Period maps and the Gauss-Manin connection

Abstract: For a family of smooth projective varieties over a number field, we have a complex period map and a $p$-adic period map, and they are both governed by the (algebraic) Gauss-Manin connection. After some preliminaries, we introduce these objects and prove some bounds on the dimensions of their images.

Reference:

$\bullet$ Lawrence and Venkatesh, Diophantine problems and $p$-adic period mappings, Section 3.

For more details:

$\bullet$ Deligne, Hodge cycles on abelian varieties, Section 2 (for Gauss-Manin connection).

$\bullet$ Katz and Oda, On the differentiation of De Rham cohomology classes with respect to parameters

$\bullet$ Voisin, Hodge Theory and Complex Algebraic Geometry I, Part III.

$\bullet$ Hotta, Takeuchi and Tanisaki, D-Modules, Perverse Sheaves, and Representation Theory (for more on Riemann-Hilbert).

algebraic geometrynumber theory

Audience: advanced learners

( slides )

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STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Fall 2025 topic: Weil conjectures.

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.

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