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

TL;DR

This paper investigates the structure of projective varieties with nef anticanonical divisors and log terminal singularities, extending known results to singular cases and analyzing their canonical fibrations and decompositions.

Contribution

It generalizes structural results for smooth varieties with nef anticanonical bundles to the klt singular case, including Albanese and MRC fibrations.

Findings

Albanese map is a locally constant fibration.

MRC fibration induces a product decomposition under certain conditions.

Extends Beauville-Bogomolov decomposition to singular varieties.

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: 1. the Albanese map for such variety is a locally constant fibration (that is, an analytic fibre bundle with connected fibres that $X$ is equal 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); 2. if the smooth locus is simply connected, the MRC fibration of such variety is an everywhere defined morphism and induces a decomposition into a product of a rationally connected variety and of 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-H\"oring (2019) to the klt case, and can be also regarded as a partial extension of the singular Beauville-Bogomolov decomposition theorem proved by successive works of Greb-Kebekus-Peternell (2016), Druel (2018), Guenencia-Greb-Kebekus (2019) and H\"oring-Peternell (2019).