Albanese morphism of log smooth klt compact K\"ahler manifold with nef log anticanonical divisor

TL;DR

This paper studies the Albanese morphism of a compact K"ahler manifold with a nef log anticanonical divisor, showing it is a submersion outside a high codimension set with connected fibers.

Contribution

It establishes new geometric properties of the Albanese map for log smooth klt compact K"ahler manifolds with nef log anticanonical divisors.

Findings

Albanese map is a submersion outside a codimension > 2 set.

Fibers of the Albanese map are connected.

Results extend understanding of the structure of such K"ahler manifolds.

Abstract

Let $(X, \omega)$ be an n-dimensional compact K\"ahler manifold. Let $D=\sum (1-\beta_j) Y_j=\sum (1-\beta_j) [s_j=0]$ a divisor with simple normal crossings with $\beta_j \in ]0,1[$ such that $-(K_X+D)$ is nef. We show that its Albanese map is submersion outside an analytic set of codimension larger than two with connected fibres.