de Rham cohomolgy and Gauss-Manin connections

Abstract: We state the equivalence of algebraic and analytic de Rham cohomologies for vector bundles with regular integrable connections and discuss a relative version. We then discuss the Gauss-Manin connection, which is obtained on the derived pushforward sheaves of an integrable connection and is a prominent example of connections considered in practice. We show that the Gauss-Manin connection is regular if the integrable connection we start with is regular. We will try to provide examples along the way to elucidate the theory.

algebraic geometrynumber theory

Audience: advanced learners

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.