Let S be a smooth projective variety and Δ\documentclass[12pt]{minimal}
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\begin{document}$$\Delta $$\end{document} a simple normal crossing Q\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Q}}$$\end{document}-divisor with coefficients in (0, 1]. For any ample Q\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Q}}$$\end{document}-line bundle L over S, we denote by E(L)\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr {E}(L)$$\end{document} the extension sheaf of the orbifold tangent sheaf TS(-log(Δ))\documentclass[12pt]{minimal}
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\begin{document}$$T_S(-\log (\Delta ))$$\end{document} by the structure sheaf OS\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {O}_S$$\end{document} with the extension class c1(L)\documentclass[12pt]{minimal}
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\begin{document}$$c_1(L)$$\end{document}. We prove the following two results: if -(KS+Δ)\documentclass[12pt]{minimal}
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\begin{document}$$-(K_S+\Delta )$$\end{document} is ample and (S,Δ)\documentclass[12pt]{minimal}
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\begin{document}$$(S, \Delta )$$\end{document} is K-semistable, then for any λ∈Q>0\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \in {\mathbb {Q}}_{>0}$$\end{document}, the extension sheaf E(λc1(-(KS+Δ)))\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr {E}({\lambda c_1(-(K_S+\Delta ))})$$\end{document} is slope semistable with respect to -(KS+Δ)\documentclass[12pt]{minimal}
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\begin{document}$$-(K_S+\Delta )$$\end{document};if KS+Δ≡0\documentclass[12pt]{minimal}
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\begin{document}$$K_S+\Delta \equiv 0$$\end{document}, then for any ample Q\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Q}}$$\end{document}-line bundle L over S, E(L)\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr {E}(L)$$\end{document} is slope semistable with respect to L. These results generalize Tian’s result where -KS\documentclass[12pt]{minimal}
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\begin{document}$$-K_S$$\end{document} is ample and Δ=∅\documentclass[12pt]{minimal}
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\begin{document}$$\Delta =\emptyset $$\end{document}. We give two applications of these results. The first is to study a question by Borbon–Spotti about the relationship between local Euler numbers and normalized volumes of log canonical surface singularities. We prove that the two invariants differ only by a factor 4 when the log canonical pair is an orbifold cone over a marked Riemann surface. In particular we complete the computation of Langer’s local Euler numbers for any line arrangements in C2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {C}}^2$$\end{document}. The second application is to derive Miyaoka–Yau-type inequalities on K-semistable log-smooth Fano pairs and Calabi–Yau pairs, which generalize some Chern-number inequalities proved by Song–Wang.
