Miyaoka-Yau equality and uniformization of log Fano pairs

TL;DR

This paper proves that log Fano pairs satisfying the Miyaoka-Yau equality have orbifold universal covers isomorphic to projective space, linking geometric inequalities to uniformization results.

Contribution

It establishes a new characterization of log Fano pairs via the Miyaoka-Yau equality and their universal covers, extending uniformization theory.

Findings

Equality case implies orbifold universal cover is projective space

Connects Miyaoka-Yau equality to uniformization of log Fano pairs

Provides conditions for uniformization in algebraic geometry

Abstract

Let $(X, \Delta)$ be a log Fano pair with standard coefficients. We show that if it satisfies the equality case in the Miyaoka-Yau inequality, then its orbifold universal cover is a projective space.