Calabi-Yau metrics with conical singularities along line arrangements

Borbon, Martin de; Spotti, Cristiano

doi:10.48550/arxiv.1712.07967

Cited by 2 publications

(3 citation statements)

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“…More generally when ∆ is simple normal crossing, the polyhomogeneity property for Kähler-Einstein metrics on (S, ∆) was announced by Rubinstein-Mazzeo. In the case of log canonical surfaces, Borbon-Spotti conjectured in [3] that the correction term associated to any point is precisely one less than the volume density of the Kähler-Einstein metric and, as mentioned in the introduction, that the volume densities should match Langer's local Euler numbers (at least for log terminal surface singularities). The main part [3] is to study the behavior of Kähler-Einstein metrics near the singularities when the boundary divisors have good configurations (more precisely when the metric cone at any point is isomorphic to the germ of the point itself ).…”

Section: Local Euler Numbers For 2-dimensional Log Canonical Conesmentioning

confidence: 99%

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

Li¹

2018

Preprint

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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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Section: Local Euler Numbers For 2-dimensional Log Canonical Conesmentioning

confidence: 99%

“…δ < 2 and δ m < δ ′ . Then (P 1 , i δ i p i ) is K-stable (called the "stable regime" in [3]) . So by Proposition 1.8, e orb (0;…”

Section: Logarithmic Miyaoka-yau Inequalities For K-semistable Pairsmentioning

confidence: 99%

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

Li¹

2018

Preprint

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“…From a different point of view, such type of results can be interpreted as regularity results for weak KE metrics. At the moment, one special doable case is to restrict to two dimensional flat tangent cones [6]; another is to assume a smoothability condition and appeal to Riemannian convergence theory and recent results for Gromov-Hausdorff limits of KE metrics [14].…”

Section: Kähler-einstein Equationsmentioning

confidence: 99%

Local models for conical Kähler-Einstein metrics

Borbon¹,

Spotti²

2018

Proc. Amer. Math. Soc.

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In this note we use the Calabi ansatz, in the context of metrics with conical singularities along a divisor, to produce regular Calabi-Yau cones and Kähler-Einstein metrics of negative Ricci with a cuspidal point. As an application, we describe singularities and cuspidal ends of the completions of the complex hyperbolic metrics on the moduli spaces of ordered configurations of points in the projective line introduced by Thurston and Deligne-Mostow.

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Calabi-Yau metrics with conical singularities along line arrangements

Cited by 2 publications

References 0 publications

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

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

Local models for conical Kähler-Einstein metrics

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