# Première semaine

### Les aspects complexes

• (1C) Les structures de Hodge par Damien Mégy
Résumé : à venir
• (2C) Les variétés complexes en famille par Jan Nagel
Résumé  Lecture 1 : Preliminaries Families of (projective algebraic or compact Kaehler) manifolds; examples (hypersurfaces, elliptic curves); Ehresmann fibration theorem; Kodaira-Spencer map; monodromy, connections and local systems; Gauss-Manin connection. Lecture 2 : Variations of Hodge structure: analytic point of view Hodge numbers are constant in a smooth family; Hodge bundles; period matrix, local period map, period domain; examples (curves, surfaces); period map is holomorphic, differential of the period map; short discussion of the global period map. Lecture 3 Variations of Hodge structure: algebraic point of view Some homological algebra (hypercohomology etc); Deligne's algebraic description of the Hodge bundles; algebraic description of the Gauss-Manin connection (Katz-Oda); comparison with analytic theory; application: algebraic description of the differential of the period map for hypersurfaces in projective space. References: - Carlson, James; Mueller-Stach, Stefan; Peters, Chris Period mappings and period domains. Cambridge Studies in Advanced Mathematics, 85. Cambridge University Press, Cambridge, 2003. - Voisin, Claire. Hodge theory and complex algebraic geometry. I, II. Cambridge Studies in Advanced Mathematics, 76. Cambridge University Press, Cambridge, 2007.
• (3C) Le théorème de Torelli pour les courbes et les surfaces K3 par Chris Peters
Résumé : Lecture 1 : Andreotti's classical beautiful proof of the Torelli theorem for curves (dating from 1958). The lecture ends with variational techniques due to Carlson, Green, Griffiths and Harris (1980), the curve case serving as a model case. Lecture 2 and 3 : A proof of the Torelli theorem for projective K3--surfaces modeled on the original proof of Piate\v ckii-Shapiro and \v Safarevi\v c (1971) but using the approach in the Kähler case as given by Burns and Rapoport (1975), with modifications and simplifications by Looijenga and Peters (1981). If time allows for it I shall briefly point out some related developments. I particularly want to say something about derived Torelli and also about Verbitsky's recent proof of Torelli for hyperkähler manifolds.

• (1p) Cohomologies p-adiques par Pierre Berthelot
Résumé : à venir
• (2p) Les théorèmes de comparaison p-adiques : énoncés par Laurent Berger
Résumé : à venir
• (3p) Les théorèmes de comparaison p-adiques : démonstration dans le cas de Fontaine-Messing par Farid Mokrane
Résumé : à venir

# Deuxième semaine

### Les aspects complexes

• (4C) Les travaux de Schmid par Philippe Eyssidieux
Résumé : à venir
• (5C) Le théorème de décomposition des images directes par Luca Migliorini
Résumé : 1. First lecture: Two classical results on surfaces: the Grauert Mumford contractibility criterion and the Zariski lemma. A quick reminder on constructible sheaves and their derived category, and an interpretation of the previous two results in terms of splitting of the direct image of the constant sheaf. 2. Second Lecture: The intersection cohomology complex. Examples: Intersection cohomology of some class of singular varieties. The complex of Cattani Kaplan Schmid and L^2 cohomology. 3. Third lecture: The Decomposition theorem. Examples application, a sketch of proof for semismall maps
• (6C) L'hyperbolicité d'espaces des modules par Stefan Kebekus
Résumé : à venir