Stringy Hodge numbers for a class of isolated singularities and for threefolds

Details Stringy Hodge numbers for a class of isolated singularities and for threefolds Stringy Hodge numbers for a class of isolated singularities and for threefolds

2006-03-13

arxiv.org/abs/math/0603312v3

Int. Math. Res. Not., Vol. 2007, article ID rnm016

Mathematics

Algebraic Geometry

13 pages, minor inaccuracies from version 2 corrected

Scientific paper

Batyrev has defined the stringy E-function for complex varieties with at most log terminal singularities. It is a rational function in two variables if the singularities are Gorenstein. Furthermore, if the variety is projective and its stringy E-function is a polynomial, Batyrev defined its stringy Hodge numbers essentially as the coefficients of this E-function, generalizing the usual notion of Hodge numbers of a nonsingular projective variety. He conjectured that they are nonnegative. We prove this for a class of `mild' isolated singularities (the allowed singularities depend on the dimension). As a corollary we obtain a proof of Batyrev's conjecture for threefolds in full generality. In these cases, we also give an explicit description of the stringy Hodge numbers and we suggest a possible generalized definition of stringy Hodge numbers if the E-function is not a polynomial.

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