Abelianity Conjecture for Special Compact Kähler 3-Folds

Using orbifold metrics of the appropriately signed Ricci curvature on orbifolds with a negative or numerically trivial canonical bundle and the two-dimensional log minimal model programme, we prove that the fundamental group of special compact Kähler 3-folds is almost abelian. This property was conjectured in all dimensions by Campana in 2004, and also for orbifolds in 2007, where the notion of specialness was introduced. We briefly recall the definition, basic properties and the role of special manifolds in classification theory.