Hodge theory and period maps (Spring 2026)

 

What is this course about?

This course introduces Hodge theory and period maps, which lie at the intersection of complex geometry, algebraic geometry, and arithmetic. Starting from the Hodge decomposition of the cohomology of smooth complex algebraic varieties, we develop variations of Hodge structure and their basic properties. We then study period domains and period maps, which encode how Hodge structures vary in families, and explore their geometric and arithmetic significance.​

 

Prerequisite

Basic knowledge of complex manifolds.

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Literature

[1] E. Looijenga, Hodge theory and period maps, 2020, available at https://webspace.science.uu.nl/~looij101/CoursenotesHodge.pdf

[2] R.O. Wells, Jr: Differential Analysis on Complex Manifolds.

[3] Ph. Griffiths, J. Harris: Principles of Algebraic Geometry. 

[4] Ph. Griffiths, On the periods of certain rational integrals I, II, Ann. of Math. 90 (1969), 460-495, 496-541.

[5] H. Clemens, Ph. Griffiths: The intermediate Jacobian of the cubic threefold.  Ann. of Math. 95 (1972), 281–356.

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Place and time

Tuesdays, 08:50-12:15; Venue: A3-1a-205

Zoom: 293 812 9202, PW: BIMSA

Starting from Mar. 10

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WeChat Group

We have created a WeChat Group for communication​. You could write to me if you would like to join.

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Material covered

Mar. 10: Hodge theory for Riemannian manifolds.

Mar. 17: Complex vs real linear algebra, complex manifolds, holomorphic vector bundles.

Mar. 24: Hermitian forms, Kahler metrics, the exterior algebra of a Hermitian vector space.

Mar. 31: On travel.

Apr. 7: On travel.

Apr. 14: Representation theory of sl(2), local identities on a Kahler manifold.

Apr. 21: Local identities on a Kahler manifold (continued), main theorem of Kahlerian Hodge theory.

Apr. 28: Effective Hodge structures of weight 1, polarizability as a restrictive condition.

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