Campana's abelianity conjecture for special compact Kahler manifolds

Theorem 1.1: every special compact Kahler manifold has virtually abelian fundamental group, in every dimension. Corollaries: compact Kahler manifolds with $\kappa=0$ have virtually abelian $\pi_1$; Campana's Conjecture S and the bimeromorphic Iitaka uniformization (finite etale covers bimeromorphic to tori when $\gamma d=n$ or the universal cover is $\mathbb C^n$); holomorphic convexity of universal covers of special manifolds. It does not prove the orbifold or quasi-projective versions in general: the companions give a two-step nilpotent bound for linear images in the open case (credited first to Cao-Deng-Hacon-Paun) and one conditional fourfold orbifold case.

Theorem 1.1: every special compact Kahler manifold has virtually abelian fundamental group, in every dimension. Corollaries: compact Kahler manifolds with $\kappa=0$ have virtually abelian $\pi_1$; Campana's Conjecture S and the bimeromorphic Iitaka uniformization (finite etale covers bimeromorphic to tori when $\gamma d=n$ or the universal cover is $\mathbb C^n$); holomorphic convexity of universal covers of special manifolds. It does not prove the orbifold or quasi-projective versions in general: the companions give a two-step nilpotent bound for linear images in the open case (credited first to Cao-Deng-Hacon-Paun) and one conditional fourfold orbifold case.