I will be running a weekly learning seminar on variations of Hodge structures in Winter 2025.

If you are interested, let me know, fill out the Schej to help us decide a meeting time, and add yourself to the mailing list for announcements.

Content

Hodge structures are an abstraction of the cohomology of compact complex projective manifolds. They are free modules, typically over $\mathbb{Z}$ or $\mathbb{\Q}$, whose base changes to $\mathbb{C}$ satisfy properties like Hodge decomposition and symmetry. Variations of Hodge structures are moduli spaces of Hodge structures, and they in turn allow us to study families of motives/varieties/manifolds. They appear naturally in geometry (both algebraic and complex/differential), but also connect to topics like Shimura varieties, prismatic F-gauges, and more.

Some concepts we hope to discuss are the Kodaira-Spencer map, Gauss-Manin connections, the Riemann-Hilbert correspondence, Griffiths transversality, period domains, and monodromy.

We will probably start by covering Chapters 9 and 10 of Complex Geometry and Hodge Theory Vol. 1, by Claire Voisin.

Prerequisites

Depends on the participants! Currently, beyond the content of the first-year courses, I plan to assume familiarity with sheaves and complex manifolds (e.g. to the level of 2.1, 2.3, 4.1, and 4.3 of Voisin).

Schedule

Mondays, from 10:30 AM to 12 PM

I will speak by default. If you would like to speak then please reach out and we can discuss scheduling, but there’s no pressure at all to do so.

Notes

A view-only link is here. Contact me if you’d like the editing link, but everyone on the mailing list should’ve received it.