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\AN{1078.32014 (02190255)}{2005}
\AU{Demailly, Jean-Pierre; Eckl, Thomas; Peternell, Thomas}
\TI{Line bundles on complex tori and a conjecture of Kodaira.}
\LA{English}
\DT{J}
\SO{Comment. Math. Helv. 80, No.2, 229-242 (2005).  ISSN 1420-8946]
\par http://www.ems-ph.org/journals/cmh/cmh.php}
\ISSN{010-257}
\AB{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\'eron-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.}
\RV{Eberhard Oeljeklaus (Bremen)}
\CC{32J27
32G05
32Q15}
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%% BEGIN ITEM
\AN{1071.14013 (02135198)}{2004}
\AU{Demailly, Jean-Pierre}
\TI{On the geometry of positive cones of projective and K\"ahler varieties.}
\LA{English}
\DT{CA}
\SO{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).}
\AB{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.}
\CC{14C30
14C20
32J27}
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%% BEGIN ITEM
\AN{1064.32019 (02144175)}{2004}
\AU{Demailly, Jean-Pierre; Paun, Mihai}
\TI{Numerical characterization of the K\"ahler cone of a compact K\"ahler manifold.}
\LA{English}
\DT{J}
\SO{Ann. Math. (2) 159, No.3, 1247-1274 (2004). 
\par http://www.math.princeton.edu/~annals/issues/issues.html}
\ISSN{003-486}
\AB{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 $S\setminus S^\prime$ under parallel transport
    with respect to the $(1,1)$-projection $\nabla^{1,1}$ of the
    Gauss-Manin connection.}
\RV{Vasile Oproiu (Ia\c{s}i)}
\CC{32Q15
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%% BEGIN ITEM
\AN{1077.32504 (02171928)}{2003}
\AU{Demailly, Jean-Pierre; Peternell, Thomas}
\TI{A Kawamata-Viehweg vanishing theorem on compact K\"ahler manifolds.}
\LA{English}
\DT{J}
\SO{J. Differ. Geom. 63, No.2, 231-277 (2003). 
\par http://www.intlpress.com/journals/JDG/}
\ISSN{022-040}
\AB{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).}
\RV{Imre Patyi (Atlanta)}
\CC{32L20
32J27}
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%% BEGIN ITEM
\AN{1053.32011 (02070181)}{2003}
\AU{Demailly, Jean-Pierre; Peternell, Thomas}
\TI{A Kawamata-Viehweg vanishing theorem on compact K\"ahler manifolds.}
\LA{English}
\DT{CA}
\SO{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. Surv. Differ. Geom. 8, 139-169 (2003). [ISBN 1-57146-114-0/hbk]}
\AB{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)] under the same title. \par

    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 $L^2\not=0$. \par
 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 $H^0(D,(-L+D)|D)=0$, done by working with Hodge index inequalities.
    \par
 Then claim~0.1 is applied to abundance for threefolds
    given in claim~0.3: \par
 If $X$ is a ${\Bbb Q}$-Gorenstein
    K\"ahler threefold with only terminal singularities and $K_X$
    nef, then $\kappa(X)\ge 0$ for the Kodaira dimension. \par
The
    paper is informative and pleasant to read.}
\RV{Imre Patyi (Atlanta)}
\CC{32L20
32J27}
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%% BEGIN ITEM
\AN{1011.32019 (01803774)}{2002}
\AU{Demailly, Jean-Pierre}
\TI{On the Frobenius integrability of certain holomorphic $p$-forms.}
\LA{English}
\DT{CA}
\SO{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). [ISBN 3-540-43259-0/hbk]}
\AB{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.}
\CC{32Q15
32J25}
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%% BEGIN ITEM
\AN{0996.14003 (01757936)}{2002}
\AU{Bertin, Jos\'e; Demailly, Jean-Pierre; Illusie, Luc; Peters, Chris}
\TI{Introduction to Hodge theory. Transl. from the French by James Lewis and Chris Peters.}
\LA{English}
\DT{B}
\SO{SMF/AMS Texts and Monographs. 8. Providence, RI: American Mathematical Society (AMS). ix, 232 p. \$ 65.00 (2002). [ISBN 0-8218-2040-0/pbk;}
\ISSN{525-230}
\AB{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.}
\RV{Werner Kleinert (Berlin)}
\CC{14C30
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%% BEGIN ITEM
\AN{1066.32012 (02062445)}{2001}
\AU{Campana, Fr\'ed\'eric; Demailly, Jean-Pierre}
\TI{$L^2$ -cohomology on the coverings of a compact complex manifold.}
\LA{French}
\DT{J}
\SO{Ark. Mat. 39, No.2, 263-282 (2001). 
\par http://www.arkivformatematik.org/}
\ISSN{004-208}
\AB{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.}
\RV{A.Diaz-Cano (Madrid)}
\CC{32C35
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%% BEGIN ITEM
\AN{0994.32021 (01675991)}{2001}
\AU{Demailly, Jean-Pierre; Koll\'ar, J\'anos}
\TI{Semi-continuity of complex singularity exponents and K\"ahler-Einstein metrics on Fano orbifolds.}
\LA{English}
\DT{J}
\SO{Ann. Sci. \'Ec. Norm. Sup\'er., IV. S\'er. 34, No.4, 525-556 (2001). 
\par http://www.sciencedirect.com/science/journal/00129593}
\ISSN{012-959}
\AB{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 $\exp(-2 \subset\varphi_j) \to\exp(-2
    \subset\varphi)$ in $L^1$-norm over a neighborhood of $K$ for
    all positive $c<c_K (\varphi)$.\par
As a consequence, a comparatively
    simple proof is given for the existence of K\"ahler-Einstein
    metrics on certain Fano orbifolds. In this way, the authors produce
    three new examples of rigid del Pezzo surfaces with quotient
    singularities which admit a K\"ahler-Einstein metric.}
\RV{Alexandr Yu.Rashkovsky (Khar'kov)}
\CC{32S05
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32U25}
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%% BEGIN ITEM
\AN{0986.00053 (01746061)}{2001}
\AU{Demailly, J.P.(ed.); G\"ottsche, L.(ed.); Lazarsfeld, R.(ed.)}
\TI{School on vanishing theorems and effective results in algebraic geometry. Lecture notes of the school held in Trieste, Italy, April 25--May 12, 2000.}
\LA{English}
\DT{B}
\SO{ICTP Lecture Notes. 6. Trieste: The Abdus Salam International Centre for Theoretical Physics. vii, 393 p. (2001). [ISBN 92-95003-09-8/pbk]
\par http://www.ictp.trieste.it/~pub\_off/lectures/vol6.html \par http://www.ictp.trieste.it/~pub\_off/lectures/}
\AB{The articles of this volume will be reviewed individually.}
\CC{00B25
14-06}
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%% BEGIN ITEM
\AN{01816813}{2001}
\AU{Demailly, Jean-Pierre}
\TI{Multiplier ideal sheaves and analytic methods in algebraic geometry.}
\LA{English}
\DT{CA}
\SO{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. ICTP Lect. Notes 6, 1-148 (2001). [ISBN 92-95003-09-8/pbk]}
\AB{Review in preparation}
\CC{14F99
32G99}
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%% BEGIN ITEM
\AN{01911897}{2001}
\AU{Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael}
\TI{Pseudo-effective line bundles on compact K\"ahler manifolds.}
\LA{English}
\DT{J}
\SO{Int. J. Math. 12, No.6, 689-741 (2001). 
\par http://www.worldscinet.com/ijm/ijm.shtml}
\ISSN{129-167}
\AB{Review in preparation}
\CC{32J27
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%% BEGIN ITEM
\AN{01972760}{2001}
\AU{Demailly, Jean Pierre}
\TI{Refoundation of mathematics in France.}
\LA{Italian}
\DT{J}
\SO{Lett. Mat. Pristem 42, 10-14 (2001).  ISSN 1123-7198]}
\ISSN{593-588}
\AB{The young and (even more) promising author of the presente study,
    Jean-Piere Demailly, approaches the problem of mathematics refoundation
    from very (highly) French positions, by proundly asserting that
    the ``nouvelles maths'' have been initiated in Frace and nowhere
    else. \par
The role played in France by the ``commission for
    the long-sum reflexion on the teaching of mathematics'' is emphasized,
    and the name of ist president -- Jean-Piere Kahane -- is mentioned.
    \par
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, expressing the
    author's daring opinions, his diraporal of here???? didacticism
    and old-fashioned techniques still in force. \par
Extremely
    ``represhing'' for the reader is the idea of having illustrated
    the text with coloured, challenging reproductions of Picasso.
    The choice of unordinity is explicit.}
\RV{Cristina Irimia (Ia\c{s}i)}
\CC{01-XX}
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%% BEGIN ITEM
\AN{1077.14516 (01700866)}{2000}
\AU{Demailly, Jean-Pierre; Ein, Lawrence; Lazarsfeld, Robert}
\TI{A subadditivity property of multiplier ideals.}
\LA{English}
\DT{J}
\SO{Mich. Math. J. 48, Spec. Vol., 137-156 (2000). 
\par http://www.math.lsa.umich.edu/mmj/}
\ISSN{026-228}
\AB{Summary: Given an effective $\bbfQ$-divisor $D$ on a smooth complex
    variety, one can associate to $D$ its multiplier ideal sheaf
    $J(D)$, which measures in a somewhat subtle way the singularities
    of $D$. Because of their strong vanishing properties, these ideals
    have come to play an increasingly important role in higher dimensional
    geometry. We prove that for two effective $\bbfQ$-divisors $D$
    and $E$, one has the ``subadditivity" relation: $J(D + E) \subseteq
    J(D) . J(E)$. We also establish several natural variants, including
    the analogous statement for the analytic multiplier ideals associated
    to plurisubharmonic functions. As an application, we give a new
    proof of a theorem of {\it T. Fujita} [Kodai Math. J. 17, No.
    1, 1-3 (1994; Zbl 0814.14006)] concerning the volume of a big
    linear series on a projective variety. The first section of the
    paper contains an overview of the construction and basic properties
    of multiplier ideals from an algebro-geometric perspective, as
    well as a discussion of the relation between some asymptotic
    algebraic constructions and their analytic counterparts.}
\CC{14E99
14J17}
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%% BEGIN ITEM
\AN{0966.32014 (01484672)}{2000}
\AU{Demailly, Jean-Pierre; El Goul, Jawher}
\TI{Hyperbolicity of generic surfaces of high degree in projective 3-space.}
\LA{English}
\DT{J}
\SO{Am. J. Math. 122, No.3, 515-546 (2000). 
\par http://muse.jhu.edu/journals/american\_journal\_of\_mathematics/toc/ajm122.3.html}
\ISSN{002-932}
\AB{The main result of this paper is to prove that a very generic
    surface $X$ in ${\Bbb P}^3$ of degree $d \ge 21$ is Kobayashi
    hyperbolic, that is there is no nonconstant holomorphic map from
    ${\Bbb C} \rightarrow X$. As a consequence of the proof, they
    also prove that the complement of a very generic curve in ${\Bbb
    P}^2$ is hyperbolic and hyperbolically imbedded for all degrees
    $d \ge 21$. We note that previously, Siu-Yeung proved the hyperbolicity
    of the complement of a generic smooth curve of high degree in
    ${\Bbb P}^2$. The approach roughly is divided into the following
    steps: First use the Riemann-Roch calculations to prove the existence
    of suitable jet differentials which vanish on an ample divisor;
    then use Ahlfors-Schwarz lemma to conclude that the image of
    $f$ sits in the base locus of the global sections of jet differentials;
    finally, it is hoped to show, by analysing the base locus carefully,
    that the base locus actually is a proper subvariety of $X$.}
\RV{Min Ru (Houston)}
\CC{32Q45
32H30}
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%% BEGIN ITEM
\AN{0962.14014 (01464658)}{2000}
\AU{Demailly, Jean-Pierre}
\TI{M\'ethodes $L^2$ et r\'esultats effectifs en g\'eom\'etrie alg\'ebrique. ($L^2$ methods and effective results in algebraic geometry.).}
\LA{French}
\DT{CA}
\SO{S\'eminaire Bourbaki. Volume 1998/99. Expos\'es 850-864. Paris: Soci\'et\'e Math\'ematique de France, Ast\'erisque. 266, 59-90 [Expos\'e No.852] (2000). }
\ISSN{303-117}
\AB{The paper is a review of analytic methods ($L^2$ Hodge theory)
    used in algebraic geometry for studying adjoint linear systems,
    vanishing theorem for algebraic vector bundles and invariance
    of plurigenera of general type families. Among the topics discussed
    in the paper are singular metrics, applications to Fujita's conjecture
    [{\it T. Fujita} in: Algebraic Geometry, Proc. Symp., Sendai
    1985, Adv. Stud. Pure Math. 10, 167-178 (1987; Zbl 0659.14002)]
    on global generation of adjoint linear systems, and analytic
    tools in Siu's proof [{\it Y.-T. Siu}, Invent. Math. 134, No.
    3, 661-673 (1998; Zbl 0955.32017)] of invariance of plurigenera
    for a family of general type.}
\RV{Taras E.Panov (Moskva)}
\CC{14F43
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%% BEGIN ITEM
\AN{0959.32019 (01500726)}{2000}
\AU{Demailly, Jean-Pierre}
\TI{On the Ohsawa-Takegoshi-Manivel $L^2$ extension theorem.}
\LA{English}
\DT{CA}
\SO{Dolbeault, P. (ed.) et al., Complex analysis and geometry. Proceedings of the international conference in honor of Pierre Lelong on the occasion of his 85th birthday, Paris, France, September 22-26, 1997. Basel: Birkh\"auser. Prog. Math. 188, 47-82 (2000). [ISBN 3-7643-6352-5/hbk;}
\ISSN{743-164}
\AB{The Ohsawa-Takegoshi-Manivel $L^2$ extension theorem addresses
    the following basic problem: Let $Y$ be a complex analytic submanifold
    of a complex manifold $X$; given a holomorphic function $f$ on
    $Y$ satisfying suitable $L^2$ conditions on $Y,$ find a holomorphic
    extension $F$ of $f$ to $X,$ together with a good $L^2$ estimate
    for $F$ on $X.$ \par
The first satisfactory solution of this
    problem has been obtained by T.~Ohsawa and K.~Takegoshi. The
    author follows here a more geometric approach due to L.~Manivel,
    which provides a more general extension theorem in the framework
    of vector bundles and higher cohomology groups. The first goal
    of this note is to simplify further Manivel's approach, as well
    as to point out a technical difficulty in Manivel's proof. The
    author uses a simplified and slightly extended version of the
    original Ohsawa-Takegoshi a priori inequality. Then the Ohsawa-Takegoshi-Manivel
    extension theorem is applied to solve several important problems
    of complex analysis or geometry. The first of these is an approximation
    theorem for plurisubharmonic functions. It is shown that the
    approximation can be made with a uniform convergence of the Lelong
    numbers of the holomorphic functions towards those of the given
    plurisubharmonic function. This result contains as a special
    case Siu's theorem on the analyticity of Lelong number sublevel
    sets. By combining some of the results provided by the proof
    of that approximation theorem with Skoda's $L^2$ estimates for
    the division of holomorphic functions, a Brian\c{c}on-Skoda type
    theorem for Nadel's multiplier ideal sheaves is obtained. Using
    this result and some ideas of R.~Lazarsfeld, it is obtained a
    new proof of a recent result of T.~Fujita: the growth of the
    number of sections of multiples of a big line bundle is given
    by the highest power of the first Chern class of the numerically
    effective part in the line bundle Zariski decomposition.}
\RV{A.V.Chernecky (Odessa)}
\CC{32D15
32U05}
%% END ITEM
%% BEGIN ITEM
\AN{0960.32011 (01440960)}{1999}
\AU{Demailly, Jean-Pierre}
\TI{Pseudoconvex-concave duality and regularization of currents.}
\LA{English}
\DT{CA}
\SO{Schneider, Michael (ed.) et al., Several complex variables. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 37, 233-271 (1999). [ISBN 0-521-77086-6/hbk;}
\ISSN{940-474}
\AB{The paper investigates some basic properties of Finsler metrics
    on holomorphic vector bundles in the perspective of obtaining
    geometric versions of the Serre duality theorem. A duality framework
    under which pseudo-convexity and pseudo-concavity properties
    get exchanged is established. These duality properties are related
    to several geometric problems, e.g., the conjecture of Hartshorne
    and Schneider.\par
Finally, a new shorter and more geometric
    proof of a basic regularization theorem for closed $(1,1)$-currents
    is shown.}
\RV{Viorel V\^aj\^aitu (Bucure\c{s}ti)}
\CC{32F10
32C30
32J25}
%% END ITEM
%% BEGIN ITEM
\AN{0910.32032 (01160673)}{1998}
\AU{Campana, Fr\'ed\'eric; Demailly, Jean-Pierre; Peternell, Thomas}
\TI{The algebraic dimension of compact complex threefolds with vanishing second Betti number.}
\LA{English}
\DT{J}
\SO{Compos. Math. 112, No.1, 77-91 (1998). 
\par http://www.springerlink.com/openurl.asp?genre=journal\&issn=0010-437X}
\ISSN{010-437}
\AB{The abstract of the authors describes the content of the paper
    quite precisely. It reads (with very small changes) as follows:
    ``This note investigates compact complex manifolds $X$ of dimension
    three with second Betti number $b_2=0$. If $X$ admits a nonconstant
    meromorphic function, then the authors prove that either $b_1(X)=1$
    and $b_3 (X)=0$ or that $b_1(X) =0$ and $b_3(X) =2$. The main
    idea is to show that $c_3(X) =0$ by means of a vanishing theorem
    for generic line bundles on $X$. As a consequence a compact complex
    threefold homeomorphic to the 6-Sphere $S^6$ cannot admit a non-constant
    meromorphic function. Furthermore they investigate the structure
    of threefolds with $b_2=0$ and algebraic dimension one, in the
    case when the algebraic reduction $X\to \bbfP_1$ is holomorphic''.}
\RV{E.Oeljeklaus (Bremen)}
\CC{32J17
14C20}
%% END ITEM
%% BEGIN ITEM
\AN{0937.32012 (01414080)}{1997}
\AU{Demailly, Jean-Pierre}
\TI{Vari\'et\'es projectives hyperboliques et \'equations diff\'erentielles alg\'ebriques. (Hyperbolic projective varieties and algebraic differential equations).}
\LA{French}
\DT{CA}
\SO{Hirzebruch, Friedrich et al., Journ\'ee en l'honneur de Henri Cartan. Paris: Soci\'et\'e Math\'ematique de France, SMF Journ. Annu. 1997, 3-17 (1997).}
\AB{From the introduction (translated from the French): ``The aim
    of the text is to offer an introduction, as elementary as possible,
    to an important result concerning the geometry of the images
    of holomorphic curves in complex algebraic varieties''.}
\CC{32Q45
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%% END ITEM
%% BEGIN ITEM
\AN{0932.00086 (01403899)}{1997}
\AU{Hirzebruch, Friedrich; Demailly, Jean-Pierre; Lannes, Jean}
\TI{Journ\'ee en l'honneur de Henri Cartan. (Conference in honor of Henri Cartan).}
\LA{French}
\DT{B}
\SO{SMF Journ\'ee Annuelle. 1997. Paris: Soci\'et\'e Math\'ematique de France, iv, 27 p. (1997).}
\AB{The articles of this volume will be reviewed individually. Contents:
    F. Hirzebruch, Learning complex analysis in M\"unster--Paris,
    Z\"urich and Princeton from 1945 to 1953 (1-2), Jean-Pierre Demailly,
    Vari\'et\'es projectives hyperboliques et \'equations diff\'erentielles
    alg\'ebriques (French) (Hyperbolic projective varieties and algebraic
    differential equations) (3-17), Jan Lannes, Divers aspects des
    op\'erations Steenrod (French) (Diverse aspects of the Steenrod
    operations) (18-27).}
\CC{00B30
00B15
32-06}
%% END ITEM
%% BEGIN ITEM
\AN{0919.32014 (01126802)}{1997}
\AU{Demailly, Jean-Pierre}
\TI{Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials.}
\LA{English}
\DT{CA}
\SO{Koll\'ar, J\'anos (ed.) et al., Algebraic geometry. Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9--29, 1995. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62(pt.2), 285-360 (1997). [ISBN 0-8218-0895-8/hbk]}
\AB{This are notes of a series of lectures delivered at the Santa
    Cruz AMS Summer School on Algebraic Geometry. They are mainly
    devoted to the study of complex varieties through a few geometric
    questions related to hyperbolicity in the sense of Kobayashi.
    A convenient framework for this is the category of ``directed
    manifolds'', that is, the category of pairs $(X,V)$ where $X$
    is a complex manifold and $V$ a holomorphic subbundle of $T_X$.
    If $X$ is compact, the pair $(X,V)$ is hyperbolic if and only
    if there are no nonconstant entire holomorphic curves $f:\bbfC\to
    X$ tangent to $V$ (Brody's criterion). The author describes a
    construction of projectivized $k$-jet bundles $P_kV$, which generalizes
    a construction made by Semple in 1954 and allows to analyze hyperbolicity
    in terms of negativity properties of the curvature.\par
An overview
    information on the lecture notes is given by their contents.\par
1.
    Hyperbolicity concepts and directed manifolds\par
2. Hyperbolicity
    and bounds for the genus of curves\par
3. The Ahlfors-Schwarz
    lemma for metrics of negative curvature\par
4. Projectivization
    of a directed manifold\par
5. Jets of curves and semple jet
    bundles\par
6. Jet differentials\par
7. $k$-Jet metrics with
    negative curvature\par
8. Algebraic criterion for the negativity
    of jet curvature\par
9. Proof of the Bloch theorem\par
10.
    Logarithmic jet bundles and a conjecture of Lang\par
11. Projective
    meromorphic connections and Wronskians \par
12. Decomposition
    of jets in irreducible representations\par
13. Riemann-Roch
    calculations and study of the base locus\par
14. Appendix: A
    vanishing theorem for holomorphic tensor fields.}
\RV{J.Eichhorn (Greifswald)}
\CC{32Q45
32L10
14J40
53C55}
%% END ITEM
%% BEGIN ITEM
\AN{0901.32019 (01086021)}{1997}
\AU{Demailly, Jean-Pierre}
\TI{Vari\'et\'es hyperboliques et \'equations diff\'erentielles alg\'ebriques. (Hyperbolic varieties and algebraic differential equations).}
\LA{French}
\DT{J}
\SO{Gaz. Math., Soc. Math. Fr. 73, 3-23 (1997). }
\ISSN{224-899}
\AB{In this survey article, the author presents the relationship
    between the existence of entire curves (i.e. holomorphic curves
    $f:\bbfC \to X)$ 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 $f:\bbfC \to X$
    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.}
\RV{Vasile Br\^{\i}nz\u{a}nescu (Bucure\c{s}ti)}
\CC{32Q45
32-02
32H30
32A22}
%% END ITEM
%% BEGIN ITEM
\AN{0898.32016 (01057186)}{1997}
\AU{Demailly, Jean-Pierre; El Goul, Jawher}
\TI{Connexions m\'eromorphes projectives partielles et vari\'et\'es alg\'ebriques hyperboliques. (Meromorphic partial projective connections and hyperbolic projective varieties).}
\LA{French. Abridged English version}
\DT{J}
\SO{C. R. Acad. Sci., Paris, S\'er. I 324, No.12, 1385-1390 (1997). 
\par http://www.sciencedirect.com/science/journal/07644442}
\ISSN{764-444}
\AB{{\it S. Kobayashi} conjectured in [Hyperbolic manifolds and holomorphic
    mappings, Marcel Dekker, NY (1970; Zbl 0207.37902)] that a generic
    hypersurface of ${\bf C}{\bf P}^n$ of sufficiently high degree
    $d$ (where the expected bound is $d\ge 2n-1$) is hyperbolic.
    The conjecture is true for ${\bf C}{\bf P}^2$, but for $n\ge
    3$ a few number of examples are known. For ${\bf C}{\bf P}^3$
    (where the expected bound is $5$) the first example of a smooth
    hyperbolic surface in ${\bf C}{\bf P}^3$ 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 ${\bf C}{\bf P}^3$ with arbitrary degree $d\ge 11$.}
\RV{Raul Iba\~nez (Bilb\~ao)}
\CC{32Q45
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53A20
53C55}
%% END ITEM
%% BEGIN ITEM
\AN{1008.32008 (00885323)}{1996}
\AU{Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael}
\TI{Compact K\"ahler manifolds with Hermitian semipositive anticanonical bundle.}
\LA{English}
\DT{J}
\SO{Compos. Math. 101, No.2, 217-224 (1996). 
\par http://www.springerlink.com/openurl.asp?genre=journal\&issn=0010-437X}
\ISSN{010-437}
\AB{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.}
\CC{32J27
53C55}
%% END ITEM
%% BEGIN ITEM
\AN{0883.14005}{1996}
\AU{Demailly, Jean-Pierre}
\TI{$L\sp 2$ vanishing theorems for positive line bundles and adjunction theory.}
\LA{English}
\DT{CA}
\SO{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). [ISBN 3-540-62038-9/pbk;}
\ISSN{075-843}
\AB{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.}
\RV{N.Bla\v{z}i\'c (Beograd)}
\CC{14F17
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14F05
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%% END ITEM
%% BEGIN ITEM
\AN{0869.65041}{1996}
\AU{Demailly, Jean-Pierre}
\TI{Analyse num\'erique et \'equations diff\'erentielles. (Numerical analysis and differential equations).Nouvelle \'ed.}
\LA{French}
\DT{B}
\SO{Grenoble: Presses Univ. de Grenoble. 309 p. (1996). [ISBN 2-7061-0715-4]}
\AB{See the review of the German translation (1994; Zbl 0869.65042).}
\CC{65L05
65L06
65D32
65H10
65-01
34-01}
%% END ITEM
%% BEGIN ITEM
\AN{0862.14004}{1996}
\AU{Demailly, Jean-Pierre}
\TI{Effective bounds for very ample line bundles.}
\LA{English}
\DT{J}
\SO{Invent. Math. 124, No.1-3, 243-261 (1996). 
\par http://link.springer.de/cgi/linkref?issn=0020-9910\&year=1996\&volume=124\&page=243}
\ISSN{020-991}
\AB{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.}
\RV{I.Coand\u{a} (Bucure\c{s}ti)}
\CC{14C20
14F05
14F17
32C20}
%% END ITEM
%% BEGIN ITEM
\AN{0859.14005}{1996}
\AU{Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael}
\TI{Holomorphic line bundles with partially vanishing cohomology.}
\LA{English}
\DT{CA}
\SO{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).}
\AB{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 $\bbfP_{n-1}$ 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 $NX(X) \otimes\bbfR$, $NX(X)$ being N\'eron-Severi group,
    i.e. the group of divisors modulo numerical equivalence. All
    these cones coincide for surfaces.}
\RV{U.N.Bhosle (Bombay)}
\CC{14F17
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14F05
14C22}
%% END ITEM
%% BEGIN ITEM
\AN{0849.14002}{1996}
\AU{Bertin, Jos\'e; Demailly, Jean-Pierre; Illusie, Luc; Peters, Chris}
\TI{Introduction \`a la th\'eorie de Hodge. (Introduction to Hodge theory).}
\LA{French. English summary}
\DT{B}
\SO{Panoramas et Synth\`eses. 3. Paris: Soci\'et\'e Math\'ematique de France. vi, 272 p. (1996). [ISBN 2-85629-049-3]}
\AB{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.}
\RV{W.Kleinert (Berlin)}
\CC{14C30
14F17
14-02
14D07
13A35
58A14
14D05
81T30
14J32}
%% END ITEM
%% BEGIN ITEM
\AN{0880.14003}{1995}
\AU{Demailly, Jean-Pierre}
\TI{Compact complex manifolds whose tangent bundles satisfy numerical effectivity properties (joint work with Thomas Peternell and Michael Schneider).}
\LA{English}
\DT{CA}
\SO{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). [ISBN 0-19-563740-2/hbk]}
\AB{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 $\bbfP^1$ (genus 0). In higher dimensions the situation
    is much more subtle and it has been a long standing conjecture
    due to Frankel to characterize $\bbfP_n$ as the only compact
    K\"ahler manifold with positive holomorphic bisectional curvature.
    Hartshorne strengthened Frankel's conjecture and asserted that
    $\bbfP_n$ 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.}
\CC{14C20
32J27
14F05}
%% END ITEM
%% BEGIN ITEM
\AN{0851.32013}{1995}
\AU{Demailly, Jean-Pierre; Passare, Mikael}
\TI{Courants r\'esiduels et classe fondamentale. (Residual currents and fundamental class).}
\LA{French}
\DT{J}
\SO{Bull. Sci. Math. 119, No.1, 85-94 (1995). }
\ISSN{007-449}
\AB{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 $Y$.}
\RV{Vasile Br\^{\i}nz\u{a}nescu (Bucure\c{s}ti)}
\CC{32C30
32C36
58A25
32C15}
%% END ITEM
%% BEGIN ITEM
\AN{0851.32015}{1995}
\AU{Demailly, Jean-Pierre}
\TI{Propri\'et\'es de semi-continuit\'e de la cohomologie et de la dimension de Kodaira-Iitaka. (Semicontinuity properties of cohomology and of Kodaira-Iitaka dimension).}
\LA{French. Abridged English version}
\DT{J}
\SO{C. R. Acad. Sci., Paris, S\'er. I 320, No.3, 341-346 (1995). 
\par http://www.sciencedirect.com/science/journal/07644442}
\ISSN{764-444}
\AB{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.}
\RV{Vasile Br\^{\i}nz\u{a}nescu (Bucure\c{s}ti)}
\CC{32C35
35G05}
%% END ITEM
%% BEGIN ITEM
\AN{0845.14004}{1995}
\AU{Demailly, Jean-Pierre}
\TI{$L\sp 2$-methods and effective results in algebraic geometry.}
\LA{English}
\DT{CA}
\SO{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). [ISBN 3-7643-5153-5/hbk]}
\AB{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 $m \ge m_0$.}
\RV{F.Kirwan (Oxford)}
\CC{14C20
14F05}
%% END ITEM
%% BEGIN ITEM
\AN{0869.65042}{1994}
\AU{Demailly, Jean-Pierre}
\TI{Gew\"ohnliche Differentialgleichungen. Theoretische und numerische Aspekte. Aus d. Franz. \"ubers. von Mathias Heckele. (Ordinary differential equations. Theoretical and numerical aspects).}
\LA{German}
\DT{B}
\SO{Wiesbaden: Vieweg. x, 318 p. DM 49.50 (1994). [ISBN 3-528-06553-2]}
\AB{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.}
\RV{R.D.Grigorieff (Berlin)}
\CC{65L05
65L06
65D32
65H10
65-01
34-01}
%% END ITEM
%% BEGIN ITEM
\AN{0861.32006}{1994}
\AU{Demailly, Jean-Pierre; Lempert, L\'aszl\'o; Shiffman, Bernard}
\TI{Algebraic approximations of holomorphic maps from Stein domains to projective manifolds.}
\LA{English}
\DT{J}
\SO{Duke Math. J. 76, No.2, 333-363 (1994). }
\ISSN{012-709}
\AB{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
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.}
\RV{I.Serb (Cluj-Napoca)}
\CC{32E10
14P20
32F45}
%% END ITEM
%% BEGIN ITEM
\AN{0827.14027}{1994}
\AU{Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael}
\TI{Compact complex manifolds with numerically effective tangent bundles.}
\LA{English}
\DT{J}
\SO{J. Algebr. Geom. 3, No.2, 295-345 (1994). }
\ISSN{056-391}
\AB{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: $\pi_1 (\widetilde X) \cong \bbfZ^{2q}$.\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 $\bbfP (E)$
    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 $\bbfP
    (E)$, where $E$ and $E^*$ are nef rank-2 vector bundles over
    a two-dimensional torus.}
\RV{D.Laksov (Stockholm)}
\CC{14J30
14C20
32J17
14F35
53C55
14E20}
%% END ITEM
%% BEGIN ITEM
\AN{0884.32023}{1993}
\AU{Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael}
\TI{K\"ahler manifolds with numerically effective Ricci class.}
\LA{English}
\DT{J}
\SO{Compos. Math. 89, No.2, 217-240 (1993).
\par http://www.springerlink.com/openurl.asp?genre=journal\&issn=0010-437X}
\ISSN{010-437}
\AB{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
    $\alpha:X\to\text{Alb}(X)$ is surjective.\par
Section 1 is devoted
    to proving the following theorem, which is the main contribution
    to Conjecture 1.\par
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
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.}
\RV{Antonella Nannicini (MR 95b:32044)}
\CC{32J27
14J40
32Q15
53C55}
%% END ITEM
%% BEGIN ITEM
\AN{01648048}{1991}
\AU{Demailly, J.-P.}
\TI{Transcendental proof of a generalized Kawamata-Viehweg vanishing theorem.}
\LA{English}
\DT{CA}
\SO{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).}
\AB{Review in preparation}
\CC{32L20
14F17
32L10}
%% END ITEM
%% BEGIN ITEM
\AN{0900.31004 (01170954)}{1986}
\AU{Demailly, Jean-Pierre}
\TI{Fonction de Green pluricomplexe et mesures pluriharmoniques.}
\LA{French}
\DT{CA}
\SO{S\'eminaire de th\'eorie spectrale et g\'eom\'etrie. Ann\'ee 1985-1986. Chamb\'ery: Univ. de Savoie, Fac. des Sciences, Service de Math. S\'emin. Th\'eor. Spectrale G\'eom., Chamb\'ery-Grenoble. 4, 131-143 (1986).}
\AB{From the introduction: ``L'objet de cet expos\'e est de montrer
    comment \`a un domaine pseudoconvexe $\Omega$ relativement compact
    dans une vari\'et\'e complexe on peut associer une fonction de
    Green g\'en\'eralis\'ee, invariante par biholomorphisme. Cette
    fonction est d\'efinie comme la solution $u_z(\zeta)$ du probl\`eme
    de Dirichlet pour l'\'equation de Monge-Amp\`ere complexe $(dd^c
    u_z)^n=0$ sur $\Omega \setminus \{z\}$, ayant un p\^ole logarithmique
    au point $z$.''\par
Le pr\'esent expos\'e est une version condens\'ee
    de l'article [Math. Z. 194, 519-564 (1987; Zbl 0595.32006)],
    o\`u le lecteur trouvera des d\'emonstrations d\'etaill\'ees
    de tous les r\'esultats mentionn\'es ici.}
\CC{31C10
32T99
32A25}
%% END ITEM
%% BEGIN ITEM
\AN{0824.53064}{1994}
\AU{Demailly, Jean-Pierre}
\TI{Regularization of closed positive currents of type (1,1) by the flow of a Chern connection.}
\LA{English}
\DT{CA}
\SO{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). [ISBN 3-528-06633-4/hbk]}
\AB{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.}
\RV{I-Hsun Tsai (Taipei)}
\CC{53C55
32C30}
%% END ITEM
%% BEGIN ITEM
\AN{0792.32006}{1993}
\AU{Demailly, Jean-Pierre}
\TI{Monge-Amp\`ere operators, Lelong numbers and intersection theory.}
\LA{English}
\DT{CA}
\SO{Ancona, Vincenzo (ed.) et al., Complex analysis and geometry. New York: Plenum Press. The University Series in Mathematics. 115-193 (1993). [ISBN 0-306-44179-9/hbk]}
\AB{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.}
\RV{E.Outerelo (Madrid)}
\CC{32C30
32W20
32U05}
%% END ITEM
%% BEGIN ITEM
\AN{0783.32013}{1993}
\AU{Demailly, Jean-Pierre}
\TI{A numerical criterion for very ample line bundles.}
\LA{English}
\DT{J}
\SO{J. Differ. Geom. 37, No.2, 323-374 (1993). }
\ISSN{022-040}
\AB{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 $\text{codim} Y\ge 2$, 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.}
\RV{J.-P.Demailly (Saint-Martin d'H\`eres)}
\CC{32J15
32L10
32C30}
%% END ITEM
%% BEGIN ITEM
\AN{0771.32011}{1993}
\AU{Demailly, Jean-Pierre}
\TI{Holomorphic Morse inequalities on q-convex manifolds.}
\LA{English}
\DT{CA}
\SO{Several complex variables, Proc. Mittag-Leffler Inst., Stockholm/Swed. 1987-88, Math. Notes 38, 245-257 (1993).}
\AB{[For the entire collection see Zbl 0759.00008.]\par
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
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 $X(m,L):=\{x\in X:ic(L)$ is non degenerate at $x$
    and with exactly $m$ negative eigenvalues\}, $X(\le m,L):=\bigcup\sb{t\le
    m}X(t,L)$ and $X(\ge m,L):=\bigcup\sb{t\ge m}X(t,L)$. 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
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.}
\RV{E.Ballico (Povo)}
\CC{32F10
32C35
32J99
32W20}
%% END ITEM
%% BEGIN ITEM
\AN{0784.32024}{1992}
\AU{Demailly, Jean-Pierre}
\TI{Singular Hermitian metrics on positive line bundles.}
\LA{English}
\DT{CA}
\SO{Complex algebraic varieties, Proc. Conf., Bayreuth/Ger. 1990, Lect. Notes Math. 1507, 87-104 (1992).}
\AB{[For the entire collection see Zbl 0745.00049.]\par
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''.}
\RV{E.J.Straube (College Station)}
\CC{32L05}
%% END ITEM
%% BEGIN ITEM
\AN{0777.32016}{1992}
\AU{Demailly, Jean-Pierre}
\TI{Regularization of closed positive currents and intersection theory.}
\LA{English}
\DT{J}
\SO{J. Algebr. Geom. 1, No.3, 361-409 (1992). }
\ISSN{056-391}
\AB{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.}
\RV{A.Yu.Rashkovsky (Khar'kov)}
\CC{32J25
32C30
32S60}
%% END ITEM
%% BEGIN ITEM
\AN{0771.32010}{1992}
\AU{Demailly, Jean-Pierre}
\TI{Courants positifs et th\'eorie de l'intersection. (Positive currents and intersection theory).}
\LA{French}
\DT{J}
\SO{Gaz. Math., Soc. Math. Fr. 53, 131-159 (1992). }
\ISSN{224-899}
\AB{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).}
\RV{E.Ballico (Povo)}
\CC{32C30
58A25
32J25
14C17}
%% END ITEM
%% BEGIN ITEM
\AN{0755.32008}{1991}
\AU{Demailly, Jean Pierre}
\TI{Holomorphic Morse inequalities.}
\LA{English}
\DT{CA}
\SO{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).}
\AB{[For the entire collection see Zbl 0732.00008.]\par
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.}
\RV{B.Nowak ({\L}\'od\'z)}
\CC{32C35
58E05
32L10
53C07}
%% END ITEM
%% BEGIN ITEM
\AN{0724.32005}{1990}
\AU{Blel, Mongi; Demailly, Jean-Pierre; Mouzali, Mokhtar}
\TI{Sur l'existence du c\^one tangent \`a un courant positif ferm\'e. (About the existence of the tangent cone with positive closed current).}
\LA{French}
\DT{J}
\SO{Ark. Mat. 28, No.2, 231-248 (1990). }
\ISSN{004-208}
\AB{Let $T$ be a positive closed current of degree p on an open neighborhood
    $\Omega$ of 0 in ${\bbfC}\sp n$. For $a\in {\bbfC}\sp*$ 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
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
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.}
\RV{H.-J.Reiffen (Osnabr\"uck)}
\CC{32C30
32B15}
%% END ITEM
%% BEGIN ITEM
\AN{0682.32017}{1990}
\AU{Demailly, Jean-Pierre}
\TI{Cohomology of q-convex spaces in top degrees.}
\LA{English}
\DT{J}
\SO{Math. Z. 204, No.2, 283-295 (1990). }
\ISSN{025-587}
\AB{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.}
\RV{J.P.Demailly}
\CC{32F10}
%% END ITEM
%% BEGIN ITEM
\AN{0681.32014}{1988}
\AU{Bedford, Eric; Demailly, Jean-Pierre}
\TI{Two counterexamples concerning the pluricomplex Green function in ${\bbfC}\sp n$.}
\LA{English}
\DT{J}
\SO{Indiana Univ. Math. J. 37, No.4, 865-867 (1988). }
\ISSN{022-251}
\AB{Given a domain $\Omega$ in ${\bbfC}\sp n$ 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''.}
\RV{M.Stoll}
\CC{32U05
32T99
31C10}
%% END ITEM
%% BEGIN ITEM
\AN{0651.32019}{1988}
\AU{Demailly, Jean-Pierre}
\TI{Vanishing theorems for tensor powers of a positive vector bundle.}
\LA{English}
\DT{CA}
\SO{Proc. 21st Int. Taniguchi Symp., Katata/Japan, Conf., Kyoto/Japan 1987, Lect. Notes Math. 1339, 86-105 (1988).}
\AB{[For the entire collection see Zbl 0638.00022.] \par
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 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,. 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
    $p=n$, one requires at least $\ell \ge 1$. 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 $p<n$, the generalized Borel-Le Potier spectral sequence
    does not degenerate at the $E\sb 1$ level, the author obtains
    a new curvature estimate for the bundle of X-relative differential
    forms on the flag manifold of E.}
\RV{P.E.Newstead}
\CC{32L20
32L05}
%% END ITEM
%% BEGIN ITEM
\AN{0647.14005}{1988}
\AU{Demailly, Jean-Pierre}
\TI{Vanishing theorems for tensor powers of an ample vector bundle.}
\LA{English}
\DT{J}
\SO{Invent. Math. 91, No.1, 203-220 (1988). 
\par http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN356556735}
\ISSN{020-991}
\AB{Let X be a compact complex manifold of dimension n and E resp.
    L an ample holomorphic vector bundle of rank r resp. an ample
    line bundle on X. The paper gives generalizations of Griffiths'
    vanishing theorem $H\sp{n,q}(X,S\sp kE\otimes \det (E\otimes
    L))=0$ for $q\ge 1$ [{\it P. A. Griffiths}, Global Analysis,
    papers in Honor of K. Kodaira, 185-251 (1969; Zbl 0201.240)]
    which shall not be repeated here and Le Potier's vanishing theorem
    $H\sp{p,q}(X,E)=0$ for $p+q\ge n+r$ [{\it J. Le Potier}, Math.
    Ann. 218, 35-53 (1975; Zbl 0313.32037)] saying that $H\sp{p,q}(X,E\sp{\otimes
    k}\otimes (\det (E))\sp{\ell}\otimes L)=0$ for $p+q\ge n+1$,
    $k\ge 1$ and $\ell \ge n-p+r-1.$ \par
The proof rests on a generalization
    of the Borel-Le Potier spectral sequence and the Kodaira-Akizuki-Nakano
    vanishing theorem for line bundles. Moreover it is shown that
    there is a canonical homomorphism $H\sp{p,q}(X,\wedge\sp 2E\otimes
    L)\to H\sp{p+1,q+1}(X,S\sp 2E\otimes L)$ which is bijective under
    some additional hypotheses. Using this the author gives a counterexample
    to a conjecture of {\it J. A. Sommese} in Math. Ann. 233, 229-256
    (1978; Zbl 0381.14007).}
\RV{H.Lange}
\CC{14F05
32L20}
%% END ITEM
%% BEGIN ITEM
\AN{0691.32009}{1987}
\AU{Demailly, Jean-Pierre}
\TI{Sur les th\'eor\`emes d'annulation et de finitude de T. Ohsawa et O. Abdelkader. (On vanishing and finiteness theorems of T. Ohsawa and O. Abdelkader).}
\LA{French}
\DT{CA}
\SO{S\'emin. d'analyse P. Lelong - P. Dolbeault - H. Skoda, Paris, 1985/86, Lect. Notes Math. 1295, 48-58 (1987).}
\AB{[For the entire collection see Zbl 0623.00006.] \par
L'objet
    de cette note est de donner une d\'emonstration aussi simple
    que possible des th\'eor\`emes d'annulation et de finitude dus
    \`a {\it T. Ohsawa} [Publ. Res. Inst. Math. Sci. 15, 853-870
    (1979; Zbl 0434.32014), Publ. Res. Inst. Math. Sci. 17, 113-126
    (1981; Zbl 0465.32007)], et des g\'en\'eralisations de ces th\'eor\`emes
    obtenues par {\it O. Abdelkader} [C. R. Acad. Sci., Paris, Ser.
    A 290, 75-78 (1980; Zbl 0442.32008) et ``Th\'eor\`emes de finitude
    pour la cohomologie d'une vari\'et\'e faiblement 1-compl\`ete
    \`a valeurs dans un fibr\'e en droites semi- positif'', Th\`ese
    Doct. d'Etat \`a l'Univ. Paris VI (1985)].}
\RV{R\'esum\'e}
\CC{32L20
32H35
32C35
32F10
53C55}
%% END ITEM
%% BEGIN ITEM
\AN{0632.32004}{1987}
\AU{Demailly, Jean-Pierre; Laurent-Thiebaut, Christine}
\TI{Formules int\'egrales pour les formes diff\'erentielles de type (p,q) dans les vari\'et\'es de Stein. (Integral formulas for differential forms of type (p,q) in Stein manifolds).}
\LA{French}
\DT{J}
\SO{Ann. Sci. \'Ec. Norm. Sup\'er., IV. S\'er. 20, No.4, 579-598 (1987). 
\par http://www.numdam.org/numdam-bin/feuilleter?id=ASENS\_1987\_4\_20\_4}
\ISSN{012-959}
\AB{The Cauchy-Green integral formula for a domain $D\subset \subset
    {\bbfC}$ with piecewise $C\sp 1$-boundary and for $f\in C\sp
    1(\bar D)$ is $$ (!)\quad f(z)=(1/2\pi i)\int\sb{\partial D}f(\zeta)d\zeta
    /(\zeta - z)\quad +\quad (1/2\pi i)\int\sb{D}{\bar \partial}f
    d\zeta \Lambda d{\bar \zeta}/(\zeta -z). $$ Over the past half-century,
    various integral formulas have been gradually developed that
    generalise (!) for several complex variables. An excellent, systematic
    exposition of this work and of some problems that can be solved
    by such methods is {\it G. M. Khenkin} and {\it J. Leiterer},
    Theory of functions on complex manifolds (1984; Zbl 0573.32001).
    The present paper contains the construction of the relevant kernels
    and integrals for differential forms of type (p,q) on Stein manifolds:
    {\S}{\S} 1 and 3 develop and extend the methods of Khenkin and
    Leiterer concerning the Bochner-Martinelli kernel; {\S}{\S} 2
    and 4 generalise the Koppelman formula and the Koppelman-Leray
    formula, respectively.}
\RV{E.J.Akutowicz}
\CC{32A30
30E20
32A25
32E10}
%% END ITEM
%% BEGIN ITEM
\AN{0629.32011}{1987}
\AU{Demailly, Jean-Pierre}
\TI{Nombres de Lelong g\'en\'eralis\'es, th\'eor\`emes d'int\'egralit\'e et d'analyticit\'e. (Generalized Lelong numbers, integrability and analyticity theorems).}
\LA{French}
\DT{J}
\SO{Acta Math. 159, 153-169 (1987). }
\ISSN{001-596}
\AB{Let X be a complex Stein space, T a closed positive current of
    bidimension (p,p) on X and $\phi:X\to [-\infty,+\infty [$ an
    exhaustive plurisubharmonic function. The author's generalized
    Lelong number $\nu$ (T,$\phi)$ is defined as the mass of the
    measure $T\wedge (dd\sp c\phi)\sp p$ carried by the polar set
    $\phi\sp{-1}(-\infty)$ and is obtained by means of the Monge-Amp\`ere
    operator of {\it E. Bedford} and {\it B. A. Taylor} [ibid. 149,
    1-40 (1982; Zbl 0547.32012)]. $\nu$ (T,$\phi)$ generalizes the
    classical {\it P. Lelong} [``Plurisubharmonic functions and positive
    differential forms'' (1969; Zbl 0195.116)] and C. O. Kiselman's
    numbers. The author establishes that $\nu$ (T,$\phi)$ depends
    only on the behaviour of $\phi$ in a neighbourhood of the poles.
    The use of $\nu$ (T,$\phi)$ allows him to obtain very simple
    proofs of classical results on Lelong numbers, e.g. that these
    numbers are invariant with respect to local coordinate transformations
    [cf. {\it Y. T. Siu}, Invent. Math. 27, 53-156 (1974; Zbl 0289.32003)]
    and also on {\it P. Thie}'s [Math. Ann. 172, 269-312 (1967; Zbl
    0158.328)] theorem showing that the Lelong number of an analytic
    set X coincides to the algebraic multiplicity of Y at x. Finally,
    the author obtains a generalization of Siu's theorem on the analyticity
    of the level sets associated to Lelong numbers, his result containing
    as a particular case a recent theorem of C. O. Kiselman on directional
    Lelong numbers.}
\RV{P.Caraman}
\CC{32E10
32U05
32C30
31C10
31C15}
%% END ITEM
%% BEGIN ITEM
\AN{0629.32026}{1987}
\AU{Demailly, Jean-Pierre}
\TI{Une preuve simple de la conjecture de Grauert-Riemenschneider. (A simple proof of the Grauert-Riemenschneider conjecture).}
\LA{French}
\DT{CA}
\SO{S\'emin. d'analyse P. Lelong - P. Dolbeault - H. Skoda, Paris 1985/86, Lect. Notes Math. 1295, 24-47 (1987).}
\AB{[For the entire collection see Zbl 0623.00006.] \par
The author's
    abstract: ``Let E be a hermitian holomorphic line bundle over
    a compact complex manifold X. We give an asymptotic upper bound
    for the dimension of cohomology groups of high tensor powers
    $E\sp k$. This bound is invariantly expressed in terms of an
    integral of the bundle curvature form. As an application, we
    find a simple proof of the Grauert- Riemenschneider conjecture,
    recently solved by Siu: if X possesses a quasi-positive line
    bundle E, then X is a Moishezon space; furthermore the quasipositivity
    hypothesis can be weakened here in an integral condition which
    does not require the bundle E to be pointwise semi- positive.''}
\RV{D.Barlet}
\CC{32J25
32J99
32L15
32L20}
%% END ITEM
%% BEGIN ITEM
\AN{0627.32022}{1987}
\AU{Demailly, Jean-Pierre}
\TI{Th\'eor\`emes d'annulation pour la cohomologie des puissances tensorielles d'un fibr\'e vectoriel positif. (Vanishing theorems for cohomology groups of tensor powers of a positive vector bundle).}
\LA{French}
\DT{J}
\SO{C. R. Acad. Sci., Paris, S\'er. I 305, 419-422 (1987). }
\ISSN{764-444}
\AB{The author cleverly combines the Kodaira-Akizuki-Nakano vanishing
    theorem with some standard geometrical constructions, in order
    to prove the following result: Let E, L be holomorphic vector
    bundles over a compact complex manifold X. Assume $rk(L)=1$ and
    $E>0$, $L\ge 0$ 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.}
\RV{M.Putinar}
\CC{32L20
32M10}
%% END ITEM
%% BEGIN ITEM
\AN{0602.31006}{1986}
\AU{Demailly, J.-P.}
\TI{Mesures de Monge-Amp\`ere et mesures pluriharmoniques. (Monge-Amp\`ere measures and pluriharmonic measures).}
\LA{French}
\DT{J}
\SO{S\'emin., \'Equations D\'eriv. Partielles 1985-1986, Expos\'e No.19, 15 p. (1986).}
\AB{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.}
\RV{D.Armitage}
\CC{31C10
32E10
32U05}
%% END ITEM
%% BEGIN ITEM
\AN{0595.32006}{1987}
\AU{Demailly, Jean-Pierre}
\TI{Mesures de Monge-Amp\`ere et mesures pluriharmoniques. (Monge-Amp\`ere measures and pluriharmonic measures).}
\LA{French}
\DT{J}
\SO{Math. Z. 194, 519-564 (1987). }
\ISSN{025-587}
\AB{Let $\Omega$ be a relatively compact open subset in a Stein manifold,
    and $n=\dim\sb{{\bbfC}}\Omega$. 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 $\Omega$ $\setminus \{z\}$
    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 $K\subset {\bbfR}\sp n$ as a barycenter
    of the extremal points of K.}
\CC{32A25
32C30
32F45
32U05
31C10
32E10}
%% END ITEM
%% BEGIN ITEM
\AN{0594.32030}{1986}
\AU{Demailly, Jean-Pierre}
\TI{Un exemple de fibr\'e holomorphe non de Stein \`a fibre ${\bbfC}\sp 2$ au-dessus du disque ou du plan. (An example of a non-Stein holomorphic fiber bundle over the disk or the plane, with fiber ${\bbfC}\sp 2)$.}
\LA{French}
\DT{CA}
\SO{S\'emin. analyse P. Lelong - P. Dobeault - H. Skoda, Ann\'ees 1983/84, Lect. Notes Math. 1198, 98-104 (1986).}
\AB{[For the entire collection see Zbl 0583.00011.] \par
It is constructed
    a simple example of a non-Stein holomorphic fiber bundle over
    the disk with fiber ${\bbfC}\sp 2$. Moreover, it is shown that
    all holomorphic functions on the bundle arise from functions
    on the base.}
\RV{A.Pankov}
\CC{32L05
32E10}
%% END ITEM
%% BEGIN ITEM
\AN{0594.32031}{1986}
\AU{Demailly, Jean-Pierre}
\TI{Sur l'identit\'e de Bochner-Kodaira-Nakano en g\'eom\'etrie hermitienne. (The Bochner-Kodaira-Nakano identity in hermitian geometry).}
\LA{French}
\DT{CA}
\SO{S\'emin. analyse P. Lelong - P. Dolbeault - H. Skoda, Ann\'ees 1983/84, Lect. Notes Math. 1198, 88-97 (1986).}
\AB{[For the entire collection see Zbl 0583.00011.] \par
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.}
\RV{A.Pankov}
\CC{32L05
32Q99
53C55}
%% END ITEM
%% BEGIN ITEM
\AN{0595.58014}{1985}
\AU{Demailly, Jean-Pierre}
\TI{Champs magn\'etiques et in\'egalit\'es de Morse pour la d"-cohomologie. (Magnetic fields and Morse inequalities for d"-cohomology).}
\LA{French}
\DT{J}
\SO{C. R. Acad. Sci., Paris, S\'er. I 301, 119-122 (1985). }
\ISSN{764-444}
\AB{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
    $h\sp q\sb k= \dim H\sp q(X,E\sp k\otimes F)$ satisfies as $k\to
    +\infty$ the asymptotic Morse inequality $$ h\sp q\sb k\le r\frac{k\sp
    n}{n!}\int\sb{X(q)}(-1)\sp q(ic(E)/2\pi)\sp n+o(k\sp n). $$ An
    analogous inequality holds for $\sum\sp{q}\sb{j=0}(-1)\sp jh\sp
    j\sb k$. As an application, the author obtains geometric characterizations
    of Moishezon spaces, improving recent results of Y. T. Siu.}
\RV{G.Roos}
\CC{58E35
32J25
32L10}
%% END ITEM
%% BEGIN ITEM
\AN{0579.32012}{1985}
\AU{Demailly, Jean-Pierre}
\TI{M\'esures de Monge-Amp\`ere et caract\'erisation g\'eom\'etrique des vari\'et\'es alg\'ebriques affines.}
\LA{French}
\DT{J}
\SO{M\'em. Soc. Math. Fr., Nouv. S\'er. 19, 124 p. (1985). 
\par http://www.numdam.org/numdam-bin/feuilleter?id=MSMF\_1985\_2\_19\_}
\ISSN{037-948}
\AB{Let X be an irreducible n-dimensional Stein space and $\phi$
    : $X\to [- \infty,R[$ a continuous psh (plurisubharmonic) exhaustion
    function. Each level set $S(r)=\{x\in X;\quad \phi (x)=r\},$
    $r<R$, is shown to carry an intrinsic positive measure $\mu\sb
    r$; the measure $\mu\sb r$ is given by the (2n-1)-form $(dd\sp
    c\phi)\sp{n-1}\wedge d\sp c\phi$ when $\phi$ is smooth, and otherwise
    $\mu\sb r$ is constructed by means of the Monge- Amp\`ere operators
    introduced by Bedford and Taylor. In this context, a general
    Lelong-Jensen formula $$ \mu\sb r(V)=\int\sp{r}\sb{- \infty}dt\int\sb{\phi
    <t}dd\sp cV\wedge (dd\sp c\phi)\sp{n- 1}+\int\sb{\phi <r}V(dd\sp
    c\phi)\sp n, $$ is proved and used to study the growth and convexity
    properties of plurisubharmonic or holomorphic functions. If $(dd\sp
    c\phi)\sp n=0$ on $\{\phi >r\sb 0\}$, the function $r\to \mu\sb
    r(V)$ is shown to be convex and increasing in the interval $]r\sb
    0,R[$. Furthermore, if the volume $\tau (r)=\int\sb{\phi <r}(dd\sp
    c\phi)\sp n$ has moderate growth, i.e. if $\tau (r)=o(r)$, then
    bounded holomorphic functions on X are constant; using Siegel's
    method, we prove also in that case that the ring of holomorphic
    functions with $\phi$- polynomial growth has a transcendance
    degree $\le n$. This last result is then applied in order to
    obtain a necessary and sufficient geometric criterion characterizing
    affine algebraic manifolds: X is algebraic iff it has finite
    Monge-Amp\`ere volume and if the Ricci-curvature of the metric
    $dd\sp c(\exp (\phi))$ is bounded below by $-dd\sp c\psi,$ where
    $\psi \le A\phi +B.$}
\CC{32C30
32U05
32A22
32J10
32J99}
%% END ITEM
%% BEGIN ITEM
\AN{0566.32005}{1985}
\AU{Demailly, Jean-Pierre}
\TI{Propagation des singularit\'es des courants positifs ferm\'es.}
\LA{French}
\DT{J}
\SO{Ark. Mat. 23, 35-52 (1985). }
\ISSN{004-208}
\AB{Let T be a closed positive current in a bounded Runge open subset
    $\Omega \subset {\bbfC}\sp n$. One studies sufficient conditions
    to be verified by the mass density of T in order that there exists
    a global extension of T to ${\bbfC}\sp n$. Assume that T is of
    bidegree (1,1) (resp. (q,q), $q>1)$, that the cohomology class
    of T in $\Omega$ is 0 and that the trace measure $\sigma\sb T(z,r)$
    of the ball B(z,r) satisfies $(*)\quad \sup\sb{z\in K}\int\sp{\epsilon}\sb{0}(\sigma\sb
    T(z,r)/r\sp{2n- 1})dr<+\infty$ (resp. $\int\sp{\epsilon}\sb{0}(\sup\sb{z\in
    K} \sigma\sb T(z,r)\sp{1/2}/r\sp n)dr<+\infty)$ for every $K\subset
    \subset \Omega$ and $\epsilon >0$ small enough. Then for every
    $\omega\sb 1\subset \subset \omega\sb 2\subset \subset \Omega$,
    one proves the existence of a closed positive current $\Theta$
    in ${\bbfC}\sp n$ which is equal to T in $\omega\sb 1$ and $C\sp{\infty}$-smooth
    in ${\bbfC}\sp n\setminus {\bar \omega}\sb 2$. Conversely, using
    Skoda - El Mir structure theorems for closed $\ge 0 currents$,
    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 $\sup\sb{z\in K} \sigma\sb
    T(z,r)\le C\sb Kr\sp{2n-2q-1}$, 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 ${\bbfC}\sp n$, a result due
    to Diederich-Fornaess which one proves again in a new and simpler
    way.}
\CC{32C30
32D15}
%% END ITEM
%% BEGIN ITEM
\AN{0565.58017}{1985}
\AU{Demailly, Jean-Pierre}
\TI{Champs magn\'etiques et in\'egalit\'es de Morse pour la d''-cohomologie. (Magnetic fields and Morse inequalities for d''-cohomology).}
\LA{French}
\DT{J}
\SO{Ann. Inst. Fourier 35, No.4, 189-229 (1985). 
\par http://annalif.ujf-grenoble.fr/aif\_1985.html}
\ISSN{373-095}
\AB{Nous d\'emontrons des in\'egalit\'es de Morse-Witten asymptotiques
    pour la dimension des groupes de cohomologie des puissances tensorielles
    d'un fibr\'e holomorphe en droites hermitien au-dessus d'une
    vari\'et\'e ${\bbfC}$-analytique compacte. La dimension du q-i\`eme
    groupe de cohomologie se trouve ainsi major\'ee par une int\'egrale
    de courbure intrins\`eque, \'etendue \`a l'ensemble des points
    d'indice q de la forme de courbure du fibr\'e. La preuve repose
    sur un th\'eor\`eme spectral qui d\'ecrit la distribution asymptotique
    des valeurs propres de l'op\'erateur de Schr\"odinger associ\'e
    \`a un champ magn\'etique assez grand. Comme application, nous
    obtenons une nouvelle d\'emonstration de la conjecture de Grauert-Riemenschneider
    sur la caract\'erisation des espaces de Moi\v{s}ezon, r\'esolue
    r\'ecemment par Siu, sous des hypoth\`eses g\'eom\'etriques plus
    g\'en\'erales qui n'exigent pas n\'ecessairement la semi-positivit\'e
    ponctuelle du fibr\'e.}
\CC{58E05}
%% END ITEM
%% BEGIN ITEM
\AN{0571.43003}{1984}
\AU{Demailly, Jean-Pierre}
\TI{Sur les transform\'ees de Fourier de fonctions continues et le th\'eor\`eme de de Leeuw-Katznelson-Kahane. (On Fourier transforms of continuous functions and a theorem of de Leeuw-Katznelson-Kahane).}
\LA{French}
\DT{J}
\SO{C. R. Acad. Sci., Paris, S\'er. I 299, 435-438 (1984). }
\ISSN{764-444}
\AB{Given any locally compact abelian group G and any function $\phi
    \in L\sp 2(\hat G)$, we prove the existence of a function $f\in
    L\sp 2(G)$ continuous and vanishing at infinity such that $\vert
    \hat f\vert \ge \vert \phi \vert$ a.e. on $\hat G.$}
\CC{43A25}
%% END ITEM
%% BEGIN ITEM
\AN{0551.32009}{1984}
\AU{Demailly, Jean-Pierre}
\TI{Sur la propagation des singularit\'es des courants positifs ferm\'es.}
\LA{English}
\DT{CA}
\SO{Analyse complexe, Proc. Journ. Fermat - Journ. SMF, Toulouse 1983, Lect. Notes Math. 1094, 53-64 (1984).}
\AB{[For the entire collection see Zbl 0539.00009.] \par
Let T be
    a closed positive current in a bounded Runge open subset $\Omega
    \subset {\bbfC}\sp n$. We study sufficient conditions to be verified
    by the mass densities of T in order that there exists a global
    extension of T to ${\bbfC}\sp n$. Assume that the cohomology
    class of T in $\Omega$ is 0 and that the trace measure $\sigma\sb
    T$ satisfies $$ \int\sp{\delta /2}\sb{0} \left(\sup\sb{z\in \Omega\sb{\delta}\setminus
    \Omega\sb{\epsilon}} (\sigma\sb T (z,r) \sp{\frac12}/r\sp n\right)
    dr < +\infty \leqno * $$ for some $\epsilon >\delta >0$, where
    $\Omega\sb{\epsilon}=\{z\in \Omega;d(z,{\bbfC}\sp n\setminus
    \Omega)>\epsilon \}.$ Then we prove the existence of a closed
    positive current $\Theta$ in ${\bbfC}\sp n$ which is equal to
    T in $\Omega\sb{\epsilon}$ and $C\sp{\infty}$-smooth in ${\bbfC}\sp
    n\setminus {\bar \Omega}\sb{\delta}$. Thus, there is no propagation
    of singularities in that case. Conversely, using Skoda-El Mir
    structure theorems for closed $\ge 0$ currents and a result of
    Diederich- Fornaess on the existence of totally real complete
    pluripolar (n-1)- submanifolds in ${\bbfC}\sp n$, 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 $$ \sup\sb{z\in\Omega\sb{\delta}\setminus\Omega\sb{\epsilon}}
    \int\sp{\delta/2}\sb{0} (\sigma\sb T(z,r)/r\sp{2n-1})dr < +\infty,
    \leqno ** $$ whereas in bidegree (q,q) there exist currents
    T such that $\sup\sb{z\in \Omega\sb{\delta}}\sigma\sb T(z,r)\le
    Cr\sp{2n-q-1},$ whose singularities propagate up to $\partial
    \Omega$ along (2n-q-1)-CR submanifolds.}
\CC{32C30
32Sxx}
%% END ITEM
%% BEGIN ITEM
\AN{0533.53054}{1983}
\AU{Demailly, Jean-Pierre; Gaveau, Bernard}
\TI{Majoration statistique de courbure d'une vari\'et\'e analytique.}
\LA{French}
\DT{CA}
\SO{S\'emin. d'Analyse P. Lelong-P. Dolbeault-H. Skoda, Ann\'ees 1981/83, Lect. Notes Math. 1028, 96-124 (1983).}
\AB{[For the entire collection see Zbl 0511.00025.] \par
Let $\Omega
    \subset {\bbfC}\sp n$ be a bounded strictly pseudoconvex open
    subset. Given an analytic map F: $\Omega\to {\bbfC}\sp p$, one
    studies the average growth of the Ricci curvature of the level
    varieties $X\sb{\zeta}=F\sp{-1}(\zeta)$. Especially, if $R=$-
    Ricci$(X{}\sb{\zeta})\ge 0$ denotes the positive Ricci (1,1)-form
    of $X\sb{\zeta}$, $\delta$ the distance to $\partial \Omega$
    and $\alpha =dd\sp c\vert z\vert\sp 2$, it is shown that the
    estimate $$ \int\sb{\zeta \in {\bbfC}\sp n}d\lambda(\zeta)\int\sb{X\sb{\zeta}}\delta\sp{p+q}[Log(1+1/\delta)]\sp{-
    q}\quad R\sp q\wedge \alpha\sp{n-p-q}\quad<\quad +\infty$$ 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 ${\bbfC}\sp n$. R is proved to verify
    the following Monge-Amp\`ere equation: $i\partial {\bar \partial}
    Log \Gamma =-(n+1)R+2\pi [Z]$ where Z is the divisor of zeros
    of $\Gamma$.}
\CC{53C55
53C65
32C30}
%% END ITEM
%% BEGIN ITEM
\AN{0529.32003}{1983}
\AU{Demailly, Jean-Pierre}
\AR{Kashiwara, M.}
\TI{Constructibilite des faisceaux de solutions des syst\`emes diff\'erentiels holonomes. (D'apres Masaki Kashiwara).}
\LA{French}
\DT{CA}
\SO{Semin. d'Analyse P. Lelong - P. Dolbeault - H. Skoda, Annees 1981/83, Lect. Notes Math. 1028, 83-95 (1983).}
\AB{See printed version}
\CC{32A45
58J15
14F10
58J10
32L05}
%% END ITEM
%% BEGIN ITEM
\AN{0529.32006}{1983}
\AU{Demailly, J.P.}
\TI{Sur la structure des courants positifs fermes.}
\LA{French}
\DT{J}
\SO{Inst. Elie Cartan, Univ. Nancy I 8, 52-62 (1983).}
\AB{See printed version}
\CC{32C30
30C80
14C30
32A30}
%% END ITEM
%% BEGIN ITEM
\AN{0507.32021}{1982}
\AU{Demailly, Jean-Pierre}
\TI{Estimations $L\sp 2 $pour l'op\'erateur (partial d) d'un fibre vectoriel holomorphe semi-positif au-dessus d'une vari\'et\'e Kaehlerienne complete.}
\LA{French}
\DT{J}
\SO{Ann. Sci. \'Ec. Norm. Sup\'er., IV. S\'er. 15, 457-511 (1982). 
\par http://www.numdam.org/numdam-bin/feuilleter?id=ASENS\_1982\_4\_15\_3}
\ISSN{012-959}
\AB{See printed version}
\CC{32L20
32L05
53C55
32L10
32U05}
%% END ITEM
%% BEGIN ITEM
\AN{0493.32003}{1982}
\AU{Demailly, J.-P.}
\TI{Formules de Jensen en plusieurs variables et applications arithm\'etiques.}
\LA{French}
\DT{J}
\SO{Bull. Soc. Math. Fr. 110, 75-102 (1982). 
\par http://www.numdam.org/numdam-bin/feuilleter?id=BSMF\_1982\_\_110\_}
\ISSN{037-948}
\AB{See printed version}
\CC{32A25
30C80
32A15
32C30
32A30}
%% END ITEM
%% BEGIN ITEM
\AN{0488.58001}{1982}
\AU{Demailly, Jean-Pierre}
\TI{Courants positifs extremaux et conjecture de Hodge.}
\LA{French}
\DT{J}
\SO{Invent. Math. 69, 347-374 (1982). }
\ISSN{020-991}
\AB{See printed version}
\CC{58A25
58A14
14C30
14C20}
%% END ITEM
%% BEGIN ITEM
\AN{0481.32010}{1982}
\AU{Demailly, J.-P.}
\TI{R\'elations entre les diff\'erentes notions de fibr\'es et de courants positifs.}
\LA{French}
\DT{CA}
\SO{Semin. P. Lelong - H. Skoda, Analyse, Annees 1980/81, et: Les fonctions plurisousharmoniques en dimension finie ou infinie, Colloq. Wimereux 1981, Lect. Notes Math. 919, 56-76 (1982).}
\AB{See printed version}
\CC{32L05
32C30
58A25}
%% END ITEM
%% BEGIN ITEM
\AN{0481.32011}{1982}
\AU{Demailly, J.-P.}
\TI{Scindage holomorphe d'un morphisme de fibr\'es vectoriels semi-positifs avec estimations $L\sp 2.$}
\LA{French}
\DT{CA}
\SO{Semin. P. Lelong - H. Skoda, Analyse, Annees 1980/81, et: Les fonctions plurisousharmoniques en dimension finie ou infinie, Colloq. Wimereux 1981, Lect. Notes Math. 919, 77-107 (1982).}
\AB{See printed version}
\CC{32L05
53C55
32D15
32T99}
%% END ITEM
%% BEGIN ITEM
\AN{0476.58001}{1982}
\AU{Demailly, Jean-Pierre}
\TI{Extremal positive currents and Hodge conjecture.}
\LA{English}
\DT{J}
\SO{Invent. Math. (to appear). }
\ISSN{020-991}
\AB{See printed version}
\CC{58A25
58A14
14C30
14C20}
%% END ITEM
%% BEGIN ITEM
\AN{0457.32005}{1982}
\AU{Demailly, Jean-Pierre}
\TI{Sur les nombres de Lelong associ\'es \`a l'image directe d'un courant positif ferm\'e.}
\LA{French}
\DT{J}
\SO{Ann. Inst. Fourier 32, No.2, 37-66 (1982). 
\par http://annalif.ujf-grenoble.fr/aif\_1982.html}
\ISSN{373-095}
\AB{See printed version}
\CC{32C30
58A25}
%% END ITEM
%% BEGIN ITEM
\AN{0454.55011}{1980}
\AU{Demailly, J.-P.; Skoda, H.}
\TI{Relations entre les notions de positivites de P. A. Griffiths et de S. Nakano pour les fibr\'es vectoriels.}
\LA{French}
\DT{CA}
\SO{Semin. P. Lelong - H. Skoda, Analyse, Annees 1978/79, Lect. Notes Math. 822, 304-309 (1980).}
\AB{See printed version}
\CC{55R25
32L05
32L15}
%% END ITEM
%% BEGIN ITEM
\AN{0414.32004}{1980}
\AU{Demailly, Jean-Pierre}
\TI{Construction d'hypersurfaces irr\'eductibles avec lieu singulier donn\'e dans $\bbfC^n$.}
\LA{French}
\DT{J}
\SO{Ann. Inst. Fourier 30, No.3, 219-236 (1980).
\par http://annalif.ujf-grenoble.fr/aif\_1980.html}
\ISSN{373-095}
\AB{See printed version}
\CC{32Sxx
14J17
32A22
32C25}
%% END ITEM
%% BEGIN ITEM
\AN{0412.32007}{1979}
\AU{Demailly, Jean-Pierre}
\TI{Fonctions holomorphes \`a croissance polynomiale sur la surface d'\'equation $e\sp x + e\sp y = 1$.}
\LA{French}
\DT{J}
\SO{Bull. Sci. Math., II. Ser. 103, 179-191 (1979). }
\ISSN{007-449}
\AB{See printed version}
\CC{32A22
32U05
32A10
32A07
32D15
32E30}
%% END ITEM
%% BEGIN ITEM
\AN{0409.32009}{1979}
\AU{Demailly, Jean-Pierre}
\TI{Fonctions holomorphes bornees ou \`a croissance polynomiale sur la courbe $e^x+e^y=1$.}
\LA{French}
\DT{J}
\SO{C. R. Acad. Sci., Paris, S\'er. A 288, 39-40 (1979). }
\ISSN{151-050}
\AB{See printed version}
\CC{32D15
32A22
32U05}
%% END ITEM
%% BEGIN ITEM
\AN{0418.32011}{1978}
\AU{Demailly, J.-P.}
\TI{Diff\'erents exemples de fibr\'es holomorphes non de Stein.}
\LA{French}
\DT{CA}
\SO{Semin. Pierre Lelong - Henri Skoda (Anal.), Annee 1976/77, Lect. Notes Math. 694, 15-41 (1978).}
\AB{See printed version}
\CC{32E10
32L05}
%% END ITEM
%% BEGIN ITEM
\AN{0372.32012}{1978}
\AU{Demailly, Jean-Pierre}
\TI{Un exemple de fibr\'e holomorphe non de Stein \`a fibre $C^2$ ayant pour base le disque ou le plan.}
\LA{French}
\DT{J}
\SO{Invent. Math. 48, 293-302 (1978). }
\ISSN{020-991}
\AB{See printed version}
\CC{32L05
32U05}
%% END ITEM
\bye