 
Summary: ON THE KUMMER CONSTRUCTION
MARCO ANDREATTA AND JAROSLAW A. WI“SNIEWSKI
Abstract. We discuss a generalization of the Kummer construction. Namely
an integral representation of a finite group produces an action on an abelian
variety and, via a crepant resolution of the quotient, this gives rise to a higher
dimensional variety with trivial canonical class and first cohomology. We use
virtual Poincar“e polynomials with coefficients in a ring of representations and
McKay correspondence to compute cohomology of such Kummer varieties.
1. Introduction
Kummer surfaces are constructed in a two step process: (1) divide an abelian surface
by an action of an involution, (2) resolve singularities of the quotient, which arise
from the fixed points of the action, by blowing them up to (2)curves. The result
of this process is a K3 surface, this is because the group action kills the fundamental
group of the abelian surface and preserves the canonical form, and also because the
resolution is crepant. The invariants of this surface can be computed by looking
at the invariants of the involution and the contribution of the resolution. This
construction is classical, see [Kum75] or [BPVdV84].
It is natural to ask about a generalization of the above procedure. This involves
dividing an abelian variety by an action of a finite group. Our set up is as follows:
· G is a finite group with an irreducible integral representation Z : G
