Seminars & Colloquia Calendar

Abstract:  Hodge structures on cohomology groups are fundamental invariants of algebraic varieties; they are parametrized by quotients D/? of periods domains by arithmetic groups. Except for a few very special cases, such quotients are never algebraic varieties, and this leads to many difficulties in the general theory. We explain how to partially remedy this situation by equipping D/? with an o-minimal structure, and show that period maps are "definable" with respect to this structure.
As a consequence, we obtain an easy proof of a result of Cattani--Deligne--Kaplan on the algebraicity of Hodge loci, a strong piece of evidence for the Hodge conjecture. The proof of the main theorem relies heavily on work of Schmid, Kashiwara, and Cattani--Kaplan--Schmid on the asymptotics of degenerations of Hodge structures.

This is joint work with B. Klingler and J. Tsimerman.