Variations of Hodge Structures

1. Hodge structures: definition and construction. Period space. Torelli theorems. Polarization. Cartan’s classification of symmetric Hermitian spaces and its applications to Hodge theory.
2. Variation of Hodge structures. Gauss-Manin connection. Griffiths transversality condition and its proof.
3. Holomorphic sectional curvature. Kobayashi metric. Generalized Schwarz-Pick lemma and its applications to variations of Hodge structures.
4. Griffiths’ theorem on rigidity of variations of Hodge structures (two polarized VHS on the same flat bundle over a compact manifold are equal if they are equal in one point). Deligne’s theorem on semisimplicity of monodromy representations.
5. Griffiths’ proof that the period map on Poincare disk is distance-decreasing. Quasi-unipotency of monodromy of a polarized VHS on a punctured disk.
6. Simpson’s construction of VHS from stable Higgs bundles and its application to uniformization (without a proof).

Claire Voisin, Hodge theory. Topics in Transcendental Algebraic Geometry, ed. by Phillip A. Griffiths.
C. A. M. Peters, J. Steenbrink “Monodromy of variations of Hodge structure”, Acta Applicanda Math. 75 (2003) 183-194.
C. A. M. Peters, “Curvature for period domains”, в книге “Complex Geometry and Lie Theory”, ed. James A. Carlson, C. Herbert Clemens, David R. Morrison, Proceedings of Symposia of Pure Mathematics vol. 53, 1992.
Phillip A. Griffiths, Periods of integrals on algebraic manifolds, III (some global differential-geometric properties of the period mapping) Publications de Mathematiques de IHES, January 1970, Volume 38, Issue 1, pp 125-180.
Vik. S. Kulikov and P. F. Kurchanov, Complex Algebraic Varieties: Periods of Integrals and Hodge Structures, in Algebraic geometry III. Complex algebraic varieties. Algebraic curves and their Jacobians, Encyclopaedia of Mathematical Sciences, Springer, 1997.