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, especially the canonical fibrations associated to them. We show that • The Albanese map for such variety is a locally constant fibration (that is, an analytic fibre bundle with connected fibres which splits into a product when passing to the universal cover 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 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-Höring (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-

Open problems on structure of positively curved projective varieties

“…The most important remaining problem is to establish the structure theorem for klt pairs (X, ∆) with nef anti-canonical divisor −(K X + ∆). Wang in [Wan20] partially solved this problem when the regular locus X reg of X is simply connected and reduced the problem in the general case to some conjectures on fundamental groups of X reg . In the case X being smooth, this problem was solved by [CCM21] in the following form: Theorem 5.1 ([CCM21, Theorem 1.3]).…”

Strictly nef divisors on singular threefolds

“…From our viewpoints, it is not satisfactory that the conditions of Theorem 1.6 seem quite restrictive. We believe that, one can consider Question 1.2 itself (avoiding Question 1.4) by analyzing the universal covering of X; see the recent series papers [Cao19], [CH19], [CCM19] and [Wan20].…”