Orbifold Stability and Miyaoka-Yau Inequality for minimal pairs

TL;DR

This paper proves the Miyaoka-Yau inequality for minimal pairs with singularities using Kähler-Einstein metrics, offering an alternative proof of the Abundance theorem for threefolds without relying on tangent sheaf positivity.

Contribution

It introduces new notions of stability and Chern classes for singular pairs and applies Kähler-Einstein metrics to establish orbifold tangent sheaf semistability and the Miyaoka-Yau inequality.

Findings

Proves slope semistability of orbifold tangent sheaves for minimal log-canonical pairs.

Establishes the Miyaoka-Yau inequality for all minimal pairs with standard coefficients.

Provides an alternative proof of the Abundance theorem for threefolds.

Abstract

After establishing suitable notions of stability and Chern classes for singular pairs, we use K\"ahler-Einstein metrics with conical and cuspidal singularities to prove the slope semistability of orbifold tangent sheaves of minimal log-canonical pairs of log general type. We then proceed to prove the Miyaoka-Yau inequality for all minimal pairs with standard coefficients. Our result in particular provides an alternative proof of the Abundance theorem for threefolds that is independent of positivity results for tangent sheaves.