The asymptotic profile of $χ_y$-genera of Hilbert schemes of points on K3 surfaces

Abstract

The Hodge numbers of the Hilbert schemes of points on algebraic surfaces are given by Göttsche's formula, which expresses the generating functions of the Hodge numbers in terms of theta and eta functions. We specialize in this paper to generating functions of the $χ_y(\mathrm{K3}^{[n]})$ genera of Hilbert schemes of $n$ points on K3 surfaces. We determine asymptotic values of the coefficients of the $χ_y$-genus for $n\to \infty$ as well as their asymptotic profile.

16 pages, comments welcome, some typos fixed, accepted for publication in "Communication in Number theory and Physics"