Hirzebruch-Riemann-Roch Formulae on Irreducible Symplectic K\"ahler Manifolds

Abstract

In this article we investigate Hirzebruch-Riemann-Roch formulae for line bundles on irreducible symplectic K\"ahler manifolds. As Huybrechts has shown, for every irreducible complex K\"ahler manifold $X$ of dimension $2n$, there are numbers $a_0, a_2, ..., a_{2n}$ such that $\chi(L) = \sum_{k = 0}^n a_{2k}/(2k)! q_X(c_1(L))^k$ for the Euler characteristic of a line bundle $L$, where $q_X: H^2(X, \mathbbm C) \to \mathbbm C$ is the Beauville-Bogomolov quadratic form of $X$. Using Rozansky-Witten classes similar to Hitchin and Sawon, we obtain a formula expressing the $a_{2k}$ in terms of Chern numbers of $X$. Furthermore, for the $n$-th generalized Kummer variety $\KA n$, we prove $\chi(L) = (n + 1) \binom{q(c_1(L)) / 2 + n} n$ by purely algebro-geometric methods, where $q$ is the form $q_X$ up to a positive rational constant. A similar formula is already known for the Hilbert scheme of zero-dimensional subschemes of length $n$ on a K3-surface. Using our results, we are able to calculate all Chern numbers of the generalized Kummer varieties $\KA n$ for $n \leq 5$. For $n \leq 4$ these results were previously obtained by Sawon.