On the elliptic genus of generalised Kummer varieties

Abstract

Borisov and Libgober ([2]) recently proved a conjecture of Dijkgraaf, Moore, Verlinde, and Verlinde (see [6]) on the elliptic genus of a Hilbert scheme of points on a surface. We show how their result can be used together with our work on complex genera of generalised Kummer varieties [17] to deduce the following formula, conjectured by Kawai and Yoshioka ([15]), on the elliptic genus of a generalised Kummer variety A[[ n >]] of dimension 2(n−1): Here is the weak Jacobi form of weight −1 and index and V(n) is the Hecke operator sending Jacobi forms of index r to Jacobi forms of index nr (see [7]).