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International Mathematics Research Notices (2007) Vol. 2007 : article ID rnm016, 14 pages, doi:10.1093/imrn/rnm016 published on May 24, 2007
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Copyright © The Author 2007. Published by Oxford University Press.

Stringy Hodge Numbers for a Class of Isolated Singularities and for Threefolds

Jan Schepers1, and Willem Veys2

1 Universiteit Leiden, Mathematisch Instituut, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
2 Katholieke Universiteit Leuven, DepartementWiskunde, Celestijnenlaan 200B, 3001 Leuven, Belgium

Correspondence: Correspondence to be sent to: Jan Schepers, Universiteit Leiden, Mathematisch Instituut, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands. e-mail: jschepers{at}math.leidenuniv.nl

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.

Communicated by Enrico Arbarello


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