Abelianity Conjecture for special threefolds

TL;DR

This paper proves that the fundamental group of special compact Kähler threefolds is almost abelian, using orbifold metrics and the Log Minimal Model Program, confirming a conjecture in complex geometry.

Contribution

It establishes the abelianity conjecture for fundamental groups of special threefolds, extending previous conjectures to this specific class using new geometric techniques.

Findings

Fundamental group of special threefolds is almost abelian.

Application of orbifold Ricci curvature metrics in classification.

Confirmation of the abelianity conjecture in dimension three.

Abstract

Using orbifold metrics of the appropriately signed Ricci curvature on orbifolds with negative or numerically trivial canonical bundle and the two-dimensional Log Minimal Model Program, we prove that the fundamental group of special compact K\"ahler threefolds is almost abelian. This property was conjectured in all dimensions in [Cam04b], and also for orbifolds in [Cam07], where the notion of specialness was introduced. We briefly recall below the definition, basic properties, and the role of special manifolds in classification theory.