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Quotients of E^n by A_{n+1} and Calabi-Yau manifolds

Kapil Paranjape·Dinakar Ramakrishnan
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Abstract

We give a simple construction, starting with any elliptic curve E, of an n-dimensional Calabi-Yau variety of Kummer type (for any n>1), by considering the quotient Y of the n-fold self-product of 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 non-existence of (non-zero) fixed points in certain representations of A_{n+1}. For n<4 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).

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Kapil Paranjape, Dinakar Ramakrishnan. 2005-04-21. Quotients of E^n by A_{n+1} and Calabi-Yau manifolds. https://arxiv.org/abs/math/0411310

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