First week

Complex aspects

  • (1C) Hodge structures by Damien Mégy
    Abstract : TBA
  • (2C) Complex manifolds in family by Jan Nagel
    Abstract : 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) Torelli theorem for curves and K3 surfaces by Chris Peters
    Abstract : 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.

p-adic aspects

  • (1p) p-adic cohomologies by Pierre Berthelot
    Abstract : TBA
  • (2p) p-adic comparison theorems: statements by Laurent Berger
    Abstract : TBA
  • (3p) p-adic comparison theorems: proof in the Fontaine-Messing setting by Farid Mokrane
    Abstract : TBA

Second week

Complex aspects

  • (4C) The work of Schmid by Philippe Eyssidieux
    Abstract : TBA
  • (5C) Decomposition theorem for direct images by Luca Migliorini
    Abstract : 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) Hyperbolicity of moduli spaces by Stefan Kebekus
    Abstract : TBA

p-adic aspects

  • (4p) Filtered (φ,N)-modules by Olivier Brinon
    Abstract : TBA
  • (5p) Ramification of cristalline representations by Shin Hattori
    Abstract : TBA
  • (6p) Projective varieties over the rationals with good reduction by Viktor Abrashkin
    Abstract : TBA

Possible schedule

FIRST WEEK SECOND WEEK
Monday 12 Tuesday 13 Wednesday 14 Thursday 15 Friday 16 Monday 19 Tuesday 20 Wednesday 21 Thursday 22 Friday 23
9h - 10h30 Welcome Course 1p Course 1p Course 3p Course 3p Welcome Course 4p Course 4p Course 6p Course 6p
11h - 12h30 Course 1C Course 2C Course 2C Course 2C Course 3C Course 4C Course 5C Course 5C Course 4C Course 6C
14h - 15h30 Course 1p Course 2p Course 2p Course 2p Course 3p Course 4p Course 5p Course 5p Course 5p Course 6p
16h - 17h30 Course 1C Course 1C Course 3C Course 3C Course 4C Course 5C Course 6C Course 6C