On the Euler characteristic of generalized Kummer varieties

Abstract

We apply the Yau-Zaslow-Beauville method to compute the Euler characteristic of the generalized Kummer varieties attached to a complex abelian surface A (a calculation also done by Göttsche and Soergel by a different method). The argument is very geometric: given a nondivisible ample line bundle L of degree n on A , we construct a projective symplectic variety J with a Lagrangian fibration whose Euler characteristic is n times the number of genus 2 curves in | L |, to wit n 2 Σ m | n m . Using the Mukai-Fourier transform, a degeneration to the case when A is a product of elliptic curves and a result of Huybrechts, we prove that J and the generalized (2 n -2)-dimensional Kummer variety of A are diffeomorphic, hence have the same Euler characteristic.