Singular foliations with trivial canonical class

Loray, Frank; Pereira, Jorge Vitorio; Touzet, Frédéric

doi:10.1007/s00222-018-0806-0

Cited by 42 publications

(53 citation statements)

References 52 publications

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“…Techniques from algebraic geometry have been extremely useful in the study of singular holomorphic foliations, see for instance [2,5,20,24,26,33,37,39]. In particular, Jouanolou classified codimension one foliations on P 3 of degrees 0 and 1 in his monograph [33]; Cerveau and Lins Neto showed in [16] that there exist six irreducible components of foliations of degree 2 on projective spaces; and Polishchuk, motivated by the study of holomorphic Poisson structures, also found in [44] a classification of foliations of degree 2 on P 3 under certain hypotheses on the singular set of the foliations.…”

Section: Introductionmentioning

confidence: 99%

Codimension One Holomorphic Distributions on the Projective Three-space

Calvo-Andrade

Corrêa

Jardim

2018

International Mathematics Research Notices

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We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck's Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation, and certain logarithmic foliations have stable tangent sheaves.

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Section: Introductionmentioning

confidence: 99%

Codimension One Holomorphic Distributions on the Projective Three-space

Calvo-Andrade

Corrêa

Jardim

2018

International Mathematics Research Notices

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“…It seems possible that the stability of F is enough to imply the algebraicity of leaves (cf. also [Tou08,LPT11,LPT13,PT13] for classification results of foliations with c 1 (F ) = 0).…”

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confidence: 99%

Algebraic integrability of foliations with numerically trivial canonical bundle

Höring

Peternell

2019

Invent. math.

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Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent work of Druel and Greb-Guenancia-Kebekus this establishes the Beauville-Bogomolov decomposition for minimal models with trivial canonical class.(a) the reflexive symmetric powers S [l] E are H-stable for every l ∈ N, and (b) the algebraic holonomy group of E (cf. Definition 2.13) is connected.Suppose further that E is pseudoeffective (cf. Definition 2.1). Then c 2 (E) · H n−2 = 0.

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“…Let G be a foliation with numerically trivial canonical class on a complex projective manifold, and assume that G has a compact leaf. Then Theorem 5.6 in [LPT18] asserts that G is regular and that there exists a foliation on X transverse to G at any point in X. In this section, we extend this result to our setting (see also [DPS01, Proposition 2.7.1] and [Dru18a, Corollary 7.22]).…”

Section: A Criterion For Regularitymentioning

confidence: 71%