Lecture class: Algebraic Geometry 3 (Winter 2026/27)

This is the continuation of the Algebraic Geometry 2 class I taught in the summer 2026.

We will continue our study of the cohomology of quasi-coherent $\mathscr O_X$-modules, and then apply those (and other scheme-theoretic) methods to study abelian varieties. An abelian variety (over a field $k$, say) is a connected smooth proper group scheme. (Interestingly, this definition implies that the group scheme is commutative, justifying the name.) They provide an interesting class of varieties whose basic properties can be fairly well understood. Nevertheless, as already the example of $1$-dimensional abelian varieties – elliptic curves – shows, the study of abelian varieties leads to some deep questions, some of whose are the theme of current research.

Date/time: Mon, 2-4pm, Tue, 10am-12pm; S-U-3.02.

Exercise group (led by Andreas Pieper): Wed, 2-4pm, N-U-4.04.

References to the literature: Much of the material is covered in my books with Wedhorn. Other standard references are the book of Hartshorne, the Stacks project, Mumford’s book on abelian varieties, and the manuscript of Edixhoven, van der Geer and Moonen on abelian varieties.