Variations of Hodge structures
(one lecture, one problem session every week,
January-February 2024)
Misha Verbitsky

A Hodge structure on a vector space is
its decomposition onto a direct sum of (p,q)-components,
similar to the Hodge decomposition on cohomology.
Usually on top of this decomposition one assumes
existence of some integer or rational structure,
and a pseudo-Riemannian metric compatible with
this rational structure, called the polarization.
A variation of Hodge structure is a flat
vector bundle, equipped with a Hodge structure
at every point (usually non-constant), in
such a way that the (0,1)-part of connection
takes the (p,q)-component to (p,q)-component
plus (p+1, q-1)-component; this is called
the Griffiths transversality condition. This
is the structure which occurs on cohomology
of a complex family of projective varieties;
however, the variations of Hodge structure
can be studied separately from the geometry
of families of projective varieties.

I will give an introduction to Hodge structures
and the variations of Hodge structures, giving
a proof of foundational results of Deligne and
Griffiths, and explain how these results may be
applied to elementary examples, such as the
abelian varieties and the K3 surfaces.

Program:

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 monodromyš
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).

Literature:

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. https://www-fourier.ujf-grenoble.fr/~peters/Articles/PubSteen2.pdf

C. A. M. Peters, ``Curvature for period domains'', in ``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, šSpringer, š1997.

Alex Wright,  Deligne's Theorem on the
semisimplicity of variations of Hodge structures
https://public.websites.umich.edu/~alexmw/other.html

V. A. Vassiliev, Ramified integrals, singularities and
lacunas, Math. Appl., 315, Kluwer Academic Publishers,
Dorderecht (Netherlands), 1995, xviii+289 pp. Russian translation:
https://files.nehudlit.ru/books/004/vetvyashchiesya-integraly.djvu

Eduard Looijenga, Topology of Algebraic Varieties
(Notes for a course taught at the YMSC, Spring 2016)
https://webspace.science.uu.nl/~looij101/Topalgvar.pdf

Simon Donaldson,
Lefschetz pencils and mapping class groups
https://www.ma.imperial.ac.uk/~skdona/MCGROUP.PDF

Site:
http://verbit.ru/IMPA/VHS-2024