o-minimal GAGA and applications to Hodge theory
Hodge structures on cohomology groups are fundamental invariants of algebraic varieties; they are parametrized by quotients D/¿ of period 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 in which any period map is definable. The algebraicity of Hodge loci is an immediate consequence via a theorem of Peterzil--Starchenko. We further prove a general GAGA type theorem in the definable category, and deduce some finer algebraization results. This is joint work with Y. Brunebarbe, B. Klingler, and J.Tsimerman.