Compact K\"ahler manifolds with nef anti-canonical bundle

TL;DR

This paper proves that compact K"ahler manifolds with nef anti-canonical bundles can be decomposed into a fibration with rationally connected fibers over a Calabi--Yau base, extending previous methods to singular spaces.

Contribution

It introduces a new approach to analyze nef anti-canonical bundles on singular K"ahler spaces, generalizing Cao--H"oring's strategy from smooth projective varieties.

Findings

Existence of a locally trivial fibration with rationally connected fibers and Calabi--Yau base.

Extension of Cao--H"oring's method to singular K"ahler spaces.

A flatness criterion for pseudo-effective sheaves with vanishing first Chern class.

Abstract

In this paper, we prove that a compact K\"ahler manifold $X$ with the nef anti-canonical bundle $-K_{X}$ admits a locally trivial fibration $\phi \colon X \to Y$, 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--H\"oring, originally developed for smooth projective varieties, to more general singular K\"ahler spaces. A key technical ingredient is a flatness criterion for pseudo-effective sheaves with vanishing first Chern class.