Quotients of E by an+1 and Calabi-Yau manifolds

We give a simple construction, for n ≥ 2, of an ndimensional 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 an+1 (in n + 1 variables). The vanishing of H(Y,OY ) for 0 < m < n follows from the lack of existence of (non-zero) fixed points in certain representations of an+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 Pn branched along a hypersurface of