Finite volume complex-hyperbolic surfaces, their toroidal compactifications, and geometric applications

TL;DR

This paper classifies smooth toroidal compactifications of nonuniform ball quotients and explores their geometric properties, revealing examples of complex surfaces with nonpositive curvature metrics that lack compatible Kähler metrics.

Contribution

It provides a classification of these compactifications and demonstrates the existence of complex surfaces with nonpositive curvature metrics without Kähler structures.

Findings

Classified smooth toroidal compactifications of nonuniform ball quotients.

Constructed examples of complex surfaces with nonpositive curvature metrics lacking Kähler metrics.

Showed differences between Riemannian and complex algebraic geometry in these spaces.

Abstract

We study the classification of smooth toroidal compactifications of nonuniform ball quotients in the sense of Kodaira and Enriques. Moreover, several results concerning the Riemannian and complex algebraic geometry of these spaces are given. In particular we show that there are compact complex surfaces which admit Riemannian metrics of nonpositive curvature, but which do not admit K\"ahler metrics of nonpositive curvature. An infinite class of such examples arise as smooth toroidal compactifications of ball quotients.