On the stability of extensions of tangent sheaves on K\"ahler-Einstein Fano / Calabi-Yau pairs

Abstract

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 show the following two results: (i) If $-(K_S+\Delta)$ is ample and $(S, \Delta)$ is K-semistable, then for any $\lambda\in \mathbb{Q}_{>0}$, the extension sheaf $\mathscr{E}({\lambda c_1(-(K_S+\Delta))})$ is slope semistable with respect to $-(K_S+\Delta)$; (ii) If $K_S+\Delta\equiv 0$, then for any ample $\mathbb{Q}$-line bundle $L$ over $S$, $\mathscr{E}(L)$ is slope semistable with respect to $L$. 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. 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.