. A well-known conjecture of T. Fujita asserts that $K_X+ (n+1)L$ is generated by global sections and $K_X+ (n+2)L$ is very ample. For $n=2$ this follows from I. Reider's theorem and the global generation part of the conjecture was proved for $n=3$ by {\it L. Ein} and {\it R. Lazarsfeld} [J. Am. Math. Soc. 6, No. 4, 875-903 (1993; Zbl 0803.14004)]. The present paper is mainly concerned with the very ampleness part of the conjecture. In a previous paper [J. Differ. Geom. 37, No. 2, 323-374 (1993; Zbl 0783.32013)] the author proved that $2K_X+12n^nL$ is very ample, using an analytic method based on the solution of a Monge-Amp\`ere equation. In the present paper, improving a method of {\it Y.-T. Siu} [Invent. Math. 124, No. 1-3, 563-571 (1996; Zbl 0853.32034)] based on a combination of the Riemann-Roch formula with the vanishing theorem of {\it A. M. Nadel} [Ann. Math., II. Ser. 132, No. 3, 549-596 (1990; Zbl 0731.53063)] the author proves that $2K_X+ mL$ is very ample for $m\ge 2+ {3n+1 \choose n}$ and that $m(K_X+ (n+2)L)$ is very ample for $m\ge {3n+1 \choose n}-2n$. The method of proof gives, as a byproduct, the well-known fact that $K_X+ (n+1)L$ is numerically effective (a result originally proved as a consequence of Mori theory). The paper also contains a refinement of a method developed by {\it Y.-T. Siu} [Ann. Inst. Fourier 43, No. 5, 1387-1405 (1993; Zbl 0803.32017)] which enables the author to obtain a better effective Matsusaka big theorem. \Reviewer I.Coand\u{a} (Bucure\c{s}ti) \Cited in 7 Documents \MSC 14C20 Divisors, linear systems, invertible sheaves\\ 14F05 Sheaves, derived categories of sheaves, etc.\\ 14F17 Vanishing theorems\\ 32C20 Normal analytic spaces \Keywords Hermitian metrics on line bundles; Fujita conjecture; ample line bundle; very ampleness; vanishing theorem; Mori theory; Matsusaka big theorem \Author Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael \Title Holomorphic line bundles with partially vanishing cohomology. (English)\\ Zbl 0859.14005 \Publi Teicher, Mina (ed.), Proceedings of the Hirzebruch 65 conference on algebraic geometry, Bar-Ilan University, Ramat Gan, Israel, May 2-7, 1993. Ramat-Gan: Bar-Ilan University, Isr. Math. Conf. Proc. 9, 165-198 (1996). Let $X$ denote a complex manifold of dimension $n$. The authors study holomorphic line bundles $L$ on $X$ with partially vanishing cohomology (or having metrics with positive eigenvalues of curvature). They define $\sigma_+(L)$ to be the smallest integer $q$ with the following property: There exists an ample divisor $D$ on $X$ and a constant $c>0$ such that $H^j(X, mL-pD)=0$ for all $j>q$ and $mp\geq 0$, $m\geq c(p+1)$. Note that $\sigma_+(L)=0$ if and only if $L$ is ample while $\sigma_+(L)=n$ if and only if $c_1(L^*)$ is in the closure of the cone of effective divisors. An ample $q$-flag is defined as a sequence $Y_q\subset Y_{q+1}\subset \cdots\subset Y_n=X$ of subvarieties $Y_k$ of $X$ such that $\dim Y_k=k$ and $Y_k$ is the image of an ample Cartier divisor in the normalization of $Y_{k+1}$. Then a line bundle $L$ is called $q$-flag positive if for some ample $q$-flag, $L\mid Y_q$ is positive. \par Vanishing theorem: If $L\in\text {Pic}X$ is $q$-flag positive then $\sigma_+(L)\leq n-q$. \par The converse of this theorem is not true in general. A counter example and a positive result (of converse) for $\bP_{n-1}$ bundles over a curve are given. The structure of projective 3-folds with $\sigma_+(-K_X)=1$, $K_X=$ canonical bundle, is investigated. One has $\sigma_+(-K_X)=0$ if and only if $X$ is Fano and $\sigma_+(-K_X)\leq 2$ if and only if $\kappa(X)= -\infty$. The authors also study various cones in $NX(X) \otimes\bR$, $NX(X)$ being N\'eron-Severi group, i.e.\ the group of divisors modulo numerical equivalence. All these cones coincide for surfaces. For the entire collection see [Zbl 0828.00035]. \Reviewer U.N.Bhosle (Bombay) \Cited in 2 Reviews \Cited in 7 Documents \MSC 14F17 Vanishing theorems\\ 32L20 Vanishing theorems (analytic spaces)\\ 14F05 Sheaves, derived categories of sheaves, etc.\\ 14C22 Picard groups \Keywords flag; holomorphic line bundles; vanishing cohomology \Author Demailly, Jean-Pierre \Title Compact complex manifolds whose tangent bundles satisfy numerical effectivity properties (joint work with Thomas Peternell and Michael Schneider). (English)\\ Zbl 0880.14003 \Publi Geometry and analysis. Papers presented at the Bombay colloquium, India, January 6--14, 1992. Oxford: Oxford University Press. Stud. Math., Tata Inst. Fundam. Res. 13, 67-86 (1995). A compact Riemann surface always has a hermitian metric with constant curvature, in particular the curvature sign can be taken to be constant: the negative sign corresponds to curves of general type (genus $\ge 2)$, while the case of zero curvature corresponds to elliptic curves (genus 1), positive curvature being obtained only for $\bP^1$ (genus 0). In higher dimensions the situation is much more subtle and it has been a long standing conjecture due to Frankel to characterize $\bP_n$ as the only compact K\"ahler manifold with positive holomorphic bisectional curvature. Hartshorne strengthened Frankel's conjecture and asserted that $\bP_n$ is the only compact complex manifold with ample tangent bundle. In his famous paper in Ann. Math., II. Ser. 110, 593-606 (1979; Zbl 0423.14006), {\it S. Mori} solved Hartshorne's conjecture by using characteristic $p$ methods. Around the same time {\it Y.-T. Siu} and {\it S.-T. Yau} [Invent. Math. 59, 189-204 (1980; Zbl 0442.53056)] gave an analytic proof of the Frankel conjecture. Combining algebraic and analytic tools Mok classified all compact K\"ahler manifolds with semi-positive holomorphic bisectional curvature. -- From the point of view of algebraic geometry, it is natural to consider the class of projective manifolds $X$ whose tangent bundle is numerically effective (nef). This has been done by Campana and Peternell and -- in case of dimension 3 -- by Zheng. In particular, a complete classification is obtained for dimension at most three. The main purpose of this work is to investigate compact (most often K\"ahler) manifolds with nef tangent or anticanonical bundles in arbitrary dimension. We first discuss some basic properties of nef vector bundles which will be needed in the sequel in the general context of compact complex manifolds. We refer to papers by {\it J.-P. Demailly}, {\it T. Peternell} and {\it M. Schneider} [Compos. Math. 89, No. 2, 217-240 (1993) and J. Algebr. Geom. 3, No. 2, 295-345 (1994; Zbl 0827.14027)] for detailed proofs. Instead, we put here the emphasis on some unsolved questions. For the entire collection see [Zbl 0868.00030]. \MSC 14C20 Divisors, linear systems, invertible sheaves\\ 32J27 Compact K\"ahler manifolds: generalizations, classification\\ 14F05 Sheaves, derived categories of sheaves, etc. \Keywords nef tangent bundles; nef anticanonical bundle; compact Riemann surface \Author Demailly, Jean-Pierre \Title $L\sp 2$-methods and effective results in algebraic geometry. (English) Zbl 0845.14004 \Publi Chatterji, S. D. (ed.), Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Z\"urich, Switzerland. Vol. II. Basel: Birkh\"auser. 817-827 (1995). Given an ample line bundle $L$ on a projective $n$-fold, it is an important question to find an integer $m_0$ such that $mL$ is ample for $m \ge m_0$. The example of curves shows that no universal bound (depending only on $n)$ exists. However T. Fujita has conjectured that if $L$ is an ample line bundle on a projective $n$-fold then $K_X + (n + 2)L$ is very ample, where $K_X$ is the canonical line bundle. Here the author explains how analytic methods lead to a universal bound $m_0 = 2 + {3n + 1 \choose n}$ such that $2K_X + mL$ is very ample for $m \ge m_0$. For the entire collection see [Zbl 0829.00015]. \Reviewer F.Kirwan (Oxford) \Cited in 1 Document \MSC 14C20 Divisors, linear systems, invertible sheaves\\ 14F05 Sheaves, derived categories of sheaves, etc. \Keywords ample line bundle \Author Demailly, Jean-Pierre; Passare, Mikael \Title Residual currents and fundamental class. (Courants r\'esiduels et classe fondamentale.) (French) Zbl 0851.32013 \Publi Bull. Sci. Math. 119, No.1, 85-94 (1995). Let $Y$ be the complex subspace of a complex manifold $X$ defined by a coherent ideal $I$, which is a locally complete intersection. The authors introduce the notion of the cohomology with supports in the infinitesimal neighbourhood of first order of $Y$ and then, they prove that the residual current $R_Y$ is intrinsically identified to a canonical element of the infinitesimal cohomology of first order with supports in $Y$ and with values in the sheaf of sections of the determinant of the determinant of the conormal bundle to $Y$. \Reviewer Vasile Br\^{\i}nz\u{a}nescu (Bucure\c{s}ti) \Cited in 1 Review \MSC 32C30 Integration on analytic sets and spaces, currents\\ 32C36 Local cohomology of analytic spaces 58A25 Currents (global analysis)\\ 32C15 Complex spaces \Keywords current; analytic spaces; cohomology with supports \Author Demailly, Jean-Pierre \Title Semicontinuity properties of cohomology and of Kodaira-Iitaka dimension. (Propri\'et\'es de semi-continuit\'e de la cohomologie et de la dimension de Kodaira-Iitaka.) (French. Abridged English version) Zbl 0851.32015 \Publi C. R. Acad. Sci., Paris, S\'er. I 320, No.3, 341-346 (1995). Let $X \to S$ be a proper and flat morphism of complex spaces and let $(X_t)$ be the fibres. Given a sheaf $E$ over $X$ of locally free ${\cal O}_X$-modules, inducing on the fibres a family of sheaves $(E_t \to X_t)$, the author shows that the cohomology group dimension $h^q(t) = h^q(X_t, E_t)$ satisfy the following semicontinuity property: for every $q \geq 0$, the sum $h^q(t) - h^{q-1}(t) + \cdots + (-1)^q h^0 (t)$ is upper semicontinuous for the Zariski topology. Then, some applications to the Kodaira-Iitaka dimension are given. \Reviewer Vasile Br\^{\i}nz\u{a}nescu (Bucure\c{s}ti) \MSC 32C35 Analytic sheaves and cohomology groups 35G05 General theory of linear higher-order PDE \Keywords proper morphisms of complex spaces; semicontinuity; Kodaira-Iitaka dimension \Author Demailly, Jean-Pierre \Title Ordinary differential equations. Theoretical and numerical aspects. (Gew\"ohnliche Differentialgleichungen. Theoretische und numerische Aspekte. Aus d. Franz. \"ubers. von Mathias Heckele.) (German) Zbl 0869.65042 \Publi Wiesbaden: Vieweg. x, 318 p. (1994). Die Besonderheit des vorliegenden Buches ist eine integrierte Darstellung der theoretischen Grundlagen und der numerischen Behandlung von Anfangswertaufgaben gew\"ohnlicher Differentialgleichungen. Der Numerikteil greift dabei thematisch noch weiter aus, indem Rundungsfehler, Polynomapproximation, Quadraturformeln und iterative Verfahren behandelt werden, mit Ausnahme des etwas knapp geratenen Kapitels Iteration sogar ziemlich ausf\"uhrlich. Ein - was den integrierten Differentialgleichungsteil betrifft - \"ahnlich aufgebautes Lehrbuch ist von {\it H. Werner} und {\it W. Arndt} [Gew\"ohnliche Differentialgleichungen. Eine Einf\"uhrung in Theorie und Praxis (1986; MR 88b.34002)] verfa{\ss}t worden.\par Das vorliegende Lehrbuch besticht durch seine pr\"azise Darstellung der behandelten Sachverhalte und die damit einhergehende Sorgfalt und Eleganz in der Behandlung der mathematischen Aspekte. Mancher Leser w\"urde sich vielleicht eine st\"arkere Betonung numerischer Gesichtspunkte w\"unschen, was der Titel des Buches aber auch nicht verspricht. Mit seiner speziellen thematischen Ausrichtung und der inhaltlichen Qualit\"at hat das Buch einen eigenen Platz in der vorliegenden umfangreichen Numerik-Lehrbuchliteratur, und es wird hoffentlich gen\"ugend viele Leser finden, die davon profitieren. \Reviewer R.D.Grigorieff (Berlin) \Cited in 2 Reviews \MSC 65L05 Initial value problems for ODE (numerical methods) 65L06 Multistep, Runge-Kutta, and extrapolation methods 65D32 Quadrature and cubature formulas (numerical methods) 65H10 Systems of nonlinear equations (numerical methods) 65-01 Textbooks (numerical analysis) 34-01 Textbooks (ordinary differential equations) \Keywords ordinary differential equation; initial value problem; roundoff errors; polynomial approximation; quadrature formulas; iterative methods; textbook \Author Demailly, Jean-Pierre \Title Regularization of closed positive currents of type (1,1) by the flow of a Chern connection. (English) Zbl 0824.53064 \Publi Skoda, Henri (ed.) et al., Contributions to complex analysis and analytic geometry. Based on a colloquium dedicated to Pierre Dolbeault, Paris, France, June 23-26, 1992. Braunschweig: Vieweg. Aspects Math. E 26, 105-126 (1994). Let $X$ be a compact complex manifold, and $T$ a closed positive current of (1,1) type. Some questions addressed in this article are related to the approximation of $T$ by smooth closed ``positive'' currents. It is easy to see a necessary condition for this approximation, namely the cohomology class $\{T \}$ should satisfy $\int\sb Y\{T\}\sp p \geq 0$ for every $p$-dimensional subvariety $Y \subset X$. Thus, in general case, one concerns the approximation of $T$ only by closed ``almost positive'' currents, as the following principal result shows.\par Let $\gamma$ be a continuous real (1,1) form such that $T \geq \gamma$, $u$ some continuous nonnegative (1,1) form, and $\omega$ a (smooth) Hermitian metric on $T\sb X$. Then under a certain curvature condition, $T$ can be approximated by closed ``almost positive'' (1,1) currents $T\sb \varepsilon$ with the following properties: (i) $T\sb \varepsilon \geq \gamma - \lambda\sb \varepsilon u - \delta\sb \varepsilon \omega$; (ii) $\lambda\sb \varepsilon (x)$ is an increasing family of continuous functions such that for all $x \in X$, $\lim\sb{\varepsilon \to 0} \lambda\sb \varepsilon (x) = \nu (T,x)$ (Lelong number of $T$ at $x$); (iii) the constants $\delta\sb \varepsilon \to 0$ as $\varepsilon \to 0$, and $\delta\sb \varepsilon > 0$, $\forall \varepsilon$. For the curvature condition above, we require $(\Theta (T\sb X) + u \otimes \text {Id}\sb{T\sb X}) (\theta \otimes \xi, \theta \otimes \xi) \geq 0$ for all $\theta$, $\xi$ of $T\sb X$, with $\langle \theta, \xi \rangle = 0$. Moreover if put $T = \alpha + {i\over \pi} \partial \overline {\partial} \psi$, for $\alpha$ a smooth (1,1) form in the same $\partial \overline {\partial}$-cohomology class as $T$, and $\psi$ an almost plurisubharmonic function, then we have the representation: $T\sb \varepsilon = \alpha + {i\over \pi} \partial \overline {\partial} \psi\sb \varepsilon$ such that $\psi\sb \varepsilon$ is smooth over $X$ and increasingly converges to $\psi$, as $\varepsilon \to 0$. It can be shown that the representation of the above $T$ involving a quasi-psh $\psi$ (i.e.\ locally the sum of a psh function and a smooth function) is always possible.\par Similar results as to the regularization of closed positive currents are treated elsewhere, e.g. [the author, J. Algebr. Geom. 1, No. 3, 361-409 (1992; Zbl 0777.32016)], where a numerical hypothesis rather than a curvature hypothesis is assumed: $c\sb 1({\cal O}\sb{T\sb X}(1)) + \pi\sp* u$ is nef on the total space of (dual) projectivized tangent bundles. This numerical condition does not seem to be directly related to the partial semipositivity curvature condition; for instance, the author remarks that for the curve case the partial semipositivity hypothesis is void. By using the present curvature hypothesis, the author felt it perhaps easier to extend to currents of higher bidegrees. For the entire collection see [Zbl 0811.00006]. \Reviewer I-Hsun Tsai (Taipei) \Cited in 1 Review \Cited in 8 Documents \MSC 53C55 Hermitian and K\"ahlerian manifolds (global differential geometry)\\ 32C30 Integration on analytic sets and spaces, currents \Keywords closed positive currents; semipositive curvature; $\partial \overline {\partial}$-cohomology; plurisubharmonic function \Author Demailly, Jean-Pierre; Lempert, L\'aszl\'o; Shiffman, Bernard \Title Algebraic approximations of holomorphic maps from Stein domains to projective manifolds. (English) Zbl 0861.32006 \Publi Duke Math. J. 76, No.2, 333-363 (1994). Let $Y,Z$ be quasi-projective algebraic varieties and let $\Omega$ be an open subset of $Y$. A map $F:\Omega \to Z$ is said to be Nash algebraic if $f$ is holomorphic and the graph of $f$ is contained in an algebraic subvariety of $Y \times Z$ of dimension equal to $\dim Y$. \par One of the main results in the paper is the following theorem concerning the approximation of holomorphic maps by Nash algebraic maps:\par {\bf Theorem 1.1.} Let $\Omega$ be a Runge domain in an affine algebraic variety $S$ and let $f:\Omega \to X$ be a holomorphic map into a quasi-projective algebraic manifold $X$. Then for every relatively compact domain $\Omega_0 \subset\!\subset\Omega$, there is a sequence of Nash algebraic maps $f_\nu: \Omega_0\to X$ such that $f_\nu \to f$ uniformly on $\Omega_0$.\par As important applications of Theorem 1.1 the authors obtain that the Kobayashi-Royden pseudometric and the Kobayashi pseudodistance on projective algebraic manifolds can be approximated in terms of algebraic curves. It is proved that a type of algebraic approximation is also possible in the case of locally free sheaves.\par Using the methods developed in the paper the authors give a more precise form of a result concerning the description of equivalent Nash algebraic vector bundle, obtained by {\it T. Tancredi} and {\it A. Tognoli} [Bull. Sci. Math., II. Ser. 117, No. 2, 173-183 (1993; Zbl 0798.32010)]. A result of {\it E. L. Stout} [Contemp. Math. 32, 259-266 (1984; Zbl 0584.32027)] on the exhaustion of Stein manifolds by Runge domains in affine algebraic manifolds is proved by substantially different methods. \Reviewer I.Serb (Cluj-Napoca) \Cited in 1 Review \Cited in 14 Documents \MSC 32E10 Stein spaces, Stein manifolds\\ 14P20 Nash functions and manifolds\\ 32F45 Invariant metrics and pseudodistances \Keywords approximation of holomorphic maps; Nash algebraic maps; quasi-projective algebraic manifold; Stein manifolds; Runge domains \References \[1] A. Andreotti and E. Vesentini, Carleman estimates for the Laplace-Beltrami equation on complex manifolds , Inst. Hautes \'Etudes Sci. Publ. Math. (1965), no. 25, 81-130. \. Zbl 0138.06604 \. DOI: 10.1007/BF02684398 \.\\ NUMDAM: PMIHES\_1965\_\_25\_\_81\_0 \.\\ NUMDAM: PMIHES\_1965\_\_27\_\_153\_0 \. EUDML: 103855 \[2] E. Bishop, Mappings of partially analytic spaces , Amer. J. Math. 83 (1961), 209-242. JSTOR: \. Zbl 0118.07701 \. DOI: 10.2307/2372953 \. http://links.jstor.org/sici? sici=0002-9327\%28196104\%2983\%3A2\%3C209\%3AMOPAS\%3E2.0.CO\%3B2-2\break\&origin=euclid \[3] M. Cornalba and P. Griffiths, Analytic cycles and vector bundles on non-compact algebraic varieties , Invent. Math. 28 (1975), 1-106. \. Zbl 0293.32026 \. DOI: 10.1007/ BF01389905 \. EUDML: 142315 \[4] J.-P. Demailly, Cohomology of $q$-convex spaces in top degrees , Math. Z. 204 (1990), no. 2, 283-295. \. Zbl 0682.32017 \. DOI: 10.1007/BF02570874 \. EUDML: 183771 \[5] J.-P. Demailly, Singular Hermitian metrics on positive line bundles , Complex algebraic varieties (Bayreuth, 1990), Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 87-104. \. Zbl 0784.32024 \. DOI: 10.1007/BFb0094512 \[6] J.-P. Demailly, Algebraic criteria for the hyperbolicity of projective manifolds , in preparation. \[7] L. van den Dries, A specialization theorem for analytic functions on compact sets , Proc. Kon. Nederl. Akad. Wetensch. 85 (1982), 391-396. \. Zbl 0526.30004 \[8] D. A. Eisenman, Intrinsic measures on complex manifolds and holomorphic mappings, Memoirs of the American Mathematical Society, No. 96, American Mathematical Society, Providence, R.I., 1970. \. Zbl 0197.05901 \[9] Y. Eliashberg and M. Gromov, Embeddings of Stein manifolds of dimension $n$ into the affine space of dimension $3n/2+1$ , Ann. of Math. (2) 136 (1992), no. 1, 123-135. JSTOR: \. Zbl 0758.32012 \. DOI: 10.2307/2946547 \. http://links.jstor.org/sici? sici=0003-486X\%28199207\%292\%3A136\%3A1\%3C123\%3AEOSMOD\%3E2.0.CO\break\%3B2-M\&origin=euclid \[10] H. Grauert, Analytische Faserungen \"uber holomorph-vollst\"andigen R\"aumen , Math. Ann. 135 (1958), 263-273. \. Zbl 0081.07401 \. DOI: 10.1007/BF01351803 \. EUDML: 160618 \[11] P. Griffiths, Topics in algebraic and analytic geometry , Princeton University Press, Princeton, N.J., 1974. \. Zbl 0302.14003 \[12] R. C. Gunning and H. Rossi, Analytic functions of several complex variables , Prentice-Hall Inc., Englewood Cliffs, N.J., 1965. \. Zbl 0141.08601 \[13] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II , Ann. of Math. (2) 79 (1964), 109-203; ibid. (2) 79 (1964), 205-326. JSTOR: \. Zbl 0122.38603 \. DOI: 10.2307/1970486 \. http://links.jstor.org/sici? sici=0003-486X\%28196403\%292\%3A79\%3A2\%3C205\%3AROSOAA\%3E2.0.CO\break\%3B2-I\&origin=euclid \[14] L. H\"ormander, An introduction to complex analysis in several variables , North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. \. Zbl 0685.32001 \[15] S. Kobayashi, Invariant distances on complex manifolds and holomorphic mappings , J. Math. Soc. Japan 19 (1967), 460-480. \. Zbl 0158.33201 \. DOI: 10.2969/jmsj/ 01940460 \[16] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings , Pure and Applied Mathematics, vol. 2, Marcel Dekker Inc., New York, 1970. \. Zbl 0207.37902 \[17] S. Lang, Hyperbolic and Diophantine analysis , Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 159-205. \. Zbl 0602.14019 \. DOI: 10.1090/S0273-0979-1986-15426-1 \[18] S. Lang, Introduction to complex hyperbolic spaces , Springer-Verlag, New York, 1987. \. Zbl 0628.32001 \[19] R. Narasimhan, Imbedding of holomorphically complete complex spaces , Amer. J. Math. 82 (1960), 917-934. JSTOR: \. Zbl 0104.05402 \. DOI: 10.2307/2372949 \. http://links.jstor.org/sici?sici=0002-9327\%28196010\%2982\%3A4\%3C917\%3AIO\break HCCS\%3E2.0.CO\%3B2-R\&origin=euclid \[20] J. Nash, Real algebraic manifolds , Ann. of Math. (2) 56 (1952), 405-421. JSTOR: \. Zbl 0048.38501 \. DOI: 10.2307/1969649 \. http://links.jstor.org/sici?sici=0003-486X\%28195211\%292\%3A56\%3A3\%3C405\%3ARAM\%3E2.0.CO\%3B2-B\&origin=\break euclid \[21] J. Noguchi and T. Ochiai, Geometric function theory in several complex variables , Translations of Mathematical Monographs, vol. 80, American Mathematical Society, Providence, RI, 1990. \. Zbl 0713.32001 \[22] K. Oka, Sur les fonctions de plusieurs variables, II. Domaines d'holomorphie , J. Sci. Hiroshima Univ. 7 (1937), 115-130. \. Zbl 0017.12204 \[23] R. Richberg, Stetige streng pseudokonvexe Funktionen , Math. Ann. 175 (1968), 257-286. \. Zbl 0153.15401 \. DOI: 10.1007/BF02063212 \. EUDML: 161667 \[24] H. L. Royden, Remarks on the Kobayashi metric , Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970), Springer, Berlin, 1971, 125-137. Lecture Notes in Math., Vol. 185. \. Zbl 0218.32012 \[25] N. Sibony, Quelques probl\`emes de prolongement de courants en analyse complexe , Duke Math. J. 52 (1985), no. 1, 157-197. \. Zbl 0578.32023 \. DOI: 10.1215/S0012-7094-85-05210-X \[26] Y. T. Siu, Every Stein subvariety admits a Stein neighborhood , Invent. Math. 38 (1976/77), no. 1, 89-100. \. Zbl 0343.32014 \. DOI: 10.1007/BF01390170 \. EUDML: 142443 \[27] E. L. Stout, Algebraic domains in Stein manifolds , Proceedings of the conference on Banach algebras and several complex variables (New Haven, Conn., 1983), Contemp. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1984, pp. 259-266. \. Zbl 0584.32027 \[28] A. Tancredi and A. Tognoli, Relative approximation theorems of Stein manifolds by Nash manifolds , Boll. Un. Mat. Ital. A (7) 3 (1989), no. 3, 343-350. \. Zbl 0691.32006 \[29] A. Tancredi and A. Tognoli, On the extension of Nash functions , Math. Ann. 288 (1990), no. 4, 595-604. \. Zbl 0699.32006 \. DOI: 10.1007/BF01444552 \. EUDML: 164754 \[30] A. Tancredi and A. Tognoli, Some remarks on the classification of complex Nash vector bundles , Bull. Sci. Math. 117 (1993), no. 2, 173-183. \. Zbl 0798.32010 \[31] A. Weil, L'int\'egrale de Cauchy et les fonctions de plusieurs variables , Math. Ann. 111 (1935), 178-182. \. Zbl 0011.12301 \. DOI: 10.1007/BF01472212 \. EUDML: 159767 \[32] H. Whitney, Local properties of analytic varieties , Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N. J., 1965, pp. 205-244. \. Zbl 0129.39402 \[33] S.-T. Yau, A general Schwarz lemma for K\"ahler manifolds , Amer. J. Math. 100 (1978), no. 1, 197-203. JSTOR: \. Zbl 0424.53040 \. DOI: 10.2307/2373880 \. http://links.jstor.org/sici?sici=0002-9327\%28197802\%29100\%3A1\%3C197\%3AAGSL\break FK\%3E2.0.CO\%3B2-H\&origin=euclid This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael \Title Compact complex manifolds with numerically effective tangent bundles. (English) Zbl 0827.14027 \Publi J. Algebr. Geom. 3, No.2, 295-345 (1994). The main result of this fundamental article is: Let $X$ be a compact K\"ahler manifold with nef tangent bundle $T_X$. Moreover, let $\widetilde X$ be a finite \'etale cover of $X$ of maximum irregularity $q = q (\widetilde X) = h^1 (\widetilde X, {\cal O}_{\widetilde X})$. Then: $\pi_1 (\widetilde X) \cong \bZ^{2q}$.\par The albanese map $\alpha : \widetilde X \to A(\widetilde X)$ is a smooth fibration over a $q$-dimensional torus with nef relative tangent bundle.\par The fibres of $\alpha$ are Fano manifolds with nef tangent bundles.\par Here a line bundle $L$ on a compact complex manifold $X$ with a fixed hermitian metric $\omega$ is nef if, for every $\varepsilon > 0$, there exists a smooth hermitian metric $h_\varepsilon$ on $L$ such that the curvature satisfies $\Theta_{h_\varepsilon} \ge - \varepsilon \omega$. A bundle $E$ on $X$ is nef if the line bundle ${\cal O}_E (1)$ on $\bP (E)$ is nef. -- Many other interesting and important results are contained in the article. It is proved that:\par Let $E$ be a vector bundle on a compact K\"ahler manifold $X$.\par If $E$ and $E^*$ are nef, then $E$ admits a filtration whose graded pieces are hermitian flat.\par If $E$ is nef, then $E$ is numerically semi-positive.\par Moreover, algebraic proofs are given for the result:\par Any Moisheson manifold with nef tangent bundle is projective.\par A compact K\"ahler $n$-fold with $T_X$ nef and $c_1 (X)^n > 0$ is Fano.\par Further the two following classification results are given:\par The smooth non-algebraic compact complex surfaces with nef tangent bundles are:\par non-algebraic tori; Kodaira surfaces; Hopf surfaces.\par Let $X$ be a non-algebraic three-dimensional compact K\"ahler manifold. Then $T_X$ is nef if and only if $X$, up to a finite \'etale cover, is either a torus or of the form $\bP(E)$, where $E$ and $E^*$ are nef rank-2 vector bundles over a two-dimensional torus. \Reviewer D. Laksov (Stockholm) \Cited in 10 Reviews \Cited in 60 Documents \MSC 14J30 Algebraic threefolds\\ 14C20 Divisors, linear systems, invertible sheaves\\ 32J17 Compact $3$-folds (analytic spaces)\\ 14F35 Homotopy theory; fundamental groups (algebraic geometry) 53C55 Hermitian and K\"ahlerian manifolds (global differential geometry)\\ 14E20 Coverings, fundamental group (mappings) \Keywords fundamental group; nef line bundle; Albanese map; Moisheson manifold; three-dimensional compact K\"ahler manifold \Author Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael \Title K\"ahler manifolds with numerically effective Ricci class. (English) Zbl 0884.32023 \Publi Compos. Math. 89, No.2, 217-240 (1993). The purpose of this paper is to contribute to the solution of the following conjectures: Let $X$ be a compact K\"ahler manifold with numerically effective (nef) anticanonical bundle $-K_X$; then:\par Conjecture 1: The fundamental group $\pi_1(X)$ of $X$ has polynomial growth.\par Conjecture 2: The Albanese map $\alpha:X\to{\rm Alb}(X)$ is surjective.\par Section 1 is devoted to proving the following theorem, which is the main contribution to Conjecture 1.\par {\bf Theorem 1:} Let $X$ be a compact K\"ahler manifold with nef anticanonical bundle; then $\pi_1(X)$ has subexponential growth.\par The main tools used in order to prove Theorem 1 are the solution of the Calabi conjecture and volume bounds for geodesic balls due to Bishop and Gage. It should be mentioned that from the proof of Theorem 1 it follows that Conjecture 1 holds in the case $-K_X$ is Hermitian semipositive (Theorem 2).\par In Section 2 the following theorem concerning Conjecture 2 is proved.\par {\bf Theorem 3:} Let $X$ be an $n$-dimensional compact K\"ahler manifold such that $-K_X$ is nef. Then the Albanese map $\alpha$ is surjective as soon as $\dim\alpha(X)$ is 0,1 or $n$, and, if $X$ is projective, also for $n-1$; moreover, if $X$ is projective and if the generic fibre $F$ of $\alpha$ has $-K_F$ big, then $\alpha$ is surjective.\par Finally, Section 3 is devoted to the study of the structure of projective 3-folds with nef anticanonical bundles; in particular Conjecture 2 is proved in dimension 3 with purely algebraic methods, except in one very special case. \Reviewer Antonella Nannicini (MR 95b:32044) \Cited in 3 Reviews \Cited in 8 Documents \MSC 32J27 Compact K\"ahler manifolds: generalizations, classification\\ 14J40 Algebraic $n$-folds ($n>4$)\\ 32Q15 K\"ahler manifolds 53C55 Hermitian and K\"ahlerian manifolds (global differential geometry) \Keywords numerically effective Ricci class; compact K\"ahler manifold; Albanese map; nef anticanonical bundles \Full_text \References \[1] Aubin, T. : Equations du type Monge-Amp\`ere sur les vari\'et\'es k\"ahl\'eriennes compactes , C.R. Acad. Sci Paris Ser. A 283 (1976) 119-121; Bull. Sci. Math. 102 (1978) 63-95. \. Zbl 0374.53022 \[2] Beauville, A. : Vari\'et\'es K\"ahl\'eriennes dont la premi\`ere classe de Chern est nulle , J. Diff. Geom. 18 (1983) 755-782. \. Zbl 0537.53056 \. DOI: 10.4310/jdg/1214438181 \[3] Bishop, R. : A relation between volume, mean curvature and diameter , Amer. Math. Soc. Not. 10 (1963) 364. \[4] Bogomolov, F.A. : On the decomposition of K\"ahler manifolds with trivial canonical class , Math. USSR Sbornik 22 (1974) 580-583. \. Zbl 0304.32016 \. DOI: 10.1070/ SM1974v022n04ABEH001706 \[5] Bogomolov, F.A. : K\"ahler manifolds with trivial canonical class , Izvestija Akad. Nauk 38 (1974) 11-21 and Math. USSR Izvestija 8 (1974) 9-20. \. Zbl 0299.32022 \. DOI: 10.1070/IM1974v008n01ABEH002093 \[6] Calabi, E. : On K\"ahler manifolds with vanishing canonical class, Alg. geometry and topology , Symposium in honor of S. Lefschetz, Princeton Univ. Press, Princeton (1957) 78-89. \. Zbl 0080.15002 \[7] Campana, F. , Peternell, Th. : On the second exterior power of tangent bundles of 3-folds , Comp. Math 83 (1992), 329-346. \. Zbl 0824.14037 \. NUMDAM: CM\_1992\_\_83\_3\_329\_0 \. EUDML: 90172 \[8] Demailly, J.-P. , Peternell, Th. , Schneider, M. : Compact complex manifolds with numerically effective tangent bundles , Preprint 1991. To appear in J. Alg. Geom. \. Zbl 0827.14027 \[9] Gage, M.E. : Upper bounds for the first eigenvalue of the Laplace-Beltrami operator , Indiana Univ. J. 29 (1980) 897-912. \. Zbl 0465.53031 \. DOI: 10.1512/iumj.1980.29.29061 \[10] Gromov, M. : Groups of polynomial growth and expanding maps , Appendix by J. Tits, Publ. I.H.E.S. 53 (1981) 53-78. \. Zbl 0474.20018 \. DOI: 10.1007/BF02698687 \. NUMDAM: PMIHES\_1981\_\_53\_\_53\_0 \. EUDML: 103974 \[11] Hartshorne, R. : Algebraic Geometry, Graduate Texts in Math ., Springer, Berlin, 1977. \. Zbl 0367.14001 \[12] Heintze, E. , Karcher, H. : A general comparison theorem with applications to volume estimates for submanifolds , Ann. Scient. Ec. Norm. Sup. 4e S\'erie, 11 (1978) 451-470. \. Zbl 0416.53027 \. DOI: 10.24033/asens.1354 \.\\ NUMDAM: ASENS\_1978\_4\_11\_4\_451\_0 \. EUDML: 82023 \[13] Kawamata, Y. : A generalization of Kodaira-Ramanujam's vanishing theorem , Math. Ann. 261 (1982) 43-46. \. Zbl 0476.14007 \. DOI: 10.1007/BF01456407 \. EUDML: 182862 \[14] Kawamata, Y. , Matsuki, K. , Matsuda, K. : Introduction to the minimal model pro-gram , Adv. Studies Pure Math. 10 (1987) 283-360. \. Zbl 0672.14006 \[15] Kobayashi, S. : On compact K\"ahler manifolds with positive definite Ricci tensor , Ann. Math 74 (1961) 570-574. \. Zbl 0107.16002 \. DOI: 10.2307/1970298 \[16] Koll\'ar, J. : Higher direct images of dualizing sheaves , Ann. Math. 123 (1986) 11-42. \. Zbl 0598.14015 \. DOI: 10.2307/1971351 \[17] Lichnerowicz, A. : Vari\'et\'es K\"ahl\'eriennes et premi\`ere classe de Chern , J. Diff. Geom. 1 (1967) 195-224. \. Zbl 0167.20004 \. DOI: 10.4310/jdg/1214428089 \[18] Lichnerowicz, A. : Vari\'et\'es K\"ahl\'eriennes \`a premi\`ere classe de Chern non n\'egative et vari\'et\'es riemanniennes \`a courbure de Ricci g\'en\'eralis\'ee non n\'egative , J. Diff. Geom. 6 (1971) 47-94. \. Zbl 0231.53063 \. DOI: 10.4310/jdg/1214430218 \[19] Lichnerowicz, A. : Vari\'et\'es K\"ahl\'eriennes \`a premi\`ere classe de Chern non n\'egative et situation analogue dans le cas riemannien, Ist. Naz. Alta Mat., Rome, Symposia Math ., vol. 10, Academic Press, New-York (1972) 3-18. \. Zbl 0267.53035 \[20] Matsushima, Y. : Recent results on holomorphic vector fields , J. Diff. Geom. 3 (1969) 477-480. \. Zbl 0201.25902 \. DOI: 10.4310/jdg/1214429068 \[21] Miyanishi, M. : Algebraic methods in the theory of algebraic threefolds , Adv. Studies in Pure Math. 1 (1983) 69-99. \. Zbl 0537.14027 \[22] Mori, S. : Threefolds whose canonical bundles are not numerically effective , Ann. Math. 116 (1982) 133-176. \. Zbl 0557.14021 \. DOI: 10.2307/2007050 \[23] Myers, S.B. : Riemannian manifolds with positive mean curvature , Duke Math. J. 8 (1941) 401-404. \. Zbl 0025.22704 \. DOI: 10.1215/S0012-7094-41-00832-3 \[24] Viehweg, E. : Vanishing theorems , J. Reine Angew. Math. 335 (1982) 1-8. \. Zbl 0485.32019 \. DOI: 10.1515/crll.1982.335.1 \. crelle:GDZPPN002199688 \. EUDML: 152458 \[25] Yau, S.T. : Calabi's conjecture and some new results in algebraic geometry , Proc. Nat. Acad. Sci. USA 74 (1977) 1789-1790. \. Zbl 0355.32028 \. DOI: 10.1073/ pnas.74.5.1798 \[26] Yau, S.T. : On the Ricci curvature of a complex K\"ahler manifold and the complex Monge-Amp\`ere equation I , Comm. Pure and Appl. Math. 31 (1978) 339-411. \. Zbl 0369.53059 \. DOI: 10.1002/cpa.3160310304 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title A numerical criterion for very ample line bundles. (English) Zbl 0783.32013 \Publi J. Differ. Geom. 37, No.2, 323-374 (1993). Let $X$ be a projective algebraic manifold of dimension $n$ and let $L$ be an ample line bundle over $X$. We give a numerical criterion ensuring that the adjoint bundle $K\sb X+L$ is very ample. The sufficient conditions are expressed in terms of lower bounds for the intersection numbers $L\sp p\cdot Y$ over subvarieties $Y$ of $X$. In the case of surfaces, our criterion gives universal bounds and is only slightly weaker than {\it I. Reider}'s criterion [Ann. Math., II. Ser. 127, No. 2, 309-316 (1988; Zbl 0663.14010)]. When $\dim X\ge 3$ and ${\rm codim} Y\ge 2$, the lower bounds for $L\sp p\cdot Y$ involve a numerical constant which depends on the geometry of $X$. By means of an iteration process, it is finally shown that $2K\sb X+mL$ is very ample for $m\ge 12n\sp n$. Our approach is mostly analytic and based on a combination of H\"ormander's $L\sp 2$ estimates for the operator $\overline\partial$, Lelong number theory and the Aubin-Calabi-Yau theorem. \Reviewer J.-P. Demailly (Saint-Martin d'H\`eres) \Cited in 4 Reviews \Cited in 54 Documents \MSC 32J15 Compact surfaces (analytic spaces)\\ 32L10 Sections of holomorphic vector bundles\\ 32C30 Integration on analytic sets and spaces, currents \Keywords very ample line bundle; plurisubharmonic function; closed positive current; Monge-Amp\`ere equation; intersection theory; numerical criterion; Lelong number; Aubin-Calabi-Yau theorem \Author Demailly, Jean-Pierre \Title Monge-Amp\`ere operators, Lelong numbers and intersection theory. (English)\\ Zbl 0792.32006 \Publi Ancona, Vincenzo (ed.) et al., Complex analysis and geometry. New York: Plenum Press. The University Series in Mathematics. 115-193 (1993). This article is a survey on the theory of Lelong numbers, viewed as a tool for studying intersection theory by complex differential geometry. The paper contains earlier works of the author [M\'em. Soc. Math. Fr., Nouv. S\'er. 19, 124 p. (1985; Zbl 0579.32012) and Acta Math. 159, 153- 169 (1987; Zbl 0629.32011)] and of {\it Y. T. Siu} [Invent. Math. 27, 53- 156 (1974; Zbl 0289.32003)]. Many results are given with complete proofs, which are shorter and simpler than the original ones. The references contain 37 items on these topics. For the entire collection see [Zbl 0772.00007]. \Reviewer E.Outerelo (Madrid) \Cited in 1 Review \Cited in 71 Documents \MSC 32C30 Integration on analytic sets and spaces, currents\\ 32W20 Complex Monge-Amp\`ere operators\\ 32U05 Plurisubharmonic functions and generalizations \Keywords Monge-Amp\`ere operators; current; plurisubharmonic functions; Lelong numbers; intersection theory \Author Demailly, Jean-Pierre \Title Holomorphic Morse inequalities on $q$-convex manifolds. (English) Zbl 0771.32011 \Publi Several complex variables, Proc. Mittag-Leffler Inst., Stockholm/Swed. 1987-88, Math. Notes 38, 245-257 (1993). \[For the entire collection see Zbl 0759.00008.] This paper is a nice report (``high level propaganda for a very interesting result and for very interesting tools'') of at that time recent work [{\it T. Bouche}, Ann. Sci. Ec. Norm. Super., IV. Ser. 22, No. 4, 501-513 (1989; Zbl 0693.32016)] which extends previous work by Demailly to the case of strongly $q$-convex manifolds. The main result of Bouche's paper is the following one.\par {\bf Theorem A:} Let $X$ be a strongly $q$-convex complex manifold with $n:=\dim(X)$, $E$ a rank $r$ vector bundle on $X$ and $L$ a line bundle on $X$ with hermitian metric such that the curvature form $ic(L)$ has at least $n-p+1$ eigenvalues $\ge 0$ outside a compact subset of $X$; set $$ \eqalign{ X(m,L)&:=\{x\in X\,:\;ic(L)~\hbox{is non degenerate at $x$ and}\cr\ &\qquad\hbox{with exactly $m$ negative eigenvalues$\}$},\cr X(\le m,L)&:=\bigcup\sb{t\le m}X(t,L),\cr X(\ge m,L)&:=\bigcup\sb{t\ge m}X(t,L).\cr} $$ Then for all $m\ge p+q-1$ the following asymptotic inequalities hold:\par $(a\sb m)$ Weak Morse inequalities $$\dim H\sp m(X,E\otimes L\sp k)\le r{k\sp n\over n!}\int\sb{X(m,L)}(-1)\sp m\left({i\over 2\pi}c(L)\right)\sp n+o(k\sp n)$$ $(b\sb m)$ Strong Morse inequalities: $$\sum\sb{m\le t\le n}(-1)\sp{t-m}\dim H\sp t(X,E\otimes L\sp k)\le r{k\sp n\over n!}\int\sb{X(\ge m,L)}(-1)\sp m\left({i\over 2\pi}c(L)\right)\sp n+o(k\sp n)$$ This note contains a sketch of the proof of this theorem. Here the main recent and very powerful techniques are explained and used (e.g. Witten's complex); the main tool for the proof of Morse inequalities is a spectral theorem for Schr\"odinger operators which describes very precisely the asymptotic distribution of eigenvalues for a suitable quadratic form. This report contains the statement, the history and the motivation of two important applications of Theorem A: a very general a priori estimate for Monge-Amp\`ere operator $(id'd'')\sp n$ on $q$-convex manifolds and the following stronger form of Grauert-Riemenschneider conjecture:\par {\bf Theorem B:} Let $X$ be a connected $n$-dimensional compact manifold; if $X$ has a hermitian line bundle $L$ such that $\int\sb{X(\le 1,L)}(ic(L)\sp n>0$, then $X$ is Moishezon. \Reviewer E. Ballico (Povo) \MSC 32F10 $q$-convexity, $q$-concavity\\ 32C35 Analytic sheaves and cohomology groups\\ 32J99 Compact analytic spaces\\ 32W20 Complex Monge-Amp\`ere operators \Keywords $q$-convex manifold; strongly $q$-convex manifold; Morse inequalities; Monge-Amp\`ere operator; Moishezon manifold; curvature form; line bundle; asymptotic extimates for cohomology \Author Demailly, Jean-Pierre \Title Regularization of closed positive currents and intersection theory. (English)\\ Zbl 0777.32016 \Publi J. Algebr. Geom. 1, No.3, 361-409 (1992). Let $X$ be a compact complex manifold and let $T=i\partial\overline\partial\psi$ be a closed positive current of bidegree (1,1) on $X$. Under some hypothesis on a lower bound for the Chern curvature of the tangent bundle $TX$, the current $T$ is proved to be the weak limit of closed currents $T\sb k={i\over\pi}\partial\overline\partial\psi\sb k$ with controlled negative parts; the functions $\psi\sb k$ decrease to $\psi$ as $k\to\infty$ and can be chosen smooth on $X$. However the presence of positive Lelong numbers of $T$ results in some loss of positivity of $T\sb k$.\par This regularization is applied to relations between effective and numerically effective divisors, and to some problems of intersection theory. \Reviewer A.Yu.Rashkovsky (Khar'kov) \Cited in 7 Reviews \Cited in 108 Documents \MSC 32J25 Transcendental methods of algebraic geometry\\ 32C30 Integration on analytic sets and spaces, currents\\ 32S60 Stratifications; constructible sheaves; intersection cohomology (analytic spaces) \Keywords Lelong number; closed positive current; intersection theory \Author Demailly, Jean-Pierre \Title Positive currents and intersection theory. (Courants positifs et th\'eorie de l'intersection.) (French) Zbl 0771.32010 \Publi Gaz. Math., Soc. Math. Fr. 53, 131-159 (1992). This is a nice tool for spreading mathematical culture and ideas among mathematicians. It starts with the notion of current (after de Rham) and ends with the use in the subject of top level research and new extremely powerful methods of the author (around 1991). In the middle it is shown how to use the notion of positive current to define and work (via the integration current of the fundamental class of a subvariety) in Intersection Theory (key words: Lelong numbers and multiplicities). The tools come from Analysis and bring (and often solve) with them several interesting problems which cannot be formulated in a purely algebraic way inside Algebraic Geometry. But these methods are very, very strong competitors even on natural very important algebraic problems. Of course, in this paper most of the proofs are omitted, but ideas and difficulties are not skipped. It is pleasant reading and even specialists in not too far fields can find here some ideas/tools useful for their job; everybody can find some recent deep idea (mostly from Demailly brain). \Reviewer E.Ballico (Povo) \Cited in 2 Documents \MSC 32C30 Integration on analytic sets and spaces, currents 58A25 Currents (global analysis)\\ 32J25 Transcendental methods of algebraic geometry\\ 14C17 Intersection theory, etc. \Keywords positive currents; intersection theory; Lelong numbers; intersection multiplicity; integration current; equimultiplicity \Author Demailly, Jean-Pierre \Title Singular Hermitian metrics on positive line bundles. (English) Zbl 0784.32024 \Publi Complex algebraic varieties, Proc. Conf., Bayreuth/Ger. 1990, Lect. Notes Math. 1507, 87-104 (1992). \[For the entire collection see Zbl 0745.00049.] We quote the author's abstract: ``The notion of a singular Hermitian metric on a holomorphic line bundle is introduced as a tool for the study of various algebraic questions. One of the main interests of such metrics is the corresponding $L\sp 2$ vanishing theorem for $\overline\partial$ cohomology, which gives a useful criterion for the existence of sections. In this context, numerically effective line bundles and line bundles with maximum Kodaira dimension are characterized by means of positivity properties of the curvature in the sense of currents. The coefficients of isolated logarithmic poles of a plurisubharmonic singular metric are shown to have a simple interpretation in terms of the constant $\varepsilon$ of Seshadri's ampleness criterion. Finally, we use singular metrics and approximations of the curvature current to prove a new asymptotic estimate for the dimension of cohomology groups with values in high multiples ${\cal O}(kL)$ of a line bundle $L$ with maximum Kodaira dimension''. \Reviewer E.J.Straube (College Station) \Cited in 13 Reviews \Cited in 56 Documents \MSC 32L05 Holomorphic fiber bundles and generalizations \Keywords plurisubharmonic weights; singular Hermitian metric; holomorphic line bundle; vanishing theorem; cohomology \Author Demailly, Jean-Pierre \Title Transcendental proof of a generalized Kawamata-Viehweg vanishing theorem. (English) Zbl 1112.32303 \Publi Berenstein, Carlos A. (ed.) et al., Geometrical and algebraical aspects in several complex variables. Papers from the conference, Cetraro, Italy, June 1989. Rende: Editoria Elettronica. Semin. Conf. 8, 81-94 (1991). For the entire collection see [Zbl 0969.00052]. \Cited in 5 Documents \MSC 32L20 Vanishing theorems (analytic spaces)\\ 14F17 Vanishing theorems\\ 32L10 Sections of holomorphic vector bundles \Author Demailly, Jean Pierre \Title Holomorphic Morse inequalities. (English) Zbl 0755.32008 \Publi Several complex variables and complex geometry, Proc. Summer Res. Inst., Santa Cruz/CA (USA) 1989, Proc. Symp. Pure Math. 52, Part 2, 93-114 (1991). \[For the entire collection see Zbl 0732.00008.] In this paper the complex analogues of the Morse inequalities for $\overline\partial$-cohomology groups with values in holomorphic vector bundles are explained, and some applications of that theory are presented. \Reviewer B.Nowak ({\L}\'od\'z) \Cited in 1 Review \Cited in 9 Documents \MSC 32C35 Analytic sheaves and cohomology groups 58E05 Abstract critical point theory\\ 32L10 Sections of holomorphic vector bundles 53C07 Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) \Keywords $\overline\partial$-cohomology groups; Morse inequalities; holomorphic vector bundles \Author Blel, Mongi; Demailly, Jean-Pierre; Mouzali, Mokhtar \Title Sur l'existence du c\^one tangent \`a un courant positif ferm\'e. (About the existence of the tangent cone with positive closed current). (French) Zbl 0724.32005 \Publi Ark. Mat. 28, No.2, 231-248 (1990). Let $T$ be a positive closed current of degree p on an open neighborhood $\Omega$ of 0 in ${\bC}\sp n$. For $a\in {\bC}\sp*$ let $h\sb a$ denote the homothety given by a and $h\sp*\sb aT$ the lifted current. If the weak limit $\lim\sb{\vert a\vert \to 0}h\sp*\sb aT$ exists it is called the tangent cone of $T$ in 0. The authors show: \par {\bf Theorem:} If for small $r\sb 0>0$ one of the following conditions a) or b) is satisfied then the tangent cone of T exists: $$ a)\quad \int\sp{r\sb 0}\sb{0}[(\sqrt{v\sb T(r)-v\sb T(r/2)})/r]dr<\infty,\quad b)\quad \int\sp{r\sb 0}\sb{0}[(v\sb T(r)-v\sb T(0))/r]dr<\infty. $$ $v\sb T(r)$ denotes the projective mass of $T$.-- \par The authors show that condition b) is optimal in a sense. \par {\bf Theorem:} If $T$ is the current of an analytic subset of pure dimension $p$ in $\Omega$ then $$ v\sb T(r)-v\sb T(0)\le Cr\sp{\epsilon} $$ for small $r>0$ and suitable numbers $C,\epsilon >0$.-- \par A conclusion of these theorems is a result of Thie and King on the existence of a tangent cone for a current induced by an analytic set. \Reviewer H.-J.Reiffen (Osnabr\"uck) \Cited in 3 Reviews \Cited in 6 Documents \MSC 32C30 Integration on analytic sets and spaces, currents\\ 32B15 Analytic subsets of affine space \Keywords positive closed current; tangent cone; current induced by an analytic set \References \[1] Barlet, D., D\'eveloppements asymptotiques des fonctions obtenues par int\'egration sur les fibres,Invent. Math. 68 (1982), 129--174. \. Zbl 0508.32003 \. DOI: 10.1007/ BF01394271 \[2] Blel, M., C\^one tangent \`a un courant positif ferm\'e de type (1, 1),C. R. Acad. Sci. Paris S\'er. I Math. 309 (1989), 543--546. \. Zbl 0688.32013 \[3] Blel, M.,C\^one tangent \`a un courant positif ferm\'e, preprint de la Facult\'e des Sciences de Monastir, 1988. \[4] Federer, H.,Geometric measure theory (Grundlehren der Mathematischen Wissen\-schaften 158), Springer-Verlag, Berlin, 1969. \. Zbl 0176.00801 \[5] Harvey, R., Holomorphic chains and their boundaries, in:Proc. Symp. Pure Math. 30--1, pp. 309--382, Am. Math. Soc., Providence, 1977. \. Zbl 0374.32002 \[6] King, J. R., The currents defined by analytic varieties,Acta Math. 127 (1971), 185--220. \. Zbl 0224.32008 \. DOI: 10.1007/BF02392053 \[7] Kiselman, C. O.,Tangents of plurisubharmonic functions, preprint Uppsala University (Sweden), December 1988. \. Zbl 0810.31006 \[8] Lelong, P., Fonctions plurisousharmoniques et formes diff\'erentielles positives,$\,$Dunod, Paris, Gordon \&\ Breach, New York, 1968. \. Zbl 0195.11603 \[9] Lelong, P. etGruman, L.,Entire functions of several complex variables (Grundlehren der Mathematischen Wissenschaften 282), Springer-Verlag, Berlin, 1982. \[10] Mouzali, M.,Conditions suffisantes pour l'existence du c\^one tangent \`a un courant positif ferm\'e, Th\`ese de 3e Cycle, Universit\'e de Grenoble I, mai 1989. \[11] Narasimhan, R.,Introduction to the theory of analytic spaces, Lecture Notes in Mathematics 25, Springer-Verlag, Berlin, 1966. \. Zbl 0168.06003 \[12] Thie, P., The Lelong number of a point of a complex analytic set,Math. Ann.,172 (1967), 269--312. \. Zbl 0158.32804 \. DOI: 10.1007/BF01351593 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Cohomology of q-convex spaces in top degrees. (English) Zbl 0682.32017 \Publi Math. Z. 204, No.2, 283-295 (1990). It is shown that every strongly q-complete subvariety of a complex analytic space has a fundamental system of strongly q-complete neighborhoods. As a consequence, we find a simple proof of Ohsawa's result that every non compact irreducible n-dimensional analytic space is strongly n-convex. An elementary proof of the existence of Hodge decomposition in top degrees for absolutely q-convex manifolds is also given. \Reviewer J.-P. Demailly \Cited in 3 Reviews \Cited in 34 Documents \MSC 32F10 $q$-convexity, $q$-concavity \Keywords strongly q-complete subvariety; strongly q-complete neighborhoods; strongly n-convex; existence of Hodge decomposition \Full_text \References \[1] [A-G] Andreotti, A., Grauert, H.: Th\'eor\`emes de finitude pour la cohomologie des espaces complexes. Bull. Soc. Math. Fr.90, 193--259 (1962) \. Zbl 0106.05501 \[2] [Ba] Barlet, D.: Convexit\'e de l'espace des cycles. Bull. Soc. Math. Fr.106, 373--397 (1978) \. Zbl 0395.32009 \[3] [G-R] Grauert, H., Riemenschneider, O.: K\"ahlersche Mannigfaltigkeiten mit hype-q-konvexem Rand. Problems in analysis: a Symposium in honor of Salomon Bochner. Princeton, Princeton University Press 1970 \[4] [G-W] Greene, R. E., Wu, H.: Embedding of open riemannian manifolds by harmonic functions. Ann. Inst. Fourier25, 215--235 (1975) \. Zbl 0307.31003 \[5] [Ma] Malgrange, B.: Existence et approximation des solutions des \'equations aux d\'eriv\'ees partielles et des \'equations de convolution. Ann. Inst. Fourier6, 271--355 (1955/1956) \. Zbl 0071.09002 \[6] [Oh1] Ohsawa, T.: A reduction theorem for cohomology groups of very stronglyq-convex K\"ahler manifolds. Invent. Math.63, 335--354 (1981)/66, 391--393 (1982) \. Zbl 0457.32007 \. DOI: 10.1007/BF01393882 \[7] [Oh2] Ohsawa, T.: Completeness of noncompact analytic spaces. Publ. R.I.M.S., Kyoto Univ.20, 683--692 (1984) \. Zbl 0568.32008 \. DOI: 10.2977/prims/1195181418 \[8] [O-T] Ohsawa, T., Takegoshi, K.: Hodge spectral sequence on pseudoconvex domains. Math. Z.197, 1--12 (1988) \. Zbl 0638.32016 \. DOI: 10.1007/BF01161626 \[9] [S1] Siu, Y. T.: Analytic sheaf cohomology groups of dimensionn ofn-dimensional noncompact complex manifolds. Pac. J. Math.28, 407--411 (1969) \[10] [S2] Siu, Y. T.: Analytic sheaf cohomology groups of dimensionn ofn-dimensional complex spaces. Trans. Am. Math. Soc.143, 77--94 (1969) \[11] [S3] Siu, Y.T.: Every Stein subvariety has a Stein neighborhood. Invent. Math.38, 89--100 (1976) \. Zbl 0343.32014 \. DOI: 10.1007/BF01390170 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Bedford, Eric; Demailly, Jean-Pierre \Title Two counterexamples concerning the pluricomplex Green function in ${\bC}\sp n$. (English) Zbl 0681.32014 \Publi Indiana Univ. Math. J. 37, No.4, 865-867 (1988). Given a domain $\Omega$ in ${\bC}\sp n$ and a point $z\in \Omega$, the pluricomplex Green function on $\Omega$ with logarithmic pole at z is given by $u\sb z(\zeta)=\sup \{v(\zeta):$ v is plurisubharmonic on $\Omega$, $v<0$, and $v(\zeta)\le \log \vert \zeta -z\vert +O(1)\}.$ In ``Capacities in complex analysis'' (1988; Zbl 0655.32001), {\it U. Cegrell} raised the following questions: \par 1. Is $u\sb z\in C\sp 2({\bar \Omega}-\{z\})?$ \par 2. Is $u\sb z$ symmetric, i.e., $u\sb z(\zeta)=u\sb{\zeta}(z)?$ \par {\it L. Lempert} [Bull. Soc. Math. Fr. 109, 427-474 (1981; Zbl 0492.32025)] has shown that if $\Omega$ is strictly convex and smoothly bounded, then the answer to both of these questions is ``Yes''. In the paper, the authors provide counterexamples to show that for strongly pseudoconvex domains, the answer to both these questions is ``No''. \Reviewer M.Stoll \Cited in 5 Documents \MSC 32U05 Plurisubharmonic functions and generalizations\\ 32T99 Pseudoconvex domains 31C10 Pluriharmonic and plurisubharmonic functions \Keywords pluricomplex Green function; strongly pseudoconvex domains \Author Demailly, Jean-Pierre \Title Vanishing theorems for tensor powers of a positive vector bundle. (English)\\ Zbl 0651.32019 \Publi Proc. 21st Int. Taniguchi Symp., Katata/Japan, Conf., Kyoto/Japan 1987, Lect. Notes Math. 1339, 86-105 (1988). \[For the entire collection see Zbl 0638.00022.] Let $E$ be a holomorphic vector bundle of rank $r$ over a compact complex manifold $X$ of dimension $n$, and suppose that $E$ is positive in the sense of Griffiths and that $p+q\ge n+1.$ Let $L$ be a semipositive line bundle and $\Gamma^aE$ an irreducible tensor power representation of ${\rm GL}(E)$ of highest weight $a=(a\sb 1,...,a\sb r)$ with $a\sb 1\ge a\sb 2\ge...\ge a\sb h>a\sb{h+1}=...=a\sb r=0$. The author shows that $H\sp{p,q}(X,\Gamma^aE\otimes (\det E)\sp{\ell}\otimes L)$ vanishes for $\ell \ge h+A(n,p,q),$ where $A(n,p,q)$ is a certain rational function of $n,p,q$. The best possible value for $A(n,p,q)$ is not known, but an example of {\it Th. Peternell}, {\it J. Le Potier} and {\it M. Schneider} [Invent. Math. 87, 573-586 (1987; Zbl 0618.14023)] shows, even when $\Gamma^aE=S^kE$ and $p=n$, at least $\ell \ge 1$ is required. The method of proof is to represent $\Gamma^aE$ as the direct image of a positive line bundle over a suitable flag manifold of $E$ and to apply a generalization of Le Potier's isomorphism theorem to this situation. In order to overcome a difficulty arising from the fact that, when $p0$, $L\ge 0$ or $E\ge 0$, $L>0$. Then there is an integer A(n,p,q), so that $$ (1)\quad H\sp{p,q}(X,Sym\sp k(E)\otimes (\det E)\sp 1\otimes L)=0, $$ whenever $1\ge A(n,p,q)$ and $p+q\ge n+1$. The constant A(n,p,q) is explicitly given, and it is shown that it is optimal for $p=n$. Relation (1) still holds for $E\sp{\otimes k}$ (with a different constant). \par These vanishing theorems strengthen similar celebrated results of Griffith and Le Potier. \Reviewer M. Putinar \Cited in 3 Documents \MSC 32L20 Vanishing theorems (analytic spaces)\\ 32M10 Homogeneous complex manifolds \Keywords positive bundle; curvature; Dolbeault cohomology; vanishing theorems \Author Demailly, Jean-Pierre \Title Mesures de Monge-Amp\`ere et mesures pluriharmoniques. (Monge-Amp\`ere measures and pluriharmonic measures). (French) Zbl 0595.32006 \Publi Math. Z. 194, 519-564 (1987). Let $\Omega$ be a relatively compact open subset in a Stein manifold, and $n=\dim\sb{{\bC}}\Omega$. Assume that $\Omega$ is hyperconvex, i.e.\ that there exists a bounded psh (plurisubharmonic) exhaustion function on $\Omega$. A ''pluricomplex Green function'' $u\sb{\Omega}$ is then naturally defined on $\Omega \times \Omega:$ For all $z\in \Omega$, $u\sb z(\zeta):= u\sb{\Omega}(z,\zeta)$ is the solution of the Dirichlet problem for the complex Monge-Amp\`ere equation $(dd\sp cu\sb z)\sp n=0$ on $\Omega\ssm\{z\}$ such that $$u\sb z(\zeta)=\log \vert \zeta - z\vert +O(1)$$ at $\zeta =z$; $u\sb{\Omega}$ is shown to be continuous outside the diagonal and invariant under biholomorphisms. Bedford and Taylor's Monge-Amp\`ere operators are used in conjunction with a general Lelong-Jensen formula previously found by the author [Mem. Soc. Math. Fr., Nouv. Ser. 19, 124 p. (1985; Zbl 0579.32012)] in order to construct an invariant pluricomplex Poisson kernel $d\mu\sb z(\zeta):= (2\pi)\sp{- n}(dd\sp cu\sb z(\zeta))\sp{n-1}\wedge d\sp cu\sb z(\zeta)\vert\sb{\partial \Omega},$ $(z,\zeta)\in \Omega \times \partial \Omega$. Each measure $\mu\sb z$ on $\partial \Omega$ is such that $\mu\sb z(V)=V(z)$ for every function V pluriharmonic on $\Omega$ and continuous on ${\bar \Omega}$; furthermore, $\mu\sb z$ is carried by the set of strictly pseudoconvex points of $\partial \Omega$ if ${\bar \Omega}$ has a $C\sp 2$ psh defining function. The principal part of the singularity of $d\mu\sb z(\zeta)$ on the diagonal of $\partial \Omega$ is then computed explicitly when $\Omega$ is strictly pseudoconvex, using an osculation of $\partial \Omega$ by balls. Through a complexification process, it is finally shown that Monge-Amp\`ere measures provide an explicit formula representing every point of a convex compact subset $K\subset {\bR}\sp n$ as a barycenter of the extremal points of K. \Cited in 7 Reviews \Cited in 72 Documents \MSC 32A25 Integral representation; canonical kernels (several complex variables)\\ 32C30 Integration on analytic sets and spaces, currents\\ 32F45 Invariant metrics and pseudodistances\\ 32U05 Plurisubharmonic functions and generalizations\\ 32C10 Pluriharmonic and plurisubharmonic functions\\ 32E10 Stein spaces, Stein manifolds \Keywords pluricomplex Green function; Lelong-Jensen formula; Monge-Amp\`ere measures; plurisubharmonic exhaustion function; hyperconvex domain; pluriharmonic measures; Choquet's theorem; barycentric representation \Full_text \References \[1] Bedford, E., Taylor, B.A.: The Dirichlet problem for the complex Monge-Amp?re equation. Invent. Math.37, 1-44 (1976) \. Zbl 0325.31013 \. DOI: 10.1007/BF01418826 \[2] Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math.,149 1-41 (1982) \. Zbl 0547.32012 \. DOI: 10.1007/BF02392348 \[3] Choquet, G.: Existence et unicité des représentations intégrales au moyen des points extrémaux dans les cônes convexes. Sém. Bourbaki, exposé no 139, 15 p. (Déc. 1956) \[4] Demailly, J.-P.: Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines, mémoire (nouvelle série) no 19, Soc. Math. de France, 1985 \[5] Diederich, K., Fornaess, J.E.: Pseudoconvex domains: bounded strictly plurisubharmonic functions. Invent. Math.39, 129-141 (1977) \. Zbl 0353.32025 \. DOI: 10.1007/ BF01390105 \[6] Fornaess, J.E.: Embedding strictly pseudoconvex domains in convex domains. Am. J. Math.98, 529-569 (1976) \. Zbl 0334.32020 \. DOI: 10.2307/2373900 \[7] Gamelin, T.W., Sibony, N.: Subharmonicity for uniform algebras. J. Funct. Anal.35, 64-108 (1980) \. Zbl 0422.46043 \. DOI: 10.1016/0022-1236(80)90081-6 \[8] Kerzman, N., Rosay, J.-P.: Fonctions plurisousharmoniques d'exhaustion born?es et domaines taut. Math. Ann.257, 171-184 (1981) \. Zbl 0461.32006 \. DOI: 10.1007/ BF01458282 \[9] Klimek, M.: Extremal plurisubharmonic functions and invariant pseudodistances. Bull. Soc. Math. France113, 123-142 (1985) \. Zbl 0584.32037 \[10] Lelong, P.: Fonctionnelles analytiques et fonctions enti?res (n variables). Presses de l'Univ. de Montr?al, S?m. de Math. Sup?rieures, ?t? 1967, no 28, Montr?al, 1968 \[11] Lempert, L.: La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. France109, 427-474 (1981) \. Zbl 0492.32025 \[12] Lempert, L.: Solving the degenerate Monge-Ampère equation with one concentrated singularity. Math. Ann.263, 515-532 (1983) \. Zbl 0531.35020 \. DOI: 10.1007/ BF01457058 \[13] Phelps, R.: Lectures on Choquet's theorem. Van Nostrand Math. Studies no 7, Princeton, New Jersey, 1966 \. Zbl 0135.36203 \[14] Richberg, R.: Stetige streng pseudokonvexe Funktionen. Math. Ann.175, 257-286 (1968) \. Zbl 0153.15401 \. DOI: 10.1007/BF02063212 \[15] Rudin, W.: Function theory in the unit ball of Cn. Grundlehren der math. Wissenschaften 241. Berlin Heidelberg New York: Springer 1980 \[16] Sibony, N.: Remarks on the Kobayashi metric, manuscrit, communication personnelle (juin 1986) \[17] Siciak, J.: On some extremal functions and their applications in the theory of analytic functions of several complex variables. Trans. Am. Math. Soc.105, 322-357 (1962) \. Zbl 0111.08102 \. DOI: 10.1090/S0002-9947-1962-0143946-5 \[18] Siciak, J.: Extremal plurisubharmonic functions in Cn. Ann. Pol. Math.39, 175-211 (1981) \. Zbl 0477.32018 \[19] Siciak, J.: Extremal plurisubharmonic functions and capacities in Cn. Sophia Kokyuroku in Math., Tokyo, 1982 \. Zbl 0579.32025 \[20] Stehlé, J.-L.: Fonctions plurisousharmoniques et convexité holomorphe de certains espaces fibrés analytiques. Sém. P. Lelong (Analyse) 1973/74, Lecture Notes in Math. no 474, 155-179 Berlin Heidelberg New York: Springer 1975 \[21] Taylor, B.A.: An estimate for an extremal plurisubharmonic function on Cn, S?m. P. Lelong, P. Dolbeault, H. Skoda (Analyse) 1982/83, Lecture Notes in Math. no 1028 318-328. Berlin Heidelberg New York: Springer 1983 \[22] Walsh, J.B.: Continuity of envelopes of plurisubharmonic functions. J. Math. Mech.18, 143-148 (1968) \. Zbl 0159.16002 \[23] Zeriahi, A.: Fonctions plurisousharmoniques extrémales, approximation et croissance des fonctions holomorphes sur des ensembles algébriques. Thèse de Doctorat-ès-Sciences, Univ. de Toulouse, 1986 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Mesures de Monge-Amp\`ere et mesures pluriharmoniques. (Monge-Amp\`ere measures and pluriharmonic measures). (French) Zbl 0602.31006 \Publi S\'emin., \'Equations D\'eriv. Partielles 1985-1986, Expos\'e No.19, 15 p. (1986). This work develops the potential theory of several complex variables in the form introduced by {\it E. Bedford} and {\it B. A. Taylor} [Invent. Math. 37, 1-44 (1976; Zbl 0315.31007); Acta Math. 149, 1-40 (1982; Zbl 0547.32012)]. In particular the author introduces a pluricomplex Green function for every bounded hyperconvex domain $\Omega$ in a Stein manifold. The Green function, with pole $z\in \Omega$, is a solution $u\sb z$ of the Dirichlet problem for the complex Monge-Amp\`ere operator with logarithmic pole at $z$ and boundary values $0$ on $\partial \Omega$. Using the functions $u\sb z$, the author also constructs pluriharmonic measures $\mu\sb z$ on $\partial \Omega$ which have properties analogous to those of harmonic measure. \par The paper concludes with an application to the geometry of convex sets: if $K$ is a compact convex subset of $R\sp n$, then through a complexification procedure, the measures $\mu\sb z$ provide a formula which allows every point of $K$ to be represented as a barycentre of the extremal points of K. \par Another paper of the author, with the same title [Math. Z. (to appear; Zbl 0595.32006)], gives a more detailed account of this work. \Reviewer D.Armitage \Cited in 2 Documents \MSC 31C10 Pluriharmonic and plurisubharmonic functions\\ 32E10 Stein spaces, Stein manifolds\\ 32U05 Plurisubharmonic functions and generalizations \Keywords several complex variables; pluricomplex Green function; hyperconvex domain; Stein manifold; Dirichlet problem; complex Monge-Amp\`ere operator; logarithmic pole; pluriharmonic measures \Author Demailly, Jean-Pierre \Title Sur l’identité de Bochner-Kodaira-Nakano en géométrie hermitienne. (The Bochner-Kodaira-Nakano identity in hermitian geometry). (French) Zbl 0594.32031 \Publi Sémin. analyse P. Lelong - P. Dolbeault - H. Skoda, Années 1983/84, Lect. Notes Math. 1198, 88-97 (1986). \[For the entire collection see Zbl 0583.00011.] It is obtained a generalized Kodaira-Nakano identity, relating the holomorphic and anti-holomorphic Laplace-Beltrami operators of a holomorphic hermitian vector bundle over a complex hermitian manifold. \Reviewer A.Pankov \Cited in 4 Documents \MSC 32L05 Holomorphic fiber bundles and generalizations\\ 32Q99 Complex manifolds 53C55 Hermitian and Kählerian manifolds (global differential geometry) \Keywords Bochner-Kodaira-Nakano identity; Laplace-Beltrami operators; holomorphic hermitian vector bundle; complex hermitian manifold \Author Demailly, Jean-Pierre \Title Un exemple de fibré holomorphe non de Stein à fibre $\bC2$ au-dessus du disque ou du plan. (An example of a non-Stein holomorphic fiber bundle over the disk or the plane, with fiber $\bC^2$). (French) Zbl 0594.32030 \Publi Sémin. analyse P. Lelong - P. Dobeault - H. Skoda, Années 1983/84, Lect. Notes Math. 1198, 98-104 (1986). \[For the entire collection see Zbl 0583.00011.] It is constructed a simple example of a non-Stein holomorphic fiber bundle over the disk with fiber $\bC^2$. Moreover, it is shown that all holomorphic functions on the bundle arise from functions on the base. \Reviewer A.Pankov \Cited in 1 Document \MSC 32L05 Holomorphic fiber bundles and generalizations\\ 32E10 Stein spaces, Stein manifolds \Keywords non-Stein holomorphic fiber bundle \Author Demailly, Jean-Pierre \Title Fonction de Green pluricomplexe et mesures pluriharmoniques. (French)\\ Zbl 0900.31004 \Publi Séminaire de théorie spectrale et géométrie. Année 1985-1986. Chambéry: Univ. de Savoie, Fac. des Sciences, Service de Math. Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 4, 131-143 (1986). From the introduction: ``L’objet de cet exposé est de montrer comment à un domaine pseudoconvexe $\Omega$ relativement compact dans une variété complexe on peut associer une fonction de Green généralisée, invariante par biholomorphisme. Cette fonction est définie comme la solution $u_z(\zeta)$ du problème de Dirichlet pour l’équation de Monge-Ampère complexe $(dd^cu_z)^n=0$ sur $\Omega\ssm\{z\}$, ayant un pôle logarithmique au point $z$.'' Le présent exposé est une version condensée de l’article [Math. Z. 194, 519-564 (1987; Zbl 0595.32006)], où le lecteur trouvera des démonstrations détaillées de tous les résultats mentionnés ici. For the entire collection see [Zbl 0825.00039]. \MSC 31C10 Pluriharmonic and plurisubharmonic functions\\ 32T99 Pseudoconvex domains\\ 32A25 Integral representation; canonical kernels (several complex variables) \Keywords generalized Green function; Monge-Ampère equation; logarithmic pole \Full_text \Author Demailly, Jean-Pierre \Title Champs magnétiques et inégalités de Morse pour la $d''$-cohomologie. (Magnetic fields and Morse inequalities for $d''$-cohomology). (French) Zbl 0595.58014 \Publi C. R. Acad. Sci., Paris, Sér. I 301, 119-122 (1985). This is an announcement of the paper which appeared under the same title in Ann. Inst. Fourier 35, No.4, 189-229 (1985; Zbl 0565.58017). If $E$ is a Hermitian line bundle over a compact complex manifold $X$, $ic(E)$ the curvature form of the canonical connection of $E$, $F$ a holomorphic fiber bundle of rank $r$ over $X$, $X(q)$ the set of points where $ic(E)$ has index $q$, the author proves that $h^qq_k=\dim H^q(X,E^k\otimes F)$ satisfies as $k\to+\infty$ the asymptotic Morse inequality $$ h^q_k\leq r{k^n\over n!}\int_{X(q)}(−1)^q(ic(E)/2\pi)^n+o(k^n). $$ An analogous inequality holds for $\sum_{j=0}^q(−1)^jh^j_k$. As an application, the author obtains geometric characterizations of Moishezon spaces, improving recent results of Y. T. Siu. \Reviewer G.Roos \Cited in 2 Reviews \Cited in 3 Documents \MSC 58E35 Variational inequalities (global problems)\\ 32J25 Transcendental methods of algebraic geometry\\ 32L10 Sections of holomorphic vector bundles \Keywords $\overline\partial$-cohomology; Schrödinger operator; Morse inequality; Moishezon spaces \Author Demailly, Jean-Pierre \Title Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques\break affines. (French) Zbl 0579.32012 \Publi Mém. Soc. Math. Fr., Nouv. Sér. 19, 124 p. (1985). Let $X$ be an irreducible $n$-dimensional Stein space and $\varphi : X\to[−\infty,R[$ a continuous psh (plurisubharmonic) exhaustion function. Each level set $S(r)=\{x\in X\,;\;\varphi(x)=r\}$, $rr_0\}$, the function $r\to\mu_r(V)$ is shown to be convex and increasing in the interval $]r_0,R[$. Furthermore, if the volume $\tau(r)=\int_{\varphi1$), that the cohomology class of $T$ in $\Omega$ is $0$ and that the trace measure $\sigma_T(z,r)$ of the ball $B(z,r)$ satisfies $$\sup_{z\in K}\int_0^\varepsilon{\sigma_T(z,r)\over r^{2n−1}}dr<+\infty\quad \hbox{resp.}\quad\int_0^\varepsilon\bigg(\sup_{z\in K}\big(\sigma_T(z,r)\big)^{1/2}/r^n\bigg)dr<+\infty, \leqno(*) $$ for every $K\subset\!\subset\Omega$ and $\varepsilon>0$ small enough. Then for every $\omega_1\subset\!\subset\omega_2\subset\!\subset\Omega$, one proves the existence of a closed positive current $\Theta$ in $\bC^n$ which is equal to $T$ in $\omega_1$ and $C^\infty$-smooth in $\bC^n\ssm \overline\omega_2$. Conversely, using the Skoda-El Mir structure theorems for closed${}\geq 0$ currents, one constructs various counterexamples to the extension problem. When $T$ has bidegree $(1,1)$, one shows that $(*)$ is essentially the best possible condition allowing extension, whereas in bidegree $(q,q)$ there exists a current $T$ such that $\sup_{z\in K}\sigma_T(z,r)\leq C_Kr^{2n−2q−1}$, whose singularities propagate up to $\partial\Omega$ along $(2n-2q-1)$-CR submanifolds. This last example rests upon the existence of totally real complete pluripolar $(n-1)$-submanifolds in $\bC^n$, a result due to Diederich-Fornaess that we reprove in a new and simpler way. \Cited in 1 Review \Cited in 3 Documents \MSC 32C30 Integration on analytic sets and spaces, currents\\ 32D15 Continuation of analytic objects (several variables) \Keywords global extension of closed positive current; propagation of singularities; density condition; bounded Runge open subset \Full_text \References \[1] Demailly, J. P., Courants positifs extrémaux et conjecture de Hodge;Inv. Math.,69, (1982), 347--374. \. Zbl 0488.58001 \. DOI: 10.1007/BF01389359 \[2] Diederich, K. andFornaess, J. E., Smooth, but not complex analytic pluripolar sets; Manuscripta Math. 37, (1982), 121--125. \. Zbl 0483.32012 \. DOI: 10.1007/BF01239949 \[3] El Mir, H Théorèmes de prolongement des courants positifs fermés; Thèse de Doctorat d’Etat soutenue à l’Université de Paris VI, novembre 1982; Acta Math. 153 (1984). \[4] Lelong, P.,Fonctions plurisousharmoniques et formes différentielles positives; Gordon and Breach, New York, et Dunod, Paris (1967). \[5] Siu, Y. T., Analyticity of sets associated to Lelong numbers and the extension of closed positive currents;Inv. Math. 27, pp. 53--156 (1974). \. Zbl 0289.32003 \. DOI: 10.1007/BF01389965 \[6] Skoda, H., Prolongement des courants positifs fermés de masse finie;Inv. Math. 66, pp. 361--376 (1982). \. Zbl 0488.58002 \. DOI: 10.1007/BF01389217 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Champs magnétiques et inégalités de Morse pour la $d''$-cohomologie. (Magnetic fields and Morse inequalities for $d''$-cohomology). (French) Zbl 0565.58017 \Publi Ann. Inst. Fourier 35, No.4, 189-229 (1985). Nous démontrons des inégalités de Morse-Witten asymptotiques pour la dimension des groupes de cohomologie des puissances tensorielles d’un fibré holomorphe en droites hermitien au-dessus d’une variété \bC-analytique compacte. La dimension du q-ième groupe de cohomologie se trouve ainsi majorée par une intégrale de courbure intrinsèque, étendue à l’ensemble des points d’indice q de la forme de courbure du fibré. La preuve repose sur un théorème spectral qui décrit la distribution asymptotique des valeurs propres de l’opérateur de Schrödinger associé à un champ magnétique assez grand. Comme application, nous obtenons une nouvelle démonstration de la conjecture de Grauert-Riemenschneider sur la caractérisation des espaces de Moišezon, résolue récemment par Siu, sous des hypothèses géométriques plus générales qui n’exigent pas nécessairement la semi-positivité ponctuelle du fibré. \Cited in 14 Reviews \Cited in 38 Documents \MSC 58E05 Abstract critical point theory \Keywords cohomology; Morse inequalities; curvature; Schrödinger operator for magnetic field \Full_text \Author Demailly, Jean-Pierre \Title Sur les transformées de Fourier de fonctions continues et le théorème de de Leeuw-Katznelson-Kahane. (On Fourier transforms of continuous functions and a theorem of de Leeuw-Katznelson-Kahane). (French) Zbl 0571.43003 \Publi C. R. Acad. Sci., Paris, Sér. I 299, 435-438 (1984). Given any locally compact abelian group $G$ and any function $\varphi\in L^2(\widehat G)$, we prove the existence of a function $f\in L^2(G)$ continuous and vanishing at infinity such that $|\widehat f|\geq|\varphi|$ a.e.\ on $\widehat G$. \MSC 43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups \Keywords Fourier transforms; locally compact abelian group \Author Demailly, Jean-Pierre \Title Sur la propagation des singularités des courants positifs fermés.$\,$(English)$\,$Zbl$\,$0551.32009 \Publi Analyse complexe, Proc. Journ. Fermat - Journ. SMF, Toulouse 1983, Lect. Notes Math. 1094, 53-64 (1984). \[For the entire collection see Zbl 0539.00009.] Let $T$ be a closed positive current in a bounded Runge open subset $\Omega\subset\bC^n$. We study sufficient conditions to be verified by the mass densities of $T$ in order that there exists a global extension of $T$ to $\bC^n$. Assume that the cohomology class of $T$ in $\Omega$ is $0$ and that the trace measure $\sigma_T$ satisfies $$ \int_0^{\delta/2}\bigg(\sup_{z\in\Omega_\delta\ssm\Omega_\varepsilon}\big(\sigma_T(z,r)\big)^{1\over 2}/r^n\bigg)dr<+\infty\leqno(*) $$ for some $\varepsilon>\delta>0$, where $\Omega\varepsilon=\{z\in\Omega\,;\;d(z,\bC^n\ssm\Omega)>\varepsilon\}$. Then we prove the existence of a closed positive current $\Theta$ in $\bC^n$ which is equal to $T$ in $\Omega_\varepsilon$ and $C^\infty$-smooth in $\bC^n\ssm\overline\Omega_\delta$. Thus, there is no propagation of singularities in that case. Conversely, using the Skoda-El Mir structure theorems for closed${}\geq 0$ currents and a result of Diederich- Fornaess on the existence of totally real complete pluripolar $(n-1)$-submanifolds in $\bC^n$, we construct specific counterexamples to the extension problem with density bounds. When $T$ has bidegree $(1,1)$, we show that the best possible sufficient condition allowing extension is $$ \sup_{z\in\Omega_\delta\ssm\Omega_\varepsilon}\int_0^{\delta/2}\Big(\sigma_T(z,r)/r^{2n−1}\Big)dr<+\infty, \leqno(**) $$ whereas in bidegree $(q,q)$ there exist currents $T$ such that $\sup_{z\in\Omega_\delta}\sigma_T(z,r)\leq Cr^{2n−q−1}$, whose singularities propagate up to $\partial\Omega$ along $(2n-q-1)$-CR submanifolds. \MSC 32C30 Integration on analytic sets and spaces, currents\\ 32Sxx Singularities (analytic spaces) \Keywords extension problem; propagation of singularities; density condition; closed positive current; bounded Runge open subset \Author Demailly, Jean-Pierre; Gaveau, Bernard \Title Majoration statistique de courbure d’une variété analytique. (French) Zbl 0533.53054 \Publi Sémin. d’Analyse P. Lelong-P. Dolbeault-H. Skoda, Années 1981/83, Lect. Notes Math. 1028, 96-124 (1983). \[For the entire collection see Zbl 0511.00025.] Let $\Omega\subset\bC^n$ be a bounded strictly pseudoconvex open subset. Given an analytic map $F: \Omega\to\bC^p$, one studies the average growth of the Ricci curvature of the level varieties $X_\zeta=F^{−1}(\zeta)$. Especially, if $R=- {\rm Ricci}(X_\zeta)\geq 0$ denotes the positive Ricci $(1,1)$-form of $X_\zeta$, $\delta$ the distance to $\partial\Omega$ and $\alpha=dd^c|z|^2$, it is shown that the estimate $$ \int_{\zeta\in\bC^n}d\lambda(\zeta)\int_{X_\zeta}\delta^{p+q}\,[\log(1+1/\delta)]^{-q}R^q\wedge\alpha^{n−p−q}<+\infty $$ holds for every $q=0,1,...,n−p$ when $F$ is bounded. A more general estimate valid for any order of growth of $F$ is also given. The proof consists essentially in integrations by parts of positive currents, using the explicit expression of $R$ in terms of the derivatives of $F$. Denote now by $\Gamma=\det R$ the Gaussian curvature of a smooth complex hypersurface in $\bC^n$. $R$ is proved to verify the following Monge-Ampère equation: $$ i\partial\overline\partial\log\Gamma=−(n+1)R+2\pi[Z] $$ where $Z$ is the divisor of zeros of $\Gamma$. \MSC 53C55 Hermitian and Kählerian manifolds (global differential geometry) 53C65 Integral geometry\\ 32C30 Integration on analytic sets and spaces, currents \Keywords growth estimate of Ricci curvature of level variety; positive current; Monge-Ampère equation; strictly pseudoconvex set; Gaussian curvature of a smooth complex hypersurface \Author Demailly, Jean-Pierre \Title Sur la structure des courants positifs fermés. (French) Zbl 0529.32006 \Publi Inst. Elie Cartan, Univ. Nancy I 8, 52-62 (1983). \Show_scanned_page \MSC 32C30 Integration on analytic sets and spaces, currents 30C80 Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)\\ 14C30 Transcendental methods, Hodge theory, Hodge conjecture\\ 32A30 Generalizations of function theory to several variables \Keywords closed positive current; Lelong number; extremal current; Hodge conjecture; Lelong-Jensen formulas; plurisubharmonic weight; Schwarz lemma; integration currents \Author Demailly, Jean-Pierre \Title Constructibilité des faisceaux de solutions des systèmes différentiels holonomes.\\(D’après Masaki Kashiwara). (French) Zbl 0529.32003 \Publi Sémin. d’Analyse P. Lelong - P. Dolbeault - H. Skoda, Années 1981/83, Lect. Notes Math. 1028, 83-95 (1983). \Show_scanned_page \MSC 32A45 Hyperfunctions (complex analysis) 58J15 Relations with hyperfunctions (PDE on manifolds)\\ 14F10 Special sheaves; D-modules; Bernstein-Sato ideals and polynomials 58J10 Differential complexes; elliptic complexes\\ 32L05 Holomorphic fiber bundles and generalizations \Keywords microdifferential operator; coherent differential module; holonomic system; constructibility of extension sheaf; stratification; differential operators; Lagrangian characteristic variety; extension theorem; D-module \Author Demailly, Jean-Pierre \Title Estimations $L^2$ pour l’opérateur $\overline\partial$ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété Kählérienne complète. (French) Zbl 0507.32021 \Publi Ann. Sci. Éc. Norm. Supér. (4) 15, 457-511 (1982). \Show_scanned_page \Cited in 5 Reviews \Cited in 38 Documents \MSC 32L20 Vanishing theorems (analytic spaces)\\ 32L05 Holomorphic fiber bundles and generalizations 53C55 Hermitian and Kählerian manifolds (global differential geometry)\\ 32L10 Sections of holomorphic vector bundles\\ 32U05 Plurisubharmonic functions and generalizations \Keywords Kaehler manifold; vanishing theorems; plurisubharmonic weights; smoothing theorems; s-positive hermitian vector bundle; weakly pseudoconvex \Full_text \References \[1] C. A. BERENSTEIN and B. A. TAYLOR , Interpolation Problems in $\bC^n$ with Applications to Harmonic Analysis (à paraître au Journal d’Analyse Math. de Jérusalem). \. Zbl 0464.42003 \[2] E. BOMBIERI , Algebraic Values of Meromorphic Maps (Invent. Math., vol. 10, p. 267-287, 1970 et 11, p. 163-166, 1970 ). MR 46 \#5328 | Zbl 0214.33702 \. Zbl 0214.33702 \. DOI: 10.1007/BF01418775 \. EUDML: 142035 \[3] J. BRIANÇON , Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de $\bC^n$~; preprint de l’Université de Nice, février 1974 (non publié). MR 49 \#5394 \[4] J. BRIANÇON et H. SKODA , Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de $\bC^n$ (C. R. Acad. Sc., t. 278, série A, 1974 , p. 949-951). MR 49 \#5394 | Zbl 0307.32007 \. Zbl 0307.32007 \[5] J.-P. DEMAILLY , Scindage holomorphe d’un morphisme de fibrés vectoriels semi-positifs avec estimations $L^2$ (à paraître au Séminaire P. Lelong, H. Skoda, 1980 - 1981 ). Zbl 0481.32011 \. Zbl 0481.32011 \[6] J.-P. DEMAILLY , Relations entre les différentes notions de fibrés et de courants positifs (à paraître au Séminaire P. Lelong, H. Skoda, 1980 - 1981 ). Zbl 0481.32010 \. Zbl 0481.32010 \[7] J.-P. DEMAILLY et H. SKODA , Relations entre les notions de positivités de P. A. Griffiths et de S. Nakano pour les fibrés vectoriels [Séminaire P. Lelong, H. Skoda (Analyse), 19e année, 1978 - 1979 , Lecture Notes, n${}^\circ$~822, 1980 , Springer-Verlag, Berlin, Heidelberg, New York]. Zbl 0454.55011 \. Zbl 0454.55011 \[8] K. DIEDERICH und R. P. PFLUG , Über Gebiete mit vollständiger Kählermetrik\break (à paraître aux Math. Annalen). Zbl 0472.32011 \. Zbl 0472.32011 \. DOI: 10.1007/ BF01458284 \. EUDML: 163551 \[9] A. DOUADY et J.-L. VERDIER , Séminaire de Géométrie analytique , E.N.S., 1972 - 1973 , Différents aspects de la positivité (Astérisque, 17, 1974 , Société Mathématique de France). \[10] H. GRAUERT , Charakterisierung der Holomorphie-gebiete durch die vollständige kählersche Metrik (Math. Annalen, t. 131, 1956 , p. 38-75). MR 17,1072a | Zbl 0073.30203 \. Zbl 0073.30203 \. DOI: 10.1007/BF01354665 \. EUDML: 160478 \[11] R. E. GREENE and H. WU , $C^\infty$ Approximation of Convex, Subharmonic, and Plurisubharmonic Functions (Ann. scient. Éc. Norm. Sup., 4e série, t. 12, 1979 , p. 47 à 84). Numdam | MR 80m:53055 | Zbl 0415.31001 \. Zbl 0415.31001 \. NUMDAM: ASENS\_1979\_4\_12\_1\_47\_0 \. EUDML: 82031 \[12] P. A. GRIFFITHS , Hermitian Differential Geometry, Chern Classes and Positive Vector Bundles~; Global Analysis , Princeton University Press, 1969 , p. 185-251. MR 41 \#2717 | Zbl 0201.24001 \. Zbl 0201.24001 \[13] L. HÖRMANDER , $L^2$ Estimates and Existence Theorem for the $\overline\partial$-Operator (Acta Math., 113, 1965 , p. 89-152). Zbl 0158.11002 \. Zbl 0158.11002 \. DOI: 10.1007/ BF02391775 \[14] L. HÖRMANDER , An Introduction to Complex Analysis, in Several Variables~; Princeton, van Nostrand Company, 1966~; 2e édition, North-Holland/American Elsevier, 1973 . Zbl 0271.32001 \. Zbl 0271.32001 \[15] L. HÖRMANDER , Generators for Some Rings of Analytic Functions (Bull. Amer. Math. Soc., vol. 73, 1967 , p. 943-949). Article | MR 37 \#1977 | Zbl 0172.41701 \. Zbl 0172.41701 \. DOI: 10.1090/S0002-9904-1967-11860-3 \. http://minidml.mathdoc.fr/cgi-bin/location?id=00221347 \[16] B. JENNANE , Extension d’une fonction définie sur une sous-variété avec contrôle de la croissance [Séminaire P. Lelong-H. Skoda (Analyse), 17e année, 1976 - 1977 , Lecture Notes in Math., n${}^\circ$~694, Springer-Verlag, Berlin, Heidelberg, New York, 1978 ]. Zbl 0403.32008 \. Zbl 0403.32008 \[17] J. J. KELLEHER and B. A. TAYLOR , Finitely Generated Ideals in Rings of Analytic Functions (Math. Ann., band 193, heft 3, 1971 ). MR 46 \#2077 | Zbl 0207.12906 \. Zbl 0207.12906 \. DOI: 10.1007/BF02052394 \. EUDML: 182723 \[18] P. LELONG , Fonctionnelles analytiques et fonctions entières (n variables)~; Montréal, les Presses de l’Université de Montréal, 1968 (Séminaire de Mathématiques supé\-rieures, été 1967 , n${}^\circ$~28). Zbl 0194.38801 \. Zbl 0194.38801 \[19] J. LE POTIER , Annulation de la cohomologie à valeurs dans un fibré vectoriel holomorphe positif de rang quelconque (Math. Ann., t. 218, 1975 , p. 35-53). MR 52 \#6044 | Zbl 0313.32037 \. Zbl 0313.32037 \. DOI: 10.1007/BF01350066 \. EUDML: 162783 \[20] S. NAKANO , Vanishing Theorems for Weakly 1-Complete Manifolds II (Publ. R.I.M.S., Kyoto University, vol. 10, 1974 , p. 101). Article | MR 52 \#3617 | Zbl 0298.32019 \. Zbl 0298.32019 \. DOI: 10.2977/prims/1195192175 \.\\ http://minidml.mathdoc.fr/cgi-bin/location?id=00260309 \[21] R. RICHBERG , Stetige streng pseudokonvexe Funktionen (Math. Ann., t. 175, 1968 , p. 257-286). MR 36 \#5386 | Zbl 0153.15401 \. Zbl 0153.15401 \. DOI: 10.1007/ BF02063212 \. EUDML: 161667 \[22] H. SKODA , Application des techniques $L^2$ à la théorie des idéaux d’une algèbre de fonctions holomorphes avec poids (Ann. scient. Éc. Norm. Sup., t. 5, fasc. 4, 1972 , p. 545-579). Numdam | MR 48 \#11571 | Zbl 0254.32017 \. Zbl 0254.32017 \. NUMDAM: ASENS\_1972\_4\_5\_4\_545\_0 \. EUDML: 81906 \[23] H. SKODA , Formulation hilbertienne du Nullstellensatz dans les algèbres de fonctions holomorphes~; paru dans l’Analyse harmonique dans le domaine complexe, Lecture Notes in Math., n${}^\circ$~336, Springer-Verlag, Berlin, Heidelberg, New York, 1973 . MR 52 \#11114 | Zbl 0259.32004 \. Zbl 0259.32004 \[24] H. SKODA , Morphismes surjectifs et fibrés linéaires semi-positifs [Séminaire P. Lelong-H. Skoda (Analyse), 17e année, 196-77, Lecture Notes in Math., n${}^\circ$~694, Springer-Verlag, Berlin, Heidelberg, New York, 1978 ]. MR 80b:32027 | Zbl 0396.32009 \. Zbl 0396.32009 \[25] H. SKODA , Morphismes surjectifs de fibrés vectoriels semi-positifs (Annales scient. Éc. Norm. Sup., 4e série, t. 11, p. 577-611, 1978 ). Numdam | MR 80j:32047 | Zbl 0403.32019 \. Zbl 0403.32019 \. NUMDAM: ASENS\_1978\_4\_11\_4\_577\_0 \. EUDML: 82026 \[26] H. SKODA , Relèvement des sections globales dans les fibrés semi-positifs [Séminaire P. Lelong-H. Skoda (Analyse), 19e année, 1978 - 1979 , Lecture Notes in Math., n${}^\circ$~822, Springer-Verlag, Berlin, Heidelberg, New York, 1980 ]. Zbl 0443.32017 \. Zbl 0443.32017 \[27] H. SKODA , Estimations $L^2$ pour l’opérateur $\overline\partial$ et applications arithmétiques [Séminaire P. Lelong (Analyse), 16e année, 1975 - 1976 , p. 314-323, Lecture Notes in Math., n${}^\circ$~538, Springer-Verlag, Berlin, Heidelberg, New York, 1977 ]. Zbl 0363.32004 \. Zbl 0363.32004 \[28] H. SKODA , Sous-ensembles analytiques d’ordre fini ou infini dans $\bC^n$ (Bull. Soc. Math. Fr., t. 100, 1972 , p. 353-408). Numdam | MR 50 \#5004 | Zbl 0246.32009 \. Zbl 0246.32009 \. NUMDAM: BSMF\_1972\_\_100\_\_353\_0 \. EUDML: 87191 \[29] A. WEIL , Variétés kählériennes , Hermann, Paris, 1957 . Zbl 0137.41103 \. Zbl 0137.41103 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Formules de Jensen en plusieurs variables et applications arithmétiques. (French) Zbl 0493.32003 \Publi Bull. Soc. Math. Fr. 110, 75-102 (1982). \Show_scanned_page \Cited in 1 Review \Cited in 12 Documents \MSC 32A25 Integral representation; canonical kernels (several complex variables) 30C80 Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)\\ 32A15 Entire functions (several variables)\\ 32C30 Integration on analytic sets and spaces, currents\\ 32A30 Generalizations of function theory to several variables \Keywords Lelong numbers; entire functions; Schwarz lemma; algebraic values of meromorphic maps; zero sets of polynomials \Full_text \References \[1] BOMBIERI (E.) .-- Algebraic values of meromorphic maps , Inventiones Math., Vol. 10, 1970 , p. 267-287 et Vol. 11, 1970 , p. 163-166. Zbl 0214.33702 \. Zbl 0214.33702 \. DOI: 10.1007/BF01418775 \. EUDML: 142035 \[2] CHUDNOVSKY (G. V.) .-- Singular points on complex hypersurfaces and multidimensional Schwarz lemma , Séminaire Delange-Pisot-Poitou, 21e année, 1979 - 1980 , Progress in Math., no 12, p. 29-69, Marie-José BERTIN, éd., Boston, Basel, Stuttgart, Birkhäuser, 1981 . Zbl 0455.32004 \. Zbl 0455.32004 \[3] DEMAILLY (J.-P.) .-- Sur les nombres de Lelong associés à l’image directe d’un courant positif fermé , à paraître aux Ann. Inst. Fourier, t. 32, fasc. 2, 1982 . Numdam | MR 84k:32011 | Zbl 0457.32005 \. Zbl 0457.32005 \. DOI: 10.5802/aif.872 \. NUMDAM: AIF\_1982\_\_32\_2\_37\_0 \. EUDML: 74541 \[4] LELONG (P.) .-- Fonctions plurisousharmoniques et formes différentielles positives , Gordon and Breach, New York, et Dunod, Paris, 1967 . Zbl 0195.11603 \. Zbl 0195.11603 \[5] LELONG (P.) .-- Sur les cycles holomorphes à coefficients positifs dans C” et un complément au théorème de E. Bombieri , C. R. Math. Rep. Acad. Sc. Canada, vol. 1, no 4, 1979 , p. 211-213. MR 80j:32017 | Zbl 0421.32005 \. Zbl 0421.32005 \[6] MOREAU (J.-C.) .-- Lemmes de Schwarz en plusieurs variables et applications arithmétiques , Séminaire Pierre Lelong-Henri Skoda, Analyse, année 1978 - 1979 , p. 174-190~; Lectures Notes in Math., no 822, Springer Verlag, 1980 . Zbl 0452.10036 \. Zbl 0452.10036 \[7] NARASIMHAN (R.) .-- Introduction to analytic spaces , lecture Notes in Math., no 25, Springer Verlag, 1966 . MR 36 \#428 | Zbl 0168.06003 \. Zbl 0168.06003 \. DOI: 10.1007/BFb0077071 \[8] SKODA (H.) .-- Estimations $L^2$ pour l’opérateur $\overline\partial$ et applications arithmétiques , Séminaire Pierre Lelong, Analyse, année 1975 - 1976 , p. 314-323~; Lecture Notes in Math., no 538, Springer Verlag, 1977 . Zbl 0363.32004 \. Zbl 0363.32004 \[9] WALDSCHMIDT (M.) .-- Propriétés arithmétiques des fonctions de plusieurs variables (II) , Séminaire Pierre Lelong, Analyse, année 1975 - 1976 , p. 108-135~; Lecture Notes in Math., no 538, Springer Verlag, 1977 . Zbl 0363.32003 \. Zbl 0363.32003 \[10] WALDSCHMIDT (M.) .-- Nombres transcendants et groupes algébriques , Asté\-risque, no 69-70, 1979 . MR 82k:10041 | Zbl 0428.10017 \. Zbl 0428.10017 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Courants positifs extr\'emaux et conjecture de Hodge. (French) Zbl 0488.58001 \Publi Invent. Math. 69, 347-374 (1982). \Show_scanned_page \Cited in 1 Review \Cited in 9 Documents \MSC 58A25 Currents (global analysis) 58A14 Hodge theory (global analysis)\\ 14C30 Transcendental methods, Hodge theory, Hodge conjecture\\ 14C20 Divisors, linear systems, invertible sheaves \Keywords extremal elements on the cone of closed strongly positive currents which are not integration currents over analytic subsets; Hodge conjecture; approximation of currents of bidegree $(1,1)$ by irreducible divisors on projective varieties or Stein manifolds \Full_text \References \[1] Bourbaki, N.: Espaces vectoriels topologiques, chap. 1 et 2, Paris: Hermann, 1964 \. Zbl 0205.34302 \[2] Demailly, J.-P.: Construction d’hypersurfaces irréductibles avec lieu singulier donné dans $\bC^n$. Ann. de l’Inst. Fourier 30, (fasc. 3) 219-236 (1980) \[3] Federer, H.: Geometric measure theory, Band 153. Berlin, Heidelberg, New York: Springer 1969 \. Zbl 0176.00801 \[4] Grauert, H.: On Levi’s problem and the imbedding of real analytic manifolds. Ann. of Math.68, (no 2) 460-472 (1958) \. Zbl 0108.07804 \. DOI: 10.2307/1970257 \[5] Harvey, R.: Holomorphic chains and their boundaries. Proceedings of Symposia in pure Mathematics of the Amer. Math. Soc., held at Williamstown, vol.30, Part 1, pp. 309-382 (1975) \[6] Harvey, R., Knapp, A.W.: Positive $(p,p)$ forms, Wirtinger’s inequality and currents. Value distribution theory. Part A: Proc. Tulane Univ. Program on Value Distribution Theory in Complex Analysis and Related Topics in differential Geometry, 1972-1973; pp. 43-62, New York Dekker 1974 \. Zbl 0287.53046 \[7] Lelong, P.: Intégration sur un ensemble analytique complexe. Bull. Soc. Math. France85, 239-262 (1957) \. Zbl 0079.30901 \[8] Lelong, P.: Fonctions plurisousharmoniques et formes différentielles positives. New York: Gordon and Breach, Paris: distribué par Dunod Editeur, 1968 \. Zbl 0195.11603 \[9] Lelong, P.: Eléments extrémaux sur le cône des courants positifs fermés. Séminaire P. Lelong (Analyse), 12e année, 1971-1972, Lecture Notes in Math., vol. 332. Berlin, Heidelberg, New York: Springer 1972 \[10] Phelps, R.: Lectures on Choquet’s theorem. Princeton, New Jersey: Van Nostrand, 1966 \. Zbl 0135.36203 \[11] Skoda, H.: Prolongement des courants positifs fermés de masse finie. Invent. Math.66, 361-376 (1982) \. Zbl 0488.58002 \. DOI: 10.1007/BF01389217 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \Author Demailly, Jean-Pierre \Title Scindage holomorphe d’un morphisme de fibr\'es vectoriels semi-positifs avec estimations $L^2$. (French) Zbl 0481.32011 \Publi Sémin. P. Lelong - H. Skoda, Analyse, Années 1980/81, et: Les fonctions plurisousharmoniques en dimension finie ou infinie, Colloq. Wimereux 1981, Lect. Notes Math. 919, 77-107 (1982). \Show_scanned_page \Cited in 1 Review \Cited in 2 Documents \MSC 32L05 Holomorphic fiber bundles and generalizations 53C55 Hermitian and Kählerian manifolds (global differential geometry)\\ 32D15 Continuation of analytic objects (several variables)\\ 32T99 Pseudoconvex domains \Keywords holomorphic vector bundles; weakly pseudoconvex Kaehlerian manifold; holomorphic retraction; Hörmander-Bombieri-Skoda theorem; holomorphic splitting; extension problem \Author Demailly, Jean-Pierre \Title Relations entre les différentes notions de fibrés et de courants positifs. (French) Zbl 0481.32010 \Publi S\'emin. P. Lelong - H. Skoda, Analyse, Années 1980/81, et: Les fonctions plurisousharmoniques en dimension finie ou infinie, Colloq. Wimereux 1981, Lect. Notes Math. 919, 56-76 (1982). \Show_scanned_page \Cited in 2 Documents \MSC 32L05 Holomorphic fiber bundles and generalizations\\ 32C30 Integration on analytic sets and spaces, currents 58A25 Currents (global analysis) \Keywords semi-positive vector bundles; currents; hermitian holomorphic vector bundle; positive differential forms \Author Demailly, Jean-Pierre \Title Extremal positive currents and Hodge conjecture. (English) Zbl 0476.58001 \Publi Invent. Math. 69 (1982) 347-374 \Show_scanned_page \Cited in 2 Reviews \MSC 58A25 Currents (global analysis) 58A14 Hodge theory (global analysis)\\ 14C30 Transcendental methods, Hodge theory, Hodge conjecture\\ 14C20 Divisors, linear systems, invertible sheaves \Keywords extremal elements on the cone of closed strongly positive currents which are not integration currents over analytic subsets; Hodge conjecture; approximation of currents of bidegree (1,1) by irreducible divisors on projective varieties or Stein manifolds \Author Demailly, Jean-Pierre \Title Sur les nombres de Lelong associés à l’image directe d’un courant positif fermé. (French) Zbl 0457.32005 \Publi Ann. Inst. Fourier 32, No.2, 37-66 (1982). \Show_scanned_page \Cited in 2 Reviews \Cited in 9 Documents \MSC 32C30 Integration on analytic sets and spaces, currents 58A25 Currents (global analysis) \Keywords Jensen formula; Lelong numbers of closed positive current; multiplicity \Full_text \Author Demailly, Jean-Pierre; Skoda, Henri \Title Relations entre les notions de positivités de P.A. Griffiths et de S. Nakano pour les fibres vectoriels. (French) Zbl 0454.55011 \Publi Sémin. P. Lelong - H. Skoda, Analyse, Années 1978/79, Lect. Notes Math. 822, 304-309 (1980). \Show_scanned_page \Cited in 4 Documents \MSC 55R25 Sphere bundles; vector space bundles\\ 32L05 Holomorphic fiber bundles and generalizations\\ 32L15 Bundle convexity \Keywords positive holomorphic hermitian vector bundle in the sense of Griffiths; positive holomorphic hermitian vector bundle in the sense of Nakano \Author Demailly, Jean-Pierre \Title Construction d’hypersurfaces irréductibles avec lieu singulier donné dans $\bC^n$. (French) Zbl 0414.32004 \Publi Ann. Inst. Fourier 30, No.3, 219-236 (1980). \Show_scanned_page \Cited in 4 Documents \MSC 32Sxx Singularities (analytic spaces)\\ 14J17 Singularities of surfaces\\ 32A22 Nevanlinna theory (local); growth estimates; other inequalities (several complex variables)\\ 32C25 Analytic subsets and submanifolds \Keywords convolution ring; irreducible hypersurface; singular locus; transcendental Bezout problem; growth order \Full_text \Author Demailly, Jean-Pierre \Title Fonctions holomorphes à croissance polynomiale sur la surface d’équation $e^x+e^y=1$. (French) Zbl 0412.32007 \Publi Bull. Sci. Math., II. Ser. 103, 179-191 (1979). \Show_scanned_page \Cited in 1 Document \MSC 32A22 Nevanlinna theory (local); growth estimates; other inequalities (several complex variables)\\ 32U05 Plurisubharmonic functions and generalizations\\ 32A10 Holomorphic functions (several variables)\\ 32A07 Special domains in Cn (Reinhardt, Hartogs, circular, tube)\\ 32D15 Continuation of analytic objects (several variables)\\ 32E30 Holomorphic and polynomial approximation (several variables), Runge pairs, interpolation \Keywords holomorphic functions with polynomial growth; extension theorem for holomorphic functions; special curve; plurisubharmonic function \Author Demailly, Jean-Pierre \Title Fonctions holomorphes bornées ou à croissance polynomiale sur la courbe $e^x+e^y=1$. (French) Zbl 0409.32009 \Publi C. R. Acad. Sci., Paris, Sér. A 288, 39-40 (1979). \Show_scanned_page \MSC 32D15 Continuation of analytic objects (several variables)\\ 32A22 Nevanlinna theory (local); growth estimates; other inequalities (several complex variables)\\ 32U05 Plurisubharmonic functions and generalizations \Keywords Extension of Holomorphic Function; Plurisubharmonic Functions; Holomorphic Function of Polynomical Growth \Author Demailly, Jean-Pierre \Title Différents exemples de fibrés holomorphes non de Stein. (French) Zbl 0418.32011 \Publi Sémin. Pierre Lelong - Henri Skoda (Anal.), Année 1976/77, Lect. Notes Math. 694, 15-41 (1978). \Show_scanned_page \Cited in 4 Reviews \Cited in 2 Documents \MSC 32E10 Stein spaces, Stein manifolds\\ 32L05 Holomorphic fiber bundles and generalizations \Keywords Steinness; Serre problem; holomorphic fiber bundle \Author Demailly, Jean-Pierre \Title Un exemple de fibré holomorphe non de Stein à fibre $\bC^2$ ayant pour base le disque ou le plan. (French) Zbl 0372.32012 \Publi Invent. Math. 48, 293-302 (1978). \Show_scanned_page \Cited in 2 Reviews \Cited in 2 Documents \MSC 32L05 Holomorphic fiber bundles and generalizations\\ 32U05 Plurisubharmonic functions and generalizations \Full_text \References \[1] Demailly, J.-P.: Différents exemples de fibrés holomorphes non de Stein, à paraître au Séminaire Lelong 1976/1977 \[2] Hörmander, L.: An introduction to complex analysis in several variables. Second edition. North Holland Publishing Company, 1973 \. Zbl 0271.32001 \[3] Lelong, P.: Fonctionnelles analytiques et fonctions entières (n variables). Montréal, les Presses de l’Université de Montréal, 1968, séminaire de Mathématiques Supérieures, Eté 1967, no 28 \[4] Serre, J.-P.: Quelques problèmes globaux relatifs aux variétés de Stein. Colloque sur les fonctions de plusieurs variables. Bruxelles, 1953 \[5] Skoda, H.: Fibrés holomorphes à base et à fibre de Stein. C.R. Acad. Sc. de Paris, 16 mai 1977, A. 1159-1202 \. Zbl 0353.32032 \[6] Skoda, H.: Fibrés holomorphes à base et à fibre de Stein, Inventiones Mathematicae, p. 97-107, Vol. 43, Fasc. 2, 1977 \. Zbl 0365.32018 \. DOI: 10.1007/BF01390000 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching. \bigskip {\number\artind\ publications authored by Jean-Pierre-Demailly since 1978, including 8 books} \end