Uniformization of varieties with log-canonical singularities

TL;DR

This paper establishes criteria for when a complex projective variety with log-canonical singularities can be uniformized as a Baily-Borel-Mok compactification of a ball quotient, linking singularity conditions with geometric uniformization.

Contribution

It provides new criteria involving the Miyaoka-Yau inequality and singularity assumptions for uniformization of varieties with log-canonical singularities.

Findings

Criteria for uniformization as Baily-Borel-Mok compactification

Connectedness of fibers in log-resolutions of log-canonical singularities

Examples showing necessity of log-canonical singularities for the results

Abstract

We study the problem of uniformizing quasi-projective varieties with logcanonical compactifications. More precisely, given a complex projective variety X with log-canonical singularities, we give criteria for X to be isomorphic to a Baily-Borel-Mok compactification of a ball quotient, asking on the one hand the equality case in a suitable Miyaoka-Yau (MY) inequality, and on the other hand some adequate assumptions on the singularities. We also give as a result of independent interest that log-resolutions of log-canonical singularities have their fibers connected by chains of special varieties in the sense of Campana; this is used in the proof to control the behaviour of the period map near the exceptional divisors of such resolutions. We also show that it is necessary to assume that the singularities are at least logcanonical: some examples of Deligne-Mostow-Deraux can be manipulated to provide examples of singular varieties satisfying the equality case in MY, while not being isomorphic to such Baily-Borel-Mok compactifications.