Minimal triangulations of Kummer varieties

By an (abstract) Kummer variety K^d we mean the d-dimensiona1 torus T^d modulo the involution ? ? — ?. The 2^d elements in T^d of order two are fixed points of the involution and therefore K^d has 2^d isolated singularities (for d ≧ 3). Any simplicial decomposition of K^d must have at least as many vertices. In this paper we describe a highly symmetrical simplicial decomposition of K^d with 2^d vertices such that the link of each vertex is a combinatorial real projective space ?P^d-1 with 2^d—1 vertices. The automorphism group of order (d + 1)! 2^d admits a natural representation in the affine group of dimension d over the field with two elements. A particular case is the classical Kummer surface with 16 nodes (d=4). In this case our 16-vertex triangulation has a close relationship with the abstract Kummer configuration 16₆.