Quotients of Calabi-Yau Generalized Fermat manifolds and Log Enriques varieties

Abstract: Enriques manifolds and Log Enriques varieties have attracted a lot of interest in the last years. In this paper we study Enriques manifolds and Log Enriques varieties that are respectively étale and quasi-étale quotients of Calabi-Yau manifolds. We produce several examples by studying quotients of generalized Fermat manifolds of Calabi-Yau type. These are complete intersections of Fermat type manifolds all of the same degree and were described recently by Hidalgo, Hughes and Leyton-Álvarez. We describe all their quotients by a fixed-point-free automorphism. Most of the quotients are Log Enriques varieties or Calabi-Yau varieties with terminal singularities. In one case we get a family of Enriques manifolds which is a subfamily of a family described previously by Boissière, Camere and the third author, which is one of the two so far known families of Enriques manifolds of Calabi-Yau type. This family is obtained as an étale quotient of order two of a Calabi-Yau manifold complete intersection of quadrics, the other family is the Hilbert scheme of $n$ points on an Enriques surface. In the last part of the paper we show that these two families are different by computing their second Betti numbers, which are equal to one and eleven respectively.