On Foliations with Semi-positive Anti-canonical Bundle

Druel, Stéphane

doi:10.1007/s00574-018-00128-7

Cited by 4 publications

(3 citation statements)

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“…A key ingredient for the proof is the following theorem, which reduces the situation to the case of algebraically integrable foliations. It follows directly from a series of Druel's works, see [Dru17,Proposition 6.1], [Dru17, Lemma 6.2] and [Dru19,Claim 4.3]. We remark that, in [Dru19, Claim 4.3], the foliation F is assumed to have semipositive anticanoncial class.…”

Section: Introductionmentioning

confidence: 97%

“…Foliations with nef anticanonical classes have been previously investigated as well (e.g. [Dru17], [Dru19] , [CH19] and [CCM19]). Particularly, for regular foliations with semipositive anticanonical class, Druel proved in [Dru19] that one can reduce the problem to the case of foliations with zero canonical class.…”

Section: Introductionmentioning

confidence: 99%

“…[Dru17], [Dru19] , [CH19] and [CCM19]). Particularly, for regular foliations with semipositive anticanonical class, Druel proved in [Dru19] that one can reduce the problem to the case of foliations with zero canonical class. An analogue statement for foliations with nef anticanonical class was conjectured by Cao and Höring.…”

Section: Introductionmentioning

confidence: 99%

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Foliations whose first class is nef

Ou¹

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Let F be a foliation on a projective manifold X with −K F nef. Assume that either F is regular, or it has a compact leaf. We prove that there is a locally trivial fibration f : X → Y, and a foliation G on Y with K G ≡ 0, such thatIntroduction 1 2. Foliations 3 3. Numerically flat vector bundles on complex manifolds 4 4. Algebraically integrable foliations with semistable leaves and nef anticanonical classes 10 5. Construction of semistable reductions 15 6. Proof of the main theorem 18 Reference 24

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Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Foliations whose first class is nef

Ou¹

2021

Preprint

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Foliation adjunction

Cascini,

Spicer

2024

Math. Ann.

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A global Weinstein splitting theorem for holomorphic Poisson manifolds

et al. 2022

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We investigate the structure of smooth holomorphic foliations with numerically flat tangent bundles on compact Kähler manifolds. Extending earlier results on non-uniruled projective manifolds by the second and fourth authors, we show that such foliations induce a decomposition of the tangent bundle of the ambient manifold, have leaves uniformized by Euclidean spaces, and have torsion canonical bundle. Additionally, we prove that smooth two-dimensional foliations with numerically trivial canonical bundle on projective manifolds are either isotrivial fibrations or have numerically flat tangent bundles. This in turn implies a global Weinstein splitting theorem for rank-two Poisson structures on projective manifolds. We also derive new Hodge-theoretic conditions for the existence of zeros of Poisson structures on compact Kähler manifolds.

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On Foliations with Semi-positive Anti-canonical Bundle

Abstract: In this note, we describe the structure of regular foliations with semi-positive anti-canonical bundle on smooth projective varieties.2010 Mathematics Subject Classification. 37F75.

Cited by 4 publications

References 11 publications

Foliations whose first class is nef

Foliations whose first class is nef

Foliation adjunction

A global Weinstein splitting theorem for holomorphic Poisson manifolds

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