On the elliptic genus of generalised Kummer varieties

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 Aⁿ>^]] 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]).The author was supported by the Deutsche Forschungsgemeinschaft.Revised version: 14 November 2003