The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors

TL;DR

This paper proves the Miyaoka-Yau inequality for certain singular projective varieties with big canonical or anticanonical divisors, extending classical results to more general singular settings.

Contribution

It introduces a non-pluripolar product on singular varieties and establishes the Miyaoka-Yau inequality in this broader context, including for K-semistable varieties.

Findings

Proved Miyaoka-Yau inequality for varieties with big canonical divisor.

Extended inequality to K-semistable varieties with big anticanonical divisor.

Developed a non-pluripolar product framework for singular varieties.

Abstract

We establish the Miyaoka-Yau inequality for $n$-dimensional projective klt varieties with big canonical divisor $K_X$: \[ (2(n+1)\widehat{c}_2(X) - n \widehat{c}_1(X)^2) \cdot \langle c_1(K_X)^{n-2} \rangle \ge 0. \] We also prove the Miyaoka-Yau inequality for K-semistable projective klt varieties with big anticanonical divisor $-K_X$. As part of our approach, we define the non-pluripolar product $\langle \alpha_1 \cdots \alpha_p \rangle$ on singular varieties, and establish the Bogomolov-Gieseker type inequality for $\langle \alpha^{n-1} \rangle$-semistable Higgs sheaves with respect to a big class $\alpha$. In addition, we investigate second Chern class inequalities in the cases where $K_X$ or $-K_X$ is nef.