Quotients of E^n by a_(n+1) and Calabi-Yau manifolds

Abstract

We give a simple construction, for n ≥ 2, of an n-dimensional Calabi-Yau variety of Kummer type by studying the quotient Y of an n-fold self-product of an elliptic curve E by a natural action of the alternating group a_(n+1) (in n + 1 variables). The vanishing of H^m (Y, O_Y) for 0 < m < n follows from the lack of existence of (non-zero) fixed points in certain representations of a_(n+1). For n ≤ 3 we provide an explicit (crepant) resolution X in characteristics different from 2, 3. The key point is that Y can be realized as a double cover of ℙ^n branched along a hypersurface of degree 2(n + 1).