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

We establish the Miyaoka-Yau inequality for $n$-dimensional projective klt varieties with big canonical divisor $K_X$: $$(2(n+1){\widehat{c}}_2(X)-n{\widehat{c}}_1(X{)}^2)\cdot ⟨c_1(K_X{)}^{n-2}⟩\ge 0.$$ 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 $\alpha$. In addition, we investigate second Chern class inequalities in the cases where $K_X$ or $-K_X$ is nef.