On the stability of extensions of tangent sheaves on Kähler–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 prove the following two results:

• (1) 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 $-(K_S+\Delta )$;

• (2) 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. 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.