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

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 $⟨{\alpha }_1\cdots {\alpha }_p⟩$ on singular varieties, and establish the Bogomolov-Gieseker type inequality for $⟨{\alpha }^{n-1}⟩$-semistable Higgs sheaves with respect to a big class $\alpha$.

v3: 45 pages; major revision. The original manuscript has been divided into two papers. The present version contains the first part, while the second part will appear separately on arXiv. v2: 62 pages; minor revision. (Revised Introduction and added Proposition 4.25.) v1: 61 pages; comments are welcome