Miyaoka-Yau inequalities and the topological characterization of certain klt varieties

TL;DR

This paper extends the topological and Chern class characterizations of certain complex varieties, like ball quotients and projective spaces, to those with klt singularities that are homeomorphic to these well-understood varieties.

Contribution

It establishes new topological characterizations for projective varieties with klt singularities, generalizing known results for smooth varieties.

Findings

Characterization of klt varieties homeomorphic to ball quotients.

Extension of Chern class equality conditions to singular varieties.

Topological identification of certain singular varieties as quotients of well-known spaces.

Abstract

Ball quotients, hyperelliptic varieties, and projective spaces are characterized by their Chern classes, as the varieties where the Miyaoka-Yau inequality becomes an equality. Ball quotients, Abelian varieties, and projective spaces are also characterized topologically: if a complex, projective manifold $X$ is homeomorphic to a variety of this type, then $X$ is itself of this type. In this paper, similar results are established for projective varieties with klt singularities that are homeomorphic to singular ball quotients, quotients of Abelian varieties, or projective spaces.