Uniformisation in dimension four: towards a conjecture of Iitaka

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.