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

Luca Fabrizio Di Cerbo

Vol. 255 (2012), No. 2, 305–315

DOI: 10.2140/pjm.2012.255.305

Abstract

We study the classification of smooth toroidal compactifications of nonuniform ball quotients in the sense of Kodaira and Enriques. 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 Kähler metrics of nonpositive curvature. An infinite class of such examples arise as smooth toroidal compactifications of ball quotients.

Keywords

manifolds with nonpositive curvature, toroidal compactifications.

Mathematical Subject Classification 2010

Primary: 14J29, 53C20, 53C55

Milestones

Received: 1 April 2011

Revised: 2 November 2011

Accepted: 25 January 2012

Published: 10 April 2012