The Miyaoka-Yau inequality and uniformisation of canonical models

TL;DR

This paper proves the Miyaoka-Yau inequality for complex projective varieties with singularities and characterizes the case of equality as the variety being a quotient of the unit ball, linking geometric inequalities to uniformization.

Contribution

It extends the Miyaoka-Yau inequality to varieties with klt singularities and establishes a uniformization result for cases of equality, connecting algebraic geometry and complex hyperbolic geometry.

Findings

Miyaoka-Yau inequality holds for varieties with klt singularities.

Equality case implies the variety is a quotient of the unit ball.

Provides a criterion for uniformization of canonical models.

Abstract

We establish the Miyaoka-Yau inequality in terms of orbifold Chern classes for the tangent sheaf of any complex projective variety of general type with klt singularities and nef canonical divisor. In case equality is attained for a variety with at worst terminal singularities, we prove that the associated canonical model is the quotient of the unit ball by a discrete group action.