On projectivized vector bundles and positive holomorphic sectional curvature

Alvarez, Angelynn; Heier, Gordon; Zheng, Fangyang

doi:10.48550/arxiv.1606.08347

Cited by 9 publications

(8 citation statements)

References 5 publications

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“…(1) ij = 0 for all 1 ≤ i, j ≤ n. Also, by (4), we know that the Chern curvature tensor satisfies the skew-symmetry (11) R xyuv = −R uvxy for any type (1, 0) tangent vectors x, y, u, and v. By (10), we know that Ric (3) = 0. By (11), we get that Ric (2) = −Ric (1) = 0.…”

Section: Manifolds With Constant Real Bisectional Curvaturementioning

confidence: 99%

On real bisectional curvature for Hermitian manifolds

Yang

Zheng

2018

Trans. Amer. Math. Soc.

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Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact Kähler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called real bisectional curvature for Hermitian manifolds. When the metric is Kähler, this is just the holomorphic sectional curvature H, and when the metric is non-Kähler, it is slightly stronger than H. We classify compact Hermitian manifolds with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case. The underlying reason for the extension is that the Schwarz lemma of Wu-Yau works the same when the target metric is only Hermitian but has nonpositive real bisectional curvature.It is a natural question to ask when will a Hermitian manifold have constant real bisectional curvature. To this end, we have the following:Theorem 1.4. Let (M n , g) be a compact Hermitian manifold whose real bisectional curvature is constantly equal to c. Then c ≤ 0. Moreover, when c = 0, then (M, g) is a balanced manifold with vanishing first, second, and third Ricci tensors, and its Chern curvature satisfies the property R xyuv = −R uvxy for any type (1, 0) complex tangent vectors x, y, u, v.We would like to propose the following conjecture:Conjecture 1.5. Let M n (n ≥ 3) be a compact Hermitian manifold with vanishing real bisectional curvature c. Then c = 0, and its Chern curvature tensor R = 0.By Boothby's theorem, compact Hermitian manifolds with vanishing Chern curvature are precisely the compact quotients of complex Lie groups equipped with left invariant metrics.Besides the constant real bisectional curvature cases, more generally, it would certainly be very interesting to try to understand the class of all compact Hermitian

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Section: Manifolds With Constant Real Bisectional Curvaturementioning

confidence: 99%

On real bisectional curvature for Hermitian manifolds

Yang

Zheng

2018

Trans. Amer. Math. Soc.

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“…Where each of (CP ki , g ki ) has nonnegative bisectional curvature and each of (N li , h li ) is a compact irreducible Hermitian symmetric spaces of rank ≥ 2 with its canonical Kähler-Einstein metric. Now consider a time t 1 < t 0 close to t = 0, it follows from Step 2 that g ∞ (t 1 ) is close to 1 2 -holmorphic pinching and also have the same decomposition as (2). Indeed the decomposition (2) is reduced to exactly the list in the conclusion of Proposition 1.2.…”

Section: Claim 22mentioning

confidence: 94%

“…It could be possible that any Kähler manifold with H > 0 is in fact projective, again it is an open question. We also remark that some generalization of Hitchin's construction of Kähler metrics of H > 0 in higher dimensions has been obtained in [2]. If Yau's conjecture is true, then how do we study the complexities of rational varieties which admit Kähler metrics with H > 0?…”

Section: The Theoremmentioning

confidence: 95%

A note on the almost-one-half holomorphic pinching

Cao

Yang

2017

Comptes Rendus. Mathématique

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Motivated by a previous work of Zheng and the second named author, we study pinching constants of compact Kähler manifolds with positive holomorphic sectional curvature. In particular we prove a gap theorem following the work of Petersen and Tao on Riemannian manifolds with almost quarter-pinched sectional curvature.Date: This is the version which the authors submitted to a journal for consideration for publication in June 2017. The reference has not been updated since then.

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“…. }, over P 1 , motivated the statement and proof of the main theorem in [AHZ16], establishing the existence of a Kähler metric of positive holomorphic sectional curvature on an arbitrary projectivized holomorphic vector bundle over a compact Kähler base manifold of positive holomorphic sectional curvature. The main result of this note is the following full-fledged positive curvature analog for Cheung's theorem.…”

Section: Introductionmentioning

confidence: 99%

Hermitian metrics of positive holomorphic sectional curvature on fibrations

Chaturvedi

Heier

2019

Math. Z.

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The main result of this note essentially is that if the base and fibers of a compact fibration carry Hermitian metrics of positive holomorphic sectional curvature, then so does the total space of the fibration. The proof is based on the use of a warped product metric as in the work by Cheung in case of negative holomorphic sectional curvature, but differs in certain key aspects, e.g., in that it does not use the subadditivity property for holomorphic sectional curvature due to Grauert-Reckziegel and Wu.

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On projectivized vector bundles and positive holomorphic sectional curvature

Cited by 9 publications

References 5 publications

On real bisectional curvature for Hermitian manifolds

On real bisectional curvature for Hermitian manifolds

A note on the almost-one-half holomorphic pinching

Hermitian metrics of positive holomorphic sectional curvature on fibrations

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