Optimal pinching for the holomorphic sectional curvature of Hitchin’s metrics on Hirzebruch surfaces

Alvarez, Angelynn; Chaturvedi, Ananya; Heier, Gordon

doi:10.1090/conm/654/13219

Cited by 15 publications

(38 citation statements)

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“…It is interesting to see that the optimal pinching constant is dimension free. It is the same constant 1 (2k+1) 2 discovered in [2], with the corresponding Kähler class on M n,k satisfies…”

mentioning

confidence: 61%

“…Where each of (N li , h li ) is a compact irreducible Hermitian symmetric spaces of rank ≥ 2 with its canonical Kähler-Einstein metric, The holomorphic pinching constant of such a metric was well-studied and it is exactly the reciprocal of its rank, see for example [13]. On the other hand, the pinching of product Kähler metrics was studied by [2], the proved that the pinching constant of a produce Kähler metric (M 1 × M 2 , g 1 × g 2 ) is λ1λ2 λ1+λ2 where λ 1 and λ 1 are pinching constants of M 1 and M 2 respectively. It is clearly that λ1λ2 λ1+λ2 = 1 2 is equivalent to λ 1 = λ 2 = 1.…”

Section: Holormophic Sectional Curvature: Preliminary Resultsmentioning

confidence: 99%

“…In general, the local holomorphic pinching constant of any U (n)-invariant Kähler metric on M n,k is bounded from above by 1 k 2 . A direct calculation enables us to generalize the result of Alvarez-Chaturvedi-Heier [2] to M n,k . It is interesting to see that the optimal pinching constant is dimension free.…”

mentioning

confidence: 90%

“…Recently, Alvarez-Chaturvedi-Heier [2] studied the pinching constants of Hitchin's examples on M 2,k .…”

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confidence: 99%

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Hirzebruch manifolds and positive holomorphic sectional curvature

Yang¹,

Zheng²

2019

Annales De l'Institut Fourier

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This paper is the first step in a systematic project to study examples of Kähler manifolds with positive holomorphic sectional curvature (H > 0). Previously Hitchin proved that any compact Kähler surface with H > 0 must be rational and he constructed such examples on Hirzebruch surfaces M 2,k = P(H k ⊕ 1 CP 1 ). We generalize Hitchin's construction and prove that any Hirzebruch manifold M n,k = P(H k ⊕ 1 CP n−1 ) admits a Kähler metric of H > 0 in each of its Kähler classes. We demonstrate that the pinching behaviors of holomorphic sectional curvatures of new examples differ from those of Hitchin's which were studied in the recent work of Alvarez-Chaturvedi-Heier. Some connections to recent works on the Kähler-Ricci flow on Hirzebruch manifolds are also discussed.It seems interesting to study the space of all Kähler metrics of H > 0 on a given Kähler manifold. We give higher dimensional examples such that some Kähler classes admit Kähler metrics with H > 0 and some do not.It is the Riemannian sectional curvature restricted on any J-invariant real 2-plane (p165 [29]). Compact Kähler manifolds with positive holomorphic sectional (H > 0) form an interesting class of Kähler manifolds. For example, these manifolds are simply-connected, by the work of Tsukamoto [38]. An averaging argument due to Berger [6] showed that H > 0 implies positive scalar curvature, which further leads to the vanishing of its pluri-canonical ring by a Bochner-Kodaira type identity, see [26]. In 1975 Hitchin [24] proved that any Hirzebruch surface M 2,k = P(H k ⊕ 1 CP 1 ) admits a Kähler metrics with H > 0. Moreover, he proved that any rational surface admits a Kähler metric with positive scalar curvature.

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confidence: 61%

Section: Holormophic Sectional Curvature: Preliminary Resultsmentioning

confidence: 99%

mentioning

confidence: 90%

“…Recently, Alvarez-Chaturvedi-Heier [2] studied the pinching constants of Hitchin's examples on M 2,k .…”

mentioning

confidence: 99%

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Hirzebruch manifolds and positive holomorphic sectional curvature

Yang¹,

Zheng²

2019

Annales De l'Institut Fourier

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“…We conclude this introduction by remarking that in the paper [ACH15] an explicit analysis of the pinching constants for Hitchin's metrics on Hirzebruch surfaces was conducted. We leave such an explicit analysis for the present higher-dimensional case to a future occasion.…”

Section: Introductionmentioning

confidence: 99%

On projectivized vector bundles and positive holomorphic sectional curvature

Alvarez

Heier

Zheng

2018

Proc. Amer. Math. Soc.

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On the zero set of the holomorphic sectional curvature

Chen,

Heier

2024

Mathematische Nachrichten

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A notable example due to Heier, Lu, Wong, and Zheng shows that there exist compact complex Kähler manifolds with ample canonical line bundle such that the holomorphic sectional curvature is negative semi‐definite and vanishes along high‐dimensional linear subspaces in every tangent space. The main result of this note is an upper bound for the dimensions of these subspaces. Due to the holomorphic sectional curvature being a real‐valued bihomogeneous polynomial of bidegree (2,2) on every tangent space, the proof is based on making a connection with the work of D'Angelo on complex subvarieties of real algebraic varieties and the decomposition of polynomials into differences of squares. Our bound involves an invariant that we call the holomorphic sectional curvature square decomposition length, and our arguments work as long as the holomorphic sectional curvature is semi‐definite, be it negative or positive.

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Optimal pinching for the holomorphic sectional curvature of Hitchin’s metrics on Hirzebruch surfaces

Cited by 15 publications

References 6 publications

Hirzebruch manifolds and positive holomorphic sectional curvature

Hirzebruch manifolds and positive holomorphic sectional curvature

On projectivized vector bundles and positive holomorphic sectional curvature

On the zero set of the holomorphic sectional curvature

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