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jeunes en arithmétique et variétés algébriques

MIXED HODGE MODULES: APPLICATIONS

program by Claude Sabbah
tatihou island (normandy)
22-26 July 2019

The Betti cohomology of a smooth projective complex variety carries a pure Hodge structure. More generally, if the variety is singular or noncompact, its cohomology is endowed with a mixed Hodge structure in the sense of Deligne. In a family of smooth projective varieties, the cohomogy groups of the members of the family form a variation of pure Hodge structures on the base of the family. What is the corresponding object for a family of varieties, some of whose members could be singular or non-compact? It is a mixed Hodge module!

The theory of mixed Hodge modules has been developped by Morihiko Saito in the 80's. It has received many applications: internal to Hodge theory (Saito), to complex geometry (Popa–Schnell), to singularities of complex varieties (Kebekus–Schnell), to representation theory (Schmid–Vilonen), and many others.

This is however a difficult theory to learn. The prerequisites are significant, and include perverse sheaves, $\mathcal{D}$-modules, and degeneration of Hodge structures. There have been for a long time very few accessible references, but this is changing thanks to the book in preparation by Sabbah and Schnell.

The aim of the workshop would be to provide an entry point to the theory, with applications in mind, and adapted to a wide range of algebraic or arithmetic geometers. The $\mathcal{D}$-module point of view will be emphasized, and the goal would be to reach the Popa–Schnell theorem: `a holomorphic 1-form on a complex variety of general type vanishes on at least one point' (Ann. of Math. 2014), or Schnell's work on $\mathcal{D}$-modules on abelian varieties (Publ. Math. IHES 2015). As in past years, we would also like to give some space to young researchers working in the area to present their results.

ORGANIZERS

Giuseppe Ancona (Université de Strasbourg) /
Olivier Benoist (DMA - École Normale Supérieure) /
Javier Fresán (École Polytechnique) /
Marco Maculan (Institut Mathématique de Jussieu) /

SCIENTIFIC COMMITTEE

Anna Cadoret (Institut Mathématiques de Jussieu)
François Charles (Université Paris-Sud XI)
Jérôme Poineau (Université de Caen Basse-Normandie)
Claude Sabbah (École Polytechnique)
Christian Schnell (Stony Brook)