For projective klt varieties with big canonical or anticanonical divisor, the Miyaoka-Yau Chern class inequality holds when intersections are taken with the non-pluripolar product.
Bogomolov-Gieseker inequality for log terminal K\"ahler threefolds
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abstract
In this article we are mainly concerned with three dimensional compact K\"ahler spaces with log terminal singularities. We establish the orbifold version of the Bogomolov-Gieseker inequality for stable $\mathbb Q$-sheaves.
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The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors
For projective klt varieties with big canonical or anticanonical divisor, the Miyaoka-Yau Chern class inequality holds when intersections are taken with the non-pluripolar product.