Manifolds with nef anticanonical bundle

TL;DR

This paper proves a conjecture about the Albanese map being submersive for certain compact Kähler manifolds with nef anticanonical bundle, especially when fibers are weak Fano or low-dimensional in the projective case.

Contribution

It confirms the conjecture for manifolds with weak Fano fibers and low-dimensional fibers in the projective setting, advancing understanding of the structure of such manifolds.

Findings

Albanese map is submersive when fibers are weak Fano.

Conjecture holds for projective manifolds with fibers of dimension at most two.

Provides new conditions under which the Albanese map is submersive.

Abstract

Let X be a compact K\"ahler manifold such that the anticanonical bundle $-K_X$ is nef. A classical conjecture claims that the Albanese map is submersive. We prove this conjecture if the general fibre is a weak Fano manifold. If X is projective we prove the conjecture also for fibres of dimension at most two.