Calabi–Yau metrics with conical singularities along line arrangements

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.

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Calabi–Yau metrics with conical singularities along line arrangements

Cited by 3 publications

References 60 publications

A Guided Tour to Normalized Volume

A Guided Tour to Normalized Volume

Some models for bubbling of (log) Kähler–Einstein metrics

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

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