Albanese map for K\"ahler manifolds with nef anticanonical bundle

TL;DR

This paper investigates the structure of the Albanese map for K"ahler manifolds with nef anticanonical bundle, establishing conditions under which the manifold is Calabi-Yau or a projectivization of a flat bundle.

Contribution

It extends understanding of the Albanese map for K"ahler manifolds with nef anticanonical bundle, including new results for fourfolds and general cases with specific fiber types.

Findings

For fourfolds with elliptic curve Albanese torus, specific structural results.

If the general fiber is Calabi-Yau, the manifold is Calabi-Yau.

If the fiber is projective space, the manifold is a projectivization of a flat bundle.

Abstract

We study the structure of the Albanese map for K\"ahler manifolds with nef anticanonical bundle. First, we give a result for fourfolds whose Albanse torus is an elliptic curve. In the general case of any dimension, we look at two cases: The general fiber of the Albanese map is a Calabi-Yau manifold or a projective space. In the first case, we show that the manifold itself must be Calabi-Yau. In the second case, we give a more topological proof of a result by Cao and H\"oring which says that the manifold must be a projectivization of a numerically flat vector bundle.