MMP for co-rank one foliations on threefolds

Cascini, Paolo; Spicer, Calum

doi:10.1007/s00222-021-01037-1

Cited by 21 publications

(67 citation statements)

References 35 publications

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“…Building on works of McQuillan, versions of the MMP for foliations on threefolds were recently established by Cascini and Spicer. We refer to [CS21] for details and references about the MMP for codimension one foliations, and [CS20] for foliations of dimension one on threefolds.…”

Section: Canonical Class Of Foliationsmentioning

confidence: 99%

Foliations on Complex Manifolds

Araujo¹,

Figueredo²

2022

Notices Amer. Math. Soc.

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In these notes we survey some aspects of the theory of holomorphic foliations on complex manifolds. The origins of the theory go back to works of Darboux, Poincaré and Painlevé, where it was developed to study solutions of ordinary differential equations on ℂ 2 . We briefly discuss some of the early works on this theory, mostly concerned with the local behavior of the leaves near the singularities. We then move the focus from local to global properties. Birational geometry has had a great influence on the development of a global theory of holomorphic foliations. After reviewing the Enriques-Kodaira classification of projective surfaces and explaining the general philosophy of the Mininal Model Program, we explore some of their recent counterparts for foliations.

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Section: Canonical Class Of Foliationsmentioning

confidence: 99%

Foliations on Complex Manifolds

Araujo¹,

Figueredo²

2022

Notices Amer. Math. Soc.

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“…To this end, we prove a stronger result: the Cone Theorem (cf. Theorem 3.9) holds for algebraically integrable foliations (see also [BM16,Spi20,CS21] for other versions of the Cone Theorem for foliations). This implies that if (F , ∆) is a foliated log pair (cf.…”

Section: Introductionmentioning

confidence: 98%

Positivity of the Moduli Part

Ambro¹,

Cascini²,

Shokurov³

et al. 2021

Preprint

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We prove the Cone Theorem for algebraically integrable foliations. As a consequence, we show that termination of flips implies the b-nefness of the moduli part of a log canonical pair with respect to a contraction, generalising the case of lc trivial fibrations. Contents 1. Introduction 1 2. Preliminaries 3 3. Cone theorem for algebraically integrable foliations 13 4. Proof of the Main Theorems 23 5. Applications 27 References 30

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“…If G is a foliation on a complex projective variety, we define its canonical class to be K G = −c 1 (G ). In analogy with the case of projective varieties, one expects the numerical properties of K G to reflect geometric aspects of G (see [Bru04], [McQ08], [Spi17], [CS18], [AD13], [AD14], [AD16], [AD17], [LPT18], [Tou08]). This led, for instance, to the birational classification of foliations by curves on surfaces with quotient singularities ( [Bru04], [McQ08]), generalizing most of the important results of the Enriques−Kodaira classification.…”

Section: Introductionmentioning

confidence: 99%

“…From the point of view of birational classification of foliations, this class of singularities is however inadequate. Indeed, the foliated analogue of the minimal model program aims in particular to reduce the birational study of mildly singular foliations with numerical dimension zero on complex projective manifolds to the study of associated minimal models, that is, mildly singular foliations with numerically trivial canonical class on klt spaces (see for instance [CS18,Theorem 1.7]). Building on the results of the present paper, it has been shown that both Theorem 1.1 and Theorem 1.3 are valid for codimension one foliations with canonical singularities on projective varieties with klt singularities (see [DO19]).…”

Section: Introductionmentioning

confidence: 99%

Codimension 1 foliations with numerically trivial canonical class on singular spaces

Druel

2021

Duke Math. J.

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In this article, we describe the structure of codimension one foliations with canonical singularities and numerically trivial canonical class on varieties with terminal singularities, extending a result of Loray, Pereira and Touzet to this context. COFECUB project Ma932/19 and the ANR project Foliage (ANR grant Nr ANR-16-CE40-0008-01).2. Notation, conventions, and used facts 2.1. Global Convention. Throughout the paper a variety is a reduced and irreducible scheme separated and of finite type over a field.Given a scheme X, we denote by X reg its smooth locus. Suppose that k = C. We will use the notions of terminal, canonical, klt, and lc singularities for pairs without further explanation or comment and simply refer to [KM98, Section 2.3] for a discussion and for their precise definitions. We refer to [KM98] and [KMM87] for standard references concerning the minimal model program.2.2. Q-factorializations and Q-factorial terminalizations.Definition 2.1. Let X be a normal complex quasi-projective variety with klt singularities. A Q-factorialization is a small birational projective morphism β : Z → X, where Z is Q-factorial with klt singularities.Fact 2.2. The existence of Q-factorializations is established in [Kol13, Corollary 1.37]. Note that we must haveDefinition 2.3. Let X be a normal complex quasi-projective variety with canonical singularities. A Q-factorial terminalization of X is a birational crepant projective morphism β : Z → X where Z is Q-factorial with terminal singularities. Fact 2.4. The existence of Q-factorial terminalizations is established in [BCHM10, Corollary 1.4.3].2.3. Projective space bundle. If E is a locally free sheaf of finite rank on a variety X, we denote by P(E ) the variety Proj X S • E , and by O P(E ) (1) its tautological line bundle. Stability.The word stable will always mean slope-stable with respect to a given movable curve class. Ditto for semistable and polystable. We refer to [HL97, Definition 1.2.12] for their precise definitions.

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MMP for co-rank one foliations on threefolds

Cited by 21 publications

References 35 publications

Foliations on Complex Manifolds

Foliations on Complex Manifolds

Positivity of the Moduli Part

Codimension 1 foliations with numerically trivial canonical class on singular spaces

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