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 $\mathcal{E}(L)$ 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 $\mathcal{E}(\lambda c_1(-(K_S+\Delta )))$ 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 $S$, $\mathcal{E}(L)$ is slope semistable with respect to $L$. These results generalize Tian's result where $-K_S$ is ample and $\Delta =\varnothing$. 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.

24 pages, comments are very welcome. v3: corrected some typos, submitted version

References in corpus (5)

• K-stability of log Fano hyperplane arrangements
• On the Yau-Tian-Donaldson conjecture for singular Fano varieties
• Generic positivity and applications to hyperbolicity of moduli spaces
• On the existence of conic Kahler-Einstein metrics
• Stability of Valuations: Higher Rational Rank

Cited by in corpus (4)

• Local models for conical Kähler-Einstein metrics
• A decomposition theorem for $\mathbb{Q}$-Fano Kähler-Einstein varieties
• On the structure of a log smooth pair in the equality case of the Bogomolov-Gieseker inequality
• Calabi-Yau metrics with cone singularities along intersecting complex lines: the unstable case