Complex manifolds with negative curvature operator

Lee, Man-Chun; Streets, Jeffrey

doi:10.48550/arxiv.1903.12645

Cited by 9 publications

(9 citation statements)

References 25 publications

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“…From a more geometric point of view, we note that while recently there have been some results on geometric flows of non-Kähler metrics converging to rigid, Kähler metrics (cf. [41,51,65]), or converging to interesting non-Kähler metrics assuming a certain symmetric ansatz ( [45,50]), Theorem 1.2 seems to be the first result showing that a natural class of non-Kähler metrics is globally attractive for a geometric flow with arbitrary initial data.…”

Section: In View Of Our Description Of Pluriclosed Flow In Terms Of M...mentioning

confidence: 99%

Non-Kähler Calabi-Yau geometry and pluriclosed flow

Garcia-Fernandez¹,

Jordan²,

Streets³

2021

Preprint

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Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow in terms of Hermitian metrics on holomorphic Courant algebroids, implying new global existence results, in particular on all complex non-Kähler surfaces of Kodaira dimension κ ≥ 0. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a classification of generalized Kähler structures on these spaces.

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Section: In View Of Our Description Of Pluriclosed Flow In Terms Of M...mentioning

confidence: 99%

Non-Kähler Calabi-Yau geometry and pluriclosed flow

Garcia-Fernandez¹,

Jordan²,

Streets³

2021

Preprint

Self Cite

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“…Remark 1.6. It is conjectured (see [14,Remark 1.7], [19,Conjectures 1.1]) that the above theorems could be extended to the Hermitian case, and there are interesting progresses, see [19,12,8].…”

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confidence: 99%

On almost quasi-negative holomorphic sectional curvature

Zhang¹,

Zheng²

2020

Preprint

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A recent celebrated theorem of Diverio-Trapani and Wu-Yau states that a compact Kähler manifold admitting a Kähler metric of quasi-negative holomorphic sectional curvature has an ample canonical line bundle, confirming a conjecture of Yau. In this note we shall introduce a natural notion of almost quasi-negative holomorphic sectional curvature and extend this theorem to compact Kähler manifolds of almost quasi-negative holomorphic sectional curvature. We also obtain a gap-type theorem for the inequality X c 1 (K X ) n > 0 in terms of the holomorphic sectional curvature. The α-invariant plays a key role in the discussions.

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“…When the metric is Kähler, this curvature is the same as the holomorphic sectional curvature H, while when the metric is not Kähler, the curvature condition is slightly stronger than H at least algebraically. This condition also appeared in the recent work by Lee and streets [7] where it is referred to as "positive (resp. negative) curvature operator".…”

Section: Introductionmentioning

confidence: 81%

Positive curvature operator, projective manifold and rational connectedness

Tang

2019

Preprint

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In his recent work [20], X. Yang proved a conjecture raised by Yau in 1982 ([25]), which states that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. In this note, we prove that any compact Hermitian manifold X with positive real bisectional curvature, its hodge number h 1,0 = h 2,0 = h n−1,0 = h n,0 = 0. In particular, if in addition X is Kähler, then X is projective. Also, it is rationally connected manifold when n = 3. This partially confirms the conjecture 1.11 [20] which is proposed by X. Yang.

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Complex manifolds with negative curvature operator

Cited by 9 publications

References 25 publications

Non-Kähler Calabi-Yau geometry and pluriclosed flow

Non-Kähler Calabi-Yau geometry and pluriclosed flow

On almost quasi-negative holomorphic sectional curvature

Positive curvature operator, projective manifold and rational connectedness

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