TY - JOUR

T1 - On the stability of extensions of tangent sheaves on Kähler–Einstein Fano/Calabi–Yau pairs

AU - Li, Chi

N1 - Funding Information:
The author is partially supported by NSF (Grant no. DMS-1405936 and DMS-1810867) and an Alfred P. Sloan research fellowship. The author would like to thank Martin de Borbon and Christiano Spotti for useful comments and the suggestion of adding the example , and to thank Henri Guenancia and Behrouz Taji for their interest and especially to Behrouz Taji for very helpful comments and suggestions about orbifold structures. His thanks also go to Yuchen Liu, Xiaowei Wang and Chenyang Xu for helpful discussions. The author would like to thank Professor Gang Tian for his interest in this work and constant support through the years. The author would also like to thank a referee for very helpful suggestions for improving the paper.
Funding Information:
The author is partially supported by NSF (Grant no. DMS-1405936 and DMS-1810867) and an Alfred P. Sloan research fellowship. The author would like to thank Martin de Borbon and Christiano Spotti for useful comments and the suggestion of adding the example 1.8 , and to thank Henri Guenancia and Behrouz Taji for their interest and especially to Behrouz Taji for very helpful comments and suggestions about orbifold structures. His thanks also go to Yuchen Liu, Xiaowei Wang and Chenyang Xu for helpful discussions. The author would like to thank Professor Gang Tian for his interest in this work and constant support through the years. The author would also like to thank a referee for very helpful suggestions for improving the paper.
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/12

Y1 - 2021/12

N2 - 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 TS(- log (Δ)) by the structure sheaf OS with the extension class c1(L). We prove the following two results: (1)if - (KS+ Δ) is ample and (S, Δ) is K-semistable, then for any λ∈ Q> 0, the extension sheaf E(λc1(- (KS+ Δ))) is slope semistable with respect to - (KS+ Δ) ;(2)if KS+ 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 - KS 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.

AB - 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 TS(- log (Δ)) by the structure sheaf OS with the extension class c1(L). We prove the following two results: (1)if - (KS+ Δ) is ample and (S, Δ) is K-semistable, then for any λ∈ Q> 0, the extension sheaf E(λc1(- (KS+ Δ))) is slope semistable with respect to - (KS+ Δ) ;(2)if KS+ 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 - KS 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.

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U2 - 10.1007/s00208-020-02099-x

DO - 10.1007/s00208-020-02099-x

M3 - Article

AN - SCOPUS:85094130428

SN - 0025-5831

VL - 381

SP - 1943

EP - 1977

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 3-4

ER -