Hirzebruch-Riemann-Roch Formulae on Irreducible Symplectic Kähler Manifolds

Abstract

In this article we investigate Hirzebruch-Riemann-Roch formulae for line bundles on irreducible symplectic Kähler manifolds. As Huybrechts has shown, for every irreducible complex Kähler 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 $\chi (L)=(n+1)\left(\genfrac{}{}{0ex}{}{q(c_1(L))/2+n}{n}\right)$ 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 $n\le 5$. For $n\le 4$ these results were previously obtained by Sawon.

24 pages, includes dbnsymb font