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

Abstract

Let S be a smooth projective variety and ∆ 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(∆)) by the structure sheaf O S with the extension class c 1 (L). We prove the following two results:) is slope semistable with respect to −(K S + ∆);(2) if K S + ∆ ≡ 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 ∆ = ∅. 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 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.

Local models for conical Kähler-Einstein metrics

“…More precisely in this section we construct regular log Calabi-Yau cones as lifts of conical Kähler-Einstein metrics on log Fano pairs. Our terminology differs from the one used by Chi Li, our log Calabi-Yau cones are referred to as log Fano cones in [19].…”

“…However, we should remark that, even if expected, the generalization of the results of Donaldson-Sun for tangent cones of limit spaces to the logarithmic setting are still missing. On the other hand, the work of Li [18] on characterizing the tangent cones fully algebro-geometrically via the infimum of the normalized volume of valuations extends immediately to the logarithmic setting and, in certain cases, such invariant can be checked to be equal to the volume density [19]. 5.2.…”