Universiteit Leiden Mathematisch Instituut
Geometry Seminar

Wednesday, 20 September 2006, 16:00 - 17:00, room 401.

Jan Schepers: Stringy Hodge numbers for a class of isolated singularities and for threefolds

Abstract:

(Joint work with W. Veys). In 1997, Batyrev defined the stringy E-function for varieties with at most log terminal singularities. For Gorenstein canonical varieties it is a rational function in two variables. If the variety Y is projective and if the stringy E-function is a polynomial, then Batyrev defined the stringy Hodge numbers of Y, essentially as the coefficients of this polynomial. They satisfy analogous properties as usual Hodge numbers of smooth projective varieties and coincide with them for smooth Y. However, for singular Y it is not at all clear that they are nonnegative. This was conjectured by Batyrev. We proved this conjecture for varieties with certain mild isolated singularities (the allowed singularities depend on the dimension). As a corollary we ob- tained the proof for threefolds in full generality. In the proofs we look at the first coefficients of the power series development of the stringy E-function, and so we do not use the polynomial condition. This suggests that one could define stringy Hodge numbers without assuming it. More complicated examples show however that one has to be careful with this approach.