Equality in the Miyaoka-Yau inequality and uniformization of non-positively curved klt pairs

TL;DR

This paper proves that for certain compact Kähler pairs with specific curvature conditions, equality in the Miyaoka-Yau inequality implies their universal covers are either the unit ball or affine space, revealing a uniformization phenomenon.

Contribution

It establishes a characterization of the universal cover of Kähler pairs under equality conditions in the orbifold Miyaoka-Yau inequality, extending uniformization results.

Findings

Universal cover is the unit ball when $K_X + riangle$ is ample.

Universal cover is the affine space when $K_X + riangle$ is numerically trivial.

Equality in the orbifold Miyaoka-Yau inequality implies a uniformization type result.

Abstract

Let $(X, \Delta)$ be a compact K\"ahler klt pair, where $K_X + \Delta$ is ample or numerically trivial, and $\Delta$ has standard coefficients. We show that if equality holds in the orbifold Miyaoka-Yau inequality for $(X, \Delta)$, then its orbifold universal cover is either the unit ball (ample case) or the affine space (numerically trivial case).