Abelianity Conjecture for Special Compact Kähler 3-Folds

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 Kähler 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.Date: November 5, 2018.

Finiteness of fundamental groups

“…[Zha95] and [AIM07, Question 0.11]). It has been confirmed for surfaces in [FKL93, GZ94, GZ95, KM99], see also [CC14] for a survey. On the other hand, Conjecture 1.2, as far as we know, was first formulated by Kollár [Kol13, 8.24].…”

The Bolza curve and some orbifold ball quotient surfaces

“…In section 2, we recall some results of Deraux on the quotient surface A/G 48 and introduce some notation. In section 3, we study properties of the surface P (1,3,8). In section 4, we introduce the Bolza curve θ and prove that A/G 48 is isomorphic to P (1,3,8).…”

“…Deraux also remarks in [14] that the invariants of A/G 48 and its singularities are the same as for the weighted projective plane P (1,3,8) and, in analogy with cases in [11] and [15] where weighted projective planes appear in the context of ball-quotient surfaces, he asks whether the two surfaces are isomorphic.…”

“…We then construct birational transformations of P (1,3,8) to P 1 × P 1 , P 2 and obtain the equations of the images M P 1 ×P 1 , M P 2 of the branch curve M 48 in these surfaces (and also M 48 ⊂ P(1, 3, 8)). In particular: Theorem B.…”

On the hyperbolicity of surfaces of general type with small c 1 2

“…Thanks to a result of [30] (see also the appendix of [9]), we have Example 11. Let (X, ∆), ∆ = i 1 − 1 m i C i , be a klt pair with X a surface, then (X, ∆) is an orbifold.…”