Pairs of foliations and Mattei-Moussu's Theorem

To J. Martinet, J.-F. Mattei, R. Moussu and J.-P. Ramis
https://doi.org/10.1016/j.bulsci.2021.103035Get rights and content

Abstract

We prove a reduction of singularities for pairs of foliations by blowing-up, and then investigate the analytic classification of the reduced models. Those reduced pairs of regular foliations are well understood. The case of a regular and a singular foliation is dealt with Mattei-Moussu's Theorem for which we provide a new proof, avoiding Gronwall's inequality. We end-up announcing results recently obtained by the first author in the case of a pair of reduced foliations sharing the same separatrices.

Introduction

A singular holomorphic foliation by curves on a surface is locally given by a holomorphic vector field with isolated zeroes. Outside of the zero set, the complex integral curves of the vector field v define a regular holomorphic foliation by curves F. Another vector field will define the same foliation if, and only if, it takes the form fv for a non vanishing holomorphic function f.

In [11], Mattei and Moussu provide a topological characterization of singular points of holomorphic foliations that admit a non constant holomorphic first integral, i.e. such that the leaves are locally defined as the level curves of a holomorphic function h. For this, they use the reduction of singularities by blow-up, and they provide a careful study of the reduced singular points. In particular, they prove that the saddle singular points are determined by their eigenvalues and holonomy map (see Theorem 3.1, and also [4]). The proof of this famous result is in two parts. They consider two saddles with the same eigenvalues {λ1,λ2} and assume that the holonomies of the separatrices associated to λ1 say are conjugated. First they use this conjugacy to construct a conjugacy of the foliations between neighborhoods of annuli contained into the separatrices; this is done by a simple and standard geometric argument. In a second step, they prove that the conjugacy extends on a neighborhood of the singular point minus the second separatrix, by controlling the boundedness by means of Gronwall's Inequality, and extends holomorphically along the separatrix by Riemann Extension Theorem. This second part of the proof is delicate and, in general, not written in full details. Here, we provide in Section 3 a complete and elementary proof avoiding Gronwall's Inequality. In fact, we compare the analytic conjugacy on the annulus with the formal conjugacy which is transversely formal on the disk: they differ by a transversely formal symmetry of the foliation on the annulus. It suffices to prove that all such symmetries extend as a transversely formal symmetry on the disk, and this allows to conclude the analytic extension of the conjugacy. Our proof remains valid in the saddle-node case when the central manifold (invariant curve tangent to the zero-eigendirection) is analytic (not only formal) although Gronwall's Inequality cannot be used to conclude in that case. We hope that this proof can be useful in other situations, like for higher dimensional saddles, for instance weakening assumptions of Reis' result in [12]. At least, this approach has been recently used by the first author to classify some natural pairs of singular foliations. Let us explain.

Mattei-Moussu's Theorem provides fibered conjugacy, that is to say, it can be thought as classification of pairs (F1,F2) of a saddle F2 with a regular foliation F1 such that one of the separatrices of F2 is a leaf of F1. We start a study of pairs of singular foliations by proving a reduction of singularities. In Theorem 4.2, we prove that, given a pair of singular foliations on a complex surface M, we can construct a proper map π:M˜M obtained by a finite sequence of blowing-up such that the singular points of the lifted pair π(F1,F2) fit with a list of simple models. Reduced pairs of regular foliations can be easily handled and are well-known, let us mention [15] for a recent contribution. Mattei-Moussu's Theorem and some results of Martinet and Ramis allow us to deal with the analytic classification of those models where only one of the two foliations is regular. In Section 5, we state a recent contribution of the first author to the case of two singular foliations: reduced pairs consist of pairs of reduced singular foliations sharing the same invariant curves. In [2], [3], the first author provides a complete analytic classification in the case the two foliations have a reduced tangency divisor (i.e. without multiplicity) along the invariant curves (see Theorem 5.1). All details will be published in the forthcoming paper [3]. In the study of reduced pairs, only remains the case of two reduced singular foliations with higher tangency along the invariant curves. This seems feasible, but more technical and left to the future.

Section snippets

Singular holomorphic foliations in dimension 2

A singular holomorphic foliation by curves F on a complex surface M is defined by coherent analytic subsheaf TFTM of rank 1 such that the quotient sheaf TM/TF is torsion free. The locus where TM/TF is not locally free is the singular locus, a discrete set. Locally, sections of TF take the form fv where v is a holomorphic vector field with isolated zeroes, and f is any holomorphic function. Any other generating vector field takes the form gv with g non vanishing. Equivalently, one can locally

Mattei-Moussu's Theorem

Theorem 3.1 Mattei-Moussu

Let F1 and F2 be two foliations of the formFi=ker(ωi),ωi=xdy(λ+O(y))ydx,i=1,2,λR<0. Assume that they are well defined near Δ={(x,0);0<|x|<r} and have the same holonomy along this leaf. Then, there exists an analytic diffeomorphism Φ(x,y)=(x,y+o(y)) conjugating the foliations: ΦF2F1, i.e. Φω2ω1=0.

Proof

The first part of the proof follows the paper [11]. Let p0Δ and H1,H2 be local first integrals of F1,F2 respectively at p0. Assume that H1 and H2 have the same monodromy φ=φγDiff(C,0) along a

Reduction of singularities for a pair of foliations

Consider a pair (F1,F2) of holomorphic singular foliations on a complex surface M. Denote by Tang(F1,F2) the tangency divisor between the two foliations: locally, if Fi=ker(ωi) where ωi is a holomorphic one-form with isolated zeroes, then Tang(F1,F2) is defined by the ideal (f) given by ω1ω2=fdxdy. We note that Tang(F1,F2) is passing through all singular points of each Fi. Recall Seidenberg's Theorem:

Theorem 4.1

Seidenberg [14]

Let F be a holomorphic singular foliation on a complex surface M. Then there is a proper map

Reduced pairs of foliations: a partial classification

This section is devoted to announcement of results of the first author in her thesis [2]. All details will appear in a forthcoming paper [3].

We consider a local pair (F1,F2) of type (6) in the list of Theorem 4.2 and assume that Tang(F1,F2)=(xy) is reduced, i.e. without multiplicity. One can easily check that this implies that it is possible to writeFi=ker(xdy(λi+o(1))ydx) with λiCQ0, and with λ1λ2 (otherwise Tang(F1,F2) cannot be reduced). Conversely, any pair (F1,F2) as above with λ1λ2

Declaration of Competing Interest

There is no competing interest.

References (17)

  • H. Reis

    Equivalence and semi-completude of foliations

    Nonlinear Anal.

    (2006)
  • M. Brunella

    Birational Theory of Foliations

    (2015)
  • A.A. Diaw

    Géométrie de certains tissus holomorphes singuliers en dimension 2

    (2019)
  • A.A. Diaw, Pairs of foliations: reduction of singularities and local classification, in...
  • P.M. Elizarov et al.

    Remarks on orbital analytic classification of germs of vector fields

    Mat. Sb. (N. S.)

    (1983)
  • F. Loray

    Sur les Théorèmes I et II de Painlevé

  • F. Loray

    Versal deformation of the analytic saddle-node. Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes. II

    Astérisque

    (2004)
  • F. Loray

    Pseudo-groupe d'une singularité de feuilletage holomorphe en dimension 2

There are more references available in the full text version of this article.

We warmly thank Jorge Vitório Pereira and Frédéric Touzet for useful discussions, and the anonymous referee for his helpful remarks. This work is supported by ANR-16-CE40-0008 “Foliage” grant, CAPES-COFECUB Ma 932/19 project, Université de Rennes 1 and Centre Henri Lebesgue CHL.

View full text