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

Let S be a smooth projective variety and Δ a simple normal crossing Q-divisor with coefficients in (0, 1]. For any ample Q-line bundle L over S, we denote by E(L) the extension sheaf of the orbifold tangent sheaf T_S(- log (Δ)) by the structure sheaf O_S with the extension class c₁(L). We prove the following two results: (1)if - (K_S+ Δ) is ample and (S, Δ) is K-semistable, then for any λ∈ Q_{> 0}, the extension sheaf E(λc₁(- (K_S+ Δ))) is slope semistable with respect to - (K_S+ Δ) ;(2)if K_S+ Δ ≡ 0 , then for any ample Q-line bundle L over S, E(L) is slope semistable with respect to L. These results generalize Tian’s result where - K_S is ample and Δ = ∅. 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. 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.