Birational Geometry of Foliations

Brunella, Marco

doi:10.1007/978-3-319-14310-1

Cited by 146 publications

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“…In Section 2 we will recall some standard facts about vector fields and foliations on complex surfaces and analytic spaces (the reader is however supposed to be familiar, for example, with the material discussed in the first Chapter of [4] and in [1, Ch. I, §8]).…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

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Vector fields, separatrices and Kato surfaces

Guillot¹

2014

Annales De l'Institut Fourier

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We prove that a singular complex surface that admits a complete holomorphic vector field that has no invariant curve through a singular point of the surface is obtained from a Kato surface by contracting some divisor (in particular, it is compact). We also prove that, in a singular Stein surface endowed with a complete holomorphic vector field, a singular point of the surface where the zeros of the vector field do not accumulate is either a quasihomogeneous or a cyclic quotient singularity. We give new proofs of some results concerning the classification of compact complex surfaces admitting holomorphic vector fields. Our proofs rely in a combinatorial description of the vector field on a resolution of the singular point based on previous work of Rebelo and the author.

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Section: Annales De L'institut Fouriermentioning

confidence: 99%

“…The only possible types of uniformizable affine structures are (−1), (n, −n) for n 2, (∞, ∞), (2, 2, ∞), (2,3,6), (2,4,4), (3,3,3) and (2, 2, 2, 2). Let Table 2.1.…”

Section: Semicompleteness In Manifoldsmentioning

confidence: 99%

Vector fields, separatrices and Kato surfaces

Guillot¹

2014

Annales De l'Institut Fourier

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“…D'autre part (cf [7,Chapter 3]), l'auto-intersection de C 0 peut être calculée en fonction des ε p à l'aide des indices de Camacho-Sad (qui correspondent aux λ(p)) : par la formule de Camacho-Sad, on a C (1) Si C ⊂ Y est une courbe qui passe par p avec multiplicité m et sĩ C ⊂Ỹ est sa transformée stricte alorsC…”

Section: éTude Des Courbes Rationnelles Invariantesunclassified

“…Après un revêtement double, on obtient les feuilletages modulaires de Hilbert construits en faisant agir Γ = PSL 2 (Z[ Pour les propriétés générales des feuilletages sur les surfaces on se réfère à [7]. Pour des propriétés particulières des feuilletages modulaires, on pourra consulter [26].…”

Section: Introductionunclassified

Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI

Cousin¹

2014

Ann. inst. Fourier

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“…(2) If the curve C is invariant by the foliation ω; we introduce the following index Z(F, C, p) for any p ∈ C as follows (see [Br00,page 24]). Let f = 0 be any local equation of the curve C at p, and let θ be a generating 1-form of the foliation F at p .…”

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confidence: 99%

Foliations with a radial Kupka set on projective spaces

Calvo-Andrade¹

2016

Bull Braz Math Soc, New Series

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Abstract. We consider the set K(n, c, x dy − y dx) of codimension one holomorphic foliations on P n , n ≥ 3, with Chern class c, and with a compact, connected Kupka set of radial transversal type. We will prove that foliations in this set, have a rational first integral and define an irreducible component of the space of foliations.0.1. Introduction. Let n ≥ 3 and c ≥ 2 be natural numbers. Consider a differential 1-form in C n+1 ω = a 0 dz 0 + · · · + a n dz n , ∈ where a j are homogeneous polynomials of degree c − 1 in variables z 0 , . . . , z n with complex coefficients. Let us assume that z 0 a 0 + · · · + z n a n = 0 so that ω descends to the complex projective space P n and defines a global section of the twisted sheaf of 1-forms Ω 1 P n (c). For a K-vector space V , we denote PV = V −{0}/K * the projective space of linear subspaces of V and π : V − {0} → PV the quotient map.Consider the projective space PH 0 (P n , Ω 1 (c)) and the subset F(n, c) := π { ω ∈ H 0 (P n , Ω 1 P n (c)) − 0 | ω ∧ dω = 0 } parameterizing 1-forms ω such that they satisfy the Frobenius integrability condition. This is the space of Chern class c foliations of codimension one on P n . It is an algebraic subset defined by quadratic equations, and has several irreducible components. For instance, let us recall (see [CL96], [GML91]) the following families of irreducible components:The rational components R n (a, b) ⊂ F(n, c) consisting of integrable 1-forms of the typewhere c = a + b is a partition with a, b natural numbers, p, q are the unique coprime numbers such that pa = qb and f, g are homogeneous polynomials of respective degree a, b.1991 Mathematics Subject Classification. 58A17, 32G99.

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Birational Geometry of Foliations

Cited by 146 publications

References 0 publications

Vector fields, separatrices and Kato surfaces

Vector fields, separatrices and Kato surfaces

Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI

Foliations with a radial Kupka set on projective spaces

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