Compact Kähler manifolds with nef anti-canonical bundle

Abstract

In this paper, we prove that a compact Kähler manifold $X$ with the nef anti-canonical bundle $-K_X$ admits a locally trivial fibration $\ddot{I}•:X\to Y$, where the fiber $F$ is a rationally connected manifold and the base $Y$ is a Calabi--Yau manifold. We introduce a suitable approach that extends the strategy of Cao--Höring, originally developed for smooth projective varieties, to more general singular Kähler spaces. A key technical ingredient is a flatness criterion for pseudo-effective sheaves with vanishing first Chern class.

44 pages, comments welcome