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.
A remark on compact K\"ahler manifolds with nef anticanonical bundles and its applications
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abstract
Let $(X, \omega_X)$ be a compact K\"ahler manifold such that the anticanonical bundle $-K_X$ is nef. We prove that the slopes of the Harder-Narasimhan filtration of the tangent bundle with respect to a polarization of the form $\omega_X^{n-1}$ are semi-positive. As an application, we give a characterization of rationally connected compact K\"ahler manifolds with nef anticanonical bundles. As another application, we give a simple proof of the surjectivity of the Albanese map.
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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.