Structure of projective varieties with nef anticanonical divisor: the case of log terminal singularities

Abstract

In this article we study the structure of klt projective varieties with nef anticanonical divisor (and more generally, varieties of semi-Fano type), especially the canonical fibrations associated to them. We show that The Albanese map for such variety is a locally constant fibration (that is, a locally trivial fibration such that X is isomorphic to the product of the universal cover of the Albanese torus by the fibre of the Albanese map quotient by a diagonal action of the fundamenatl group of the Albanese torus). If the smooth locus is simply connected, the MRC fibration of such variety is an everywhere defined morphism and induces a decomposition of this variety into a product of a rationally connected variety by a projective variety with trivial canonical divisor. These generalize the corresponding results for smooth projective varieties with nef anticanonical bundle in Cao (2019) and Cao-Höring (2019) to the klt case, and partially reduce the structure problem of these varieties to the singular Beauville-Bogomolov decomposition theorem proved by successive works of Greb-Kebekus-Peternell (2016), Druel (2018), Guenencia-Greb-Kebekus (2019) and Höring-Peternell (2019).