Uniformisation in dimension four: towards a conjecture of Iitaka

Abstract

Let X be a compact Kähler manifold whose universal covering is ${\mathbb{C}}^n$. A conjecture of Iitaka claims that X is a torus, up to finite étale 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.

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