Stringy Hodge numbers and Virasoro algebra

Abstract

Let $X$ be an arbitrary smooth $n$-dimensional projective variety. It was discovered by Libgober and Wood that the product of the Chern classes $c_1(X)c_{n-1}(X)$ depends only on the Hodge numbers of $X$. This result has been used by Eguchi, Jinzenji and Xiong in their approach to the quantum cohomology of $X$ via a representation of the Virasoro algebra with the central charge $c_n(X)$. In this paper we define for singular varieties $X$ a rational number $c_{st}^{1,n-1}(X)$ which is a stringy version of the number $c_1c_{n-1}$ for smooth $n$-folds. We show that the number $c_{st}^{1,n-1}(X)$ can be expressed in the same way using the stringy Hodge numbers of $X$. Our results provides an evidence for the existence of an approach to quantum cohomology of singular varieties $X$ via a representation of the Virasoro algebra whose central charge is the rational number $e_{st}(X)$ which equals the stringy Euler number of $X$.