Uniformisation in dimension four: towards a conjecture of Iitaka

TL;DR

This paper advances the understanding of Iitaka's conjecture by proving it in certain cases for four-dimensional compact Kähler manifolds with universal cover $C^n$, and links it to the non-vanishing conjecture in the projective case.

Contribution

The paper proves Iitaka's conjecture in specific cases for four-dimensional manifolds and connects it to the non-vanishing conjecture in the projective setting.

Findings

Iitaka's conjecture holds in various cases in dimension four.

In the projective case, Iitaka's conjecture follows from the non-vanishing conjecture.

The results contribute to the classification of Kähler manifolds with universal cover $C^n$.

Abstract

Let X be a compact K\"ahler manifold whose universal covering is $\mathbb C^n$. A conjecture of Iitaka claims that X is a torus, up to finite \'etale cover. We prove this conjecture in various cases in dimension four. We also show that in the projective case Iitaka's conjecture is a consequence of the non-vanishing conjecture.