Higher dimensional Enriques varieties with even index (Q2846898)

Initiated by the work of \textit{K. O'Grady} [J. Algebr. Geom. 12, No. 3, 435--505 (2003; Zbl 1068.53058)] higher-dimensional holomorphic symplectic manifolds have been studied by many authors over the last decade. More recently, \textit{S. Boissière, M. Nieper-Wißkirchen} and \textit{A. Sarti} [J. Math. Pures Appl. (9) 95, No. 5, 553--563 (2011; Zbl 1215.14046)] and \textit{K. Oguiso} and \textit{S. Schröer} [J. Reine Angew. Math. 661, 215--235 (2011; Zbl 1272.14026)] started to discuss the structure of finite étale quotients of compact simply connected Kähler manifolds with trivial canonical bundle. These quotients are called Enriques manifolds since they are considered as higher-dimensional analogues of Enriques surfaces which are étale quotients of \(K3\) surfaces. While symplectic manifolds typically have some non-algebraic deformations, most of the known examples of Enriques manifolds are projective.NEWLINENEWLINEIn the paper under review, the author proves a classification result for a certain class of Enriques manifolds: let \(Y\) be a compact Kähler manifold of dimension \(2n-2\) such that \(K_Y\) is torsion of order \(n\), the fundamental group has order \(n\) and the holomorphic Euler characteristic is equal to one. If \(n=2m\) with \(m\) an odd prime, the manifold \(Y\) is the quotient of a product of a Calabi-Yau manifold and an irreducible holomorphic symplectic manifold of dimension \(2m-2\). If \(n=4\) then \(Y\) is the quotient either of an irreducible holomorphic symplectic manifold or of two Calabi-Yau manifolds of even dimension. As a consequence of this classification the author proves that all these manifolds are projective.