chrisEmeritus Professor, Université de Grenoble I (Institut Fourier)

        email: chris.peters <at> ujf-grenoble.fr

      Guest Researcher Technical University of Eindhoven (Discrete Maths)

         email: c.a.m.peters <at> tue.nl

 


Recent and upcoming travel


Curriculum:

Scientific career
  • PhD: University of Leiden, May 1974.
  • Assistant Univ. Leiden: Sept 1969-May 1974
  • Research Fellow: University of Harvard, 1974/1975  
  • Assistant Professor Univ. Leiden: Sept 1975--March 1979, 
  • Professor, University of Leiden (UHD): March  1979-Jan 1994,
  • Professor, University of Grenoble-I: Feb. 1 1994- Aug 2013.  
  • Emeritus, University of Grenoble-I:  Sept 1 2013 - Sept 1 2014.
  • Retired, University of Grenoble-I: from Sept 1 2014 on.
  • Guest Researcher, University of Eindhoven: from Sept 1 2013 on.

 Some temporary positions
  •   Member of the Institute for Advanced Study, Princeton: fall 1978,  
  •   Invited Professorship, University of Nancy: may 1983,
  •   Invited Professorship, University of Salt Lake City: spring 1989.   
  •   Invited Professorship, University of California at Los Angeles: spring 1992.   
  •   Invited Professorship, University of Grenoble-I: spring 1993.  
  •   Invited Professorship, Tata institute (Mumbai): fall  2007.

Students
  • R. Brand 1980, co-directed  with A. Van de Ven.
  • D. Epema 1983,  Leiden, co-directed with J. Murre.
  • T. Kalker  1986, Leiden, co-directed with A. Van de Ven.
  • L. van Gastel  1989, Utrecht,   co-directed with F. Oort.
  • R. Brussee 1992, Leiden, co-directed with A. Van de Ven.
  • J. Spandaw 1993, Leiden, co-directed with A. Van de Ven.
  • J. Mulder  1993, Leiden, co-directed with J. Murre.
  • R. Laterveer 1997 Chargé de recherche  (CNRS), Strasbourg.
  • J. Nagel 1997 co-directed with J. Murre (Leiden);  Prof.  Univ. Bourgogne at Dijon.
  • M. Halic 1999;   Associate prof. KFUPM, Dahran Saudi Arabia.
  • M. Leoni 1999 ; Prof. agréé (teacher secondary school at highest level).
  • G. Li 2011-2014; did not complete thesis.


Further details



Research:

  • Here is a non-technical  description  of my field of research  Complex Algebraic Geometry, i.e the study of algebraic varieties. These are the solutions of a finite set of polynomials. In complex geometry one only looks at complex solutions: the circle x2+y2=1 should be the same as x2+y2=-1 which has no real solutions. Also, in order not to lose solutions at infinity, one looks at homogeneous equations. In our example, replacing x2+y2=1 by x2+y2=z2 one adds the two ``circle points'' (1,i,0) and (1,-i,0) at infinity. Finally, self-crossings such as for two intersectiong lines should be avoided. The objects thus described are called ``projective manifolds''. They can be viewed as algebraic objects (solution sets of equations), geometric or topological objects (``manifolds''), and as such one can study them by means of algebraic, topological, differential geometric and complex-analytic tools. The first basic invariant is the number of free parameters, the dimension: our example, the circle is 1-dimensional, essentially described by the angle parameter (in complex geometry this must be replaced by a complex parameter). A circle is an example of an algebraic curve or Riemann surface (its dimension is 1). A variety of dimension 2 is called an algebraic surface. More generally, a complex surface is a complex manifold of dimension 2, i.e it locally looks like an open subset of C2. Within the (large) field of complex algebraic geometry my specialization comprizes several subfields, the two most important of which are the fields of complex surfaces and Hodge theory.
    • The classification of algebraic surfaces is largely due to Castelnuovo (1865-1952) and Enriques(1871-1946). Their classification has been extended to include the non-algebraic surfaces by Kodaira (1915-1995). There is much recent activity, including by physicists who are using detailed results about K3-surfaces for string theory (for instance this prepublication).
    • One of the goals of Hodge theory is to measure and describe what happens if you introduce extra ``parameters'' in the equations. In our example, one might want to vary the radius of the circle. A related question is to describe the subsets of a given variety which themselves are given by supplementary polynomial equations; these are called ``subvarieties, or more generally ``algebraic cycles''. The theory has been named after Sir William Hodge (1903-1975) who boosted research in this area after he formulated a conjecture. In fact it is one of the millenium prize questions posed by the Clay Foundation.

      Hodge theory has links to other fields. It has for instance been discovered that Hodge theory is a natural tool to attack the so called ``mirror conjecture'', a conjecture posed by mathematical physicists. See for instance this book.

  • Webpages related to my research:


Publications



 


  1. The local Torelli theorem I,  Complete intersections, Math. Ann 217  (1974), 1-16.
  2. The local Torelli theorem II, Cyclic branched coverings, Ann. Sc. Norm. Pisa, Cl. Sc. Ser IV, II (1976) 321-339.
  3. (with  J. SIMONIS) A secant formula, Quart. J. math.  27  (1976) 182--189.  
  4. On two types of surfaces with vanishing geometric genus, Inv. Math. 32 (1976) 33--47. 
  5. On certain examples of surfaces with  pg =0 due to Burniat, Nagoya Math. J., 66  (1977) 109-119.
  6. (with D. LIEBERMAN, R. WILSKER) A theorem of local Torelli-type, Math. Ann.   231  (1977) 39-45. 
  7. Holomorphic automorphisms of compact Kähler surfaces and their induced  actions in cohomology, Inv. Math.  52  (1979) 143-148.  
  8. On automorphisms of compact Kähler surfaces,  in Journées de géométrie algébrique d'Angers (juillet 1979) Sijthoff-Noordhoff (1980) 249-267.
  9. (with  E. LOOIJENGA) Torelli theorems of Kähler K3-surfaces, Comp. Math.  42  (1981) 145-186.
  10. (with F. OORT) A Campedelli surface with torsion group Z/ 2Z , Indag. Math.  3  (1981) 399-407.
  11. (with  W. BARTH) Automorphisms of Enriques surfaces, Inv. Math.  73  (1983)  383-411. 
  12. (with  F. BEUKERS) A family of K3-surfaces and  ζ(3), J. für reine u. angew. Math.  351  (1984) 42-54.
  13. A criterion for flatness of Hodge bundles over curves and geometric applications, Math. Ann. 268  (1984) 1-19.
  14. Some applications of the Lefschetz fixed point theorems in (complex) algebraic geometry, in  The Lefschetz Centennial Conference, I: Proc. Alg. geometry, Cont. Math.  58 (part 1) (1985) AMS, Providence, R.I., p.213--221.
  15. Monodromy and Picard-Fuchs equations for families of K3-surfaces and elliptic curves, Ann. ENS. 19 (1986) 583--607.
  16. On Arakelov's finiteness theorem for higher dimensional varieties.  Rend. Sem. Mat. Univ. Politec. Torino, 1986.
  17. Some remarks on Reider's article  "On  the infinitesimal  Torelli  theorem for certain irregular surfaces", Math. Ann. 281, 315--324 (1988). 
  18. (With J.  STIENSTRA) A pencil of K3-surfaces related to Apéry's recurrence for ζ(3) and Fermi surfaces for potential zero.  Arithmetic of Complex Manifolds, Proc. Erlangen 1988, Springer Lect. Notes Math.  1399, pp. 110-127, 1989. 
  19. Rigidity for variations of Hodge structure and Arakelov-type finiteness theorems, Comp. Math. 75, 113--126 (1990). 
  20. Maximal rank non-rigid variations of Hodge structure of weight one and two, in   Complex algebraic varieties. Proc. Bayreuth 1990, Springer Lecture Notes 1507, 157-162, Berlin etc., Springer Verlag (1992).
  21. (with  J. TOP, M. van der VLUGT) The Hasse zeta function of a K3 surface related to the number of words of weight 5 in the Melas codes.  J. reine angew. Math.  432  (1992) 151-176.
  22. (With U. PERSSON) Homeomorphic non-diffeomorphic surfaces with small invariants. Manuscripta Math.  79  (1993) 173-182.
  23. (With U. PERSSON) Some aspects of the topology of algebraic surfaces. In: Israel Mathematical Conference Proceedings (IMCP) AMS (1995).
  24. (With U. PERSSON, X, GIAO),  Geography of Spin surfaces, Topology 35  (1996) 845--862.  (publised version)
  25. Lawson homology for varieties with small Chow groups and the induced filtration of the Griffiths group, Math. Zeitschrift 234 (2000) 209-223 (pdf)
  26. With J. STEENBRINK) Degeneration of the Leray spectral sequence for certain geometric quotients math. AG/0112093 Moscou Math. Journal  3 (2003) 1-11.
  27. (With J. STEENBRINK) Monodromy of variations of Hodge structure, Acta Applicanda Math. 75 (2003) 183-194.
  28. (With J. STEENBRINK) Hodge Number Polynomials for Nearby and Vanishing Cohomology,  pp. 597-611  (2006)  in Algebraic Cycles and Motives. Also as preprint available.
  29. (With  J. STEENBRINK) Hodge Theory: the search for purity, in Proceedings of the Trieste Singularity Summer School and Workshop ICTP, Trieste, Italy    15 August - 3 September 2005, edited by Jean-Paul Brasselet, World Scientific, Singapore etc., 556--583 (2007).
  30.  (With F. GALLUZZI et G. LOMBARDO) Automorphs of indefinite  binary  quadratic forms  and   K3-surfaces with Picard number 2.  Rendiconti Mat. Torino, 68 (2010) 57-77, preprint ArXiv mathAG/0804.0725
  31.  Bloch-type conjectures and an example of a threefold of general type.  Contemporary Mathematics 12  (2010) 587-605. 
  32.  (With   Morihiko SAITO)  Lowest Weights in Cohomology of Variations of Hodge Structure.  Nagoya Math. J. 206 (2012), 1-24. Here for the complete version and here arxiv:0708.0130v2   for a first version.
  33.  (with   V. SRINIVAS and S. MÜLLER-STACH)  Abelian varieties   and theta functions as invariants for   compact Riemannian manifolds;  constructions inspired by superstring theory.  Journal de mathématiques pures et appliquées  97 (2012) (pdf); see also the video  presentation on the occasion of Alberto Contes 70th birthday.





       

Teaching: some course notes

PhD-courses

  • Motivic Aspects of Hodge theory (the Hodge characteristic) course notes for the summer school in Istanbul 2014
  • Torelli theorems  course notes for Springschool Rennes 2014
  • Complex Surfaces  long version with background, suitable for PhD students;
  • Complex Surfaces short version

Third year (all in french)

Magistère:

Master 1 (all in french)

 

Some pictures


Last modified: September  2014