On the stability of extensions of tangent sheaves on Kähler–Einstein Fano/Calabi–Yau pairs

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Let S be a smooth projective variety and $$\Delta $$ a simple normal crossing $${\mathbb {Q}}$$ -divisor with coefficients in (0, 1]. For any ample $${\mathbb {Q}}$$ -line bundle L over S, we denote by $$\mathscr {E}(L)$$ the extension sheaf of the orbifold tangent sheaf $$T_S(-\log (\Delta ))$$ by the structure sheaf $$\mathcal {O}_S$$ with the extension class $$c_1(L)$$ . We prove the following two results: These results generalize Tian’s result where $$-K_S$$ is ample and $$\Delta =\emptyset $$ . 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 $${\mathbb {C}}^2$$ . 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.