On Kummer Construction

We discuss a generalization of Kummer construction which, on the base of an integral representation of a finite group and local resolution of its quotient, produces a higher dimensional variety with trivial canonical class. We use virtual Poincaré polynomials with coefficients in a ring of representations and McKay correspondence to compute cohomology of such Kummer varieties. 1. The construction. 1.1. The set-up. 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 which is because the group action kills the fundamental group of the abelian surface and preserves the canonical form, and also 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, [Kum75] or [BPVdV84]. We try to investigate Kummer construction in a general context. Our set up is as follows: • G is a finite group with an integral representation (usually irreducible) ρZ : G→ GL(r,Z) whose fixed point set is {0}, • A is a complex abelian variety of dimension d, with neutral element, addition and substraction denoted by 0 and ±, and if d is odd then we assume additionally det(ρZ) = 1, that is ρZ : G→ SL(r,Z) The first step of the generalized Kummer construction is achieved by the induced action (1.1) ρA = ρZ ⊗Z A : G −→ Aut(A) which is obtained by identification A = Z ⊗Z A. In other words, G acts on A with integral matrices coming from the representation ρZ. If a is the tangent space of A at identity, or Lie algebra of holomorphic vector fields tangent to A, and exp : a→ A the exponential map then the tangent action ρa = ρZ ⊗Z a : G → GL(a) splits into d copies of complexified representation ρC = ρZ ⊗Z C. The representation ρa is tangent to ρA: for g ∈ G and p ∈ A the derivative of the map ρA(g) : A → A at p is ρa(g) : TpA = a → Tg(p) = a. 1991 Mathematics Subject Classification. 20C10, 14E15, 14F10, 14J28, 14J32. Research of the first author was supported by grants of Italian Mur-PRIN. Research of the second author was financed by grants of Polish MNiSzW (N N201 2653 33). Moreover, the second author thanks University of Trento for supporting his visits. Many thanks to Andrzej Weber for