Residues for flags of holomorphic foliations

Let F be a singular holomorphic foliation, of codimension k, on a complex compact manifold such that its singular set has codimension ≥ k + 1. In this work we determinate Baum-Bott residues for F with respect to homogeneous symmetric polynomials of degree k + 1. We drop the Baum-Bott's generic hypothesis and we show that the residues can be expressed in terms of the Grothendieck residue of an one-dimensional foliation on a (k + 1)-dimensional disc transversal to a (k + 1)-codimensional component of the singular set of F . Also, we show that Cenkl's algorithm for non-expected dimensional singularities holds dropping the Cenkl's regularity assumption.

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“…We believe that this algorithm can be adapted to the context of residues for flags of foliations [4].…”

Section: Cenkl Algorithm For Singularities With Non-expected Dimensionmentioning

confidence: 99%

Determination of Baum–Bott residues of higher codimensional foliations

Corrêa

Lourenço

et al. 2019

Asian Journal of Mathematics

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“…Indeed, Lemma 4.3 implies that h 0 (T F (−1)) = 0 since T F does not split as a sum of line bundles; therefore, a nonzero section of T F induces a sub-foliation G of T F of dimension one and degree 1. Since Sing(F ) is 0-dimensional, it follows from [6,Corollary 3] that Sing(F ) must be contained on the 0-dimensional component S 0 (G ) of Sing(G ). However, Sing(F ) has length 5, while, following [47], the length of S 0 (G ) is at most 4.…”

Section: Classification Of Degree 1 Distributionsmentioning

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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“…An explicit calculation of the residues is difficult in general, see [5,29,9,19,32,38,10,39]. If the foliation F has dimension one with isolated singularities, Baum and Bott in [6] show that residues can be expressed in terms of a Grothendieck residue, i.e, for each x ∈ Sing(F ) we have…”

Section: Characteristic Classes and Residuesmentioning

confidence: 99%

“…Let F be a one-dimenional foliation on a projective manifold X and C ⊂ X a curve invariant by F . Esteves and Kleiman in [55, Proposition 5.2] prove the following formula (10) 2p…”

Section: Poincaré and Painlevé Problems For Foliations And Pfaff Systemsmentioning

confidence: 99%

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Analytic varieties invariant by foliations and Pfaff systems

Corrêa

2021

Preprint

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In this work we shall present a survey on problems and results on singular holomorphic foliations and Pfaff systems on complex manifolds assuming that these objects possess invariant analytic varieties. We will focus on recent results which have been motivated by classical works of Darboux, Poincaré and Painlevé on the problem of algebraic integration of singular polynomial differential equations. We present results on Poincaré and Painlevé problem of bounding the degree and the genus of analytic varieties invariant by holomorphic foliations and Pfaff systems. We shall discuss the general ideas of the theory of integrability characterizing the existence of meromorphic first integrals for complex analytic Pfaff equations.

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Residues for flags of holomorphic foliations

Cited by 5 publications

References 20 publications

Determination of Baum–Bott residues of higher codimensional foliations

Determination of Baum–Bott residues of higher codimensional foliations

Codimension One Holomorphic Distributions on the Projective Three-space

Analytic varieties invariant by foliations and Pfaff systems

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