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.

  1. Hodge structures. The infinitesimal period relation. Variation of Hodge structure.
  2. (polarized and unpolarized) Period domains. Period mappings. The horizontal subbundle.
  3. Kobayashi pseudometric. The generalized Schwarz lemma. Distance decreasing properties and negatively curved spaces.
  4. Negativity of the horizontal tangent bundle.
  5. The nilpotent orbit theorem (statement). Relation with Deligne's canonical extension. Algebraicity of Hodge bundles. Regularity of Gauss-Manin connections.
  6. 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.
  7. The statement of SL2-orbit theorem. Applications to Landman's theorem. The solution of the monodromy-weight problem in the realm of Hodge theory.
  8. 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

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