Algebraic Geometry Seminar

Batyrev introduced stringy Hodge numbers as a generalization of classical Hodge numbers to singular varieties, motivated by mirror symmetry. Unlike ordinary Hodge numbers, stringy Hodge numbers are not defined as dimensions of vector spaces, and their non-negativity is far from obvious. Batyrev conjectured that the stringy Hodge numbers of a projective variety with Gorenstein canonical singularities are non-negative whenever its stringy E-function is a polynomial.

In this talk, I will introduce stringy Hodge numbers and discuss their connections to motivic integration and birational geometry. I will then explain recent joint work with Matthew Satriano in which we settle Batyrev’s conjecture: it holds in dimensions at most four but fails in every dimension at least five. More generally, we give a complete classification of the stringy Hodge numbers that are universally non-negative in each dimension.