# The Hodge theory seminar

## Summar 2016

**Reference.** Schmid, W. (1973). Variation of Hodge structure: the singularities of
the period mapping. Invent. Math., 22, 211–319.

**Time.** Summer 2016, May 20 - July 15. Meeting weekly.

**Schedule.**

- Hodge structures. The infinitesimal period relation. Variation of Hodge structure.
- (polarized and unpolarized) Period domains. Period mappings. The horizontal subbundle.
- Kobayashi pseudometric. The generalized Schwarz lemma. Distance decreasing properties and negatively curved spaces.
- Negativity of the horizontal tangent bundle.
- The nilpotent orbit theorem (statement). Relation with Deligne's canonical extension. Algebraicity of Hodge bundles. Regularity of Gauss-Manin connections.
- Linear algebra of filtered spaces (gradings, splittings), relation with Lie theory. Mixed Hodge structure and splittings of mixed Hodge structure. Deligne's canonical splitting. ℝ-splitting mixed Hodge structure.
- The statement of SL
_{2}-orbit theorem. Applications to Landman's theorem. The solution of the monodromy-weight problem in the realm of Hodge theory. - Estimates of the weight filtration. The theorem of the fixed part and motivic applications (rigidity theorem, semisimplity of variation of Hodge structure).

## Fall 2016

**Time.** Tuesday afternoon 2pm – 4pm

- Proof of the nilpotent orbit theorem (I)
- Proof of the nilpotent orbit theorem (II)
- Period domains for CVHS; curvature properties of the period domain.
- Complex nilpotent orbit theorem (first try)
- The proof of the complex nilpotent orbit theorem (one variable case)
- The proof of the complex nilpotent orbit theorem (many variables)
- The nilpotent orbit theorem via Jensen's formula