Equivalence and semi-completude of foliations

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Abstract

Holomorphic vector fields of Siegel type with an isolated singularity at the origin are considered. It is proved that those vector fields, under suitable conditions always verified in dimension 3, admit a semi-complete representative. The method used gives new proof of the extension of the Theorem of Mattei–Moussu.

Introduction

Let X be a holomorphic vector field in C2 with an isolated singularity at the origin and such that the eigenvalues of its linear part are both non zero. Then the foliation associated to X admits a semi-complete representative [11]. We can ask if this result is still valid for higher dimensions.

In C2, the Siegel Domain is a thin set (it has zero measure). Contrary to the C2 case, the interior of the Siegel Domain is non empty for Cn, when n3; thus the Siegel Domain represents an important set for the problem above. As the conclusions for the Poincaré Domain are easy, we will focus on the Siegel Domain.

Let X:(Cn,0)(Cn,0), n3, be a holomorphic vector field, where λ1,,λn represent the eigenvalues of DX(0), verifying:

  • (a)

    the origin is an isolated singularity,

  • (b)

    X is of Siegel type (0 belongs to the convex hull of {λ1,,λn}),

  • (c)

    all eigenvalues of DX(0) are non zero and there exists a straight line through the origin, in the complex plane, separating λ1 from the other eigenvalues,

  • (d)

    up to a change of coordinates, X=i=1nλixi(1+fi(x))/xi, where x=(x1,,xn) and fi(0)=0 for all i.

Up to multiplication by a constant we can assume that λ1=1.

In this paper only vector fields with an isolated singularity at the origin are considered, even if not explicitly stated.

It is important to remark that if n=3 and X is a vector field with an isolated singularity at the origin and of strict Siegel type (the convex hull of {λ1,,λn} contains a neighbourhood of the origin), then (c) and (d) are immediately satisfied [1]; in particular, there exists at least one eigenvalue λi such that the angle between λi and the other eigenvalues is greater than π/2 (eigenvalues viewed as vectors in R2).

Let X be a holomorphic vector field on (C2,0), with an isolated singularity at the origin and of “strict Siegel type” (λ1/λ2R-, where λ1 and λ2 are the eigenvalues of DX(0)). Then X admits two separatrices. Let Σ be a transversal section to one of the separatrices and F the foliation associated to X. It is well known that the saturated of Σ by F together with the other separatrix contains a neighbourhood of the origin [9]. For higher dimension we obtain:

Proposition

Let X be a holomorphic vector field verifying (a), (b), (c) and (d) and S the separatrix tangent to the eigenspace associated to λ1. The saturated of a transversal section to S, at a point sufficiently close to the origin, together with the invariant manifold transverse to that separatrix contains a neighbourhood of the origin.

The method used in the proof of the proposition allows us to prove:

Theorem

Let F be the foliation associated to a holomorphic vector field X verifying (a), (b), (c) and (d). Then F admits a semi-complete vector field as its representative, in a neighbourhood of the singularity.

In particular, any vector field on (C3,0) of strict Siegel type admits a semi-complete representative.

It is obvious that if X and Y, two holomorphic vector fields, are analytically equivalent in a neighbourhood of a singularity then the holonomies relatively to the separatrices, if they exist, are analytically conjugated (there exist holomorphic vector fields, defined on a neighbourhood of the origin of Cn, n3, without separatrices [5], [7]).

Mattei and Moussu proved the reciprocal of this result for linearization of vector fields on (C2,0) of strict Siegel type (if the eigenvalues of the linear part, at the singularity, are in the Poincaré Domain and do not verify any resonance relation between them, then both X and the holonomy are linearizable) and, later, Mattei proved that if the holonomies are analytically conjugated then the vector fields are analytically equivalent [8].

An extension, in a sense, of the results of Mattei and Moussu is here obtained:

Theorem

Let X and Y be two vector fields verifying (a), (b), (c) and (d). Denote by h1X and h1Y the holonomies of X and Y relatively to the separatrices of X and Y tangent to the eigenspace associated to the first eigenvalue, respectively. Then if h1X and h1Y are analytically conjugated, X and Y are analytically equivalent.

This result has already been proved by Elizarov and Il’Yashenko [3]. However, our proof is, in our opinion, simpler then theirs.

Section snippets

Vector fields of Siegel type

Let X:UCnCn and Y:VCnCn be holomorphic vector fields with a singularity at the origin. We say that X is analytically conjugated to Y in a neighbourhood of the origin if there exists a holomorphic diffeomorphism H:V1U1, where 0U1U, 0V1V, such that H(0)=0 and Y=(DH)-1(XH).We say that X and Y are analytically equivalent if X is analytically conjugated to fY, for some holomorphic function f verifying f(0)0.

The integral curves of any vector field X define a foliation of complex dimension

Semi-completude of vector fields of strict Siegel type

The definition of a semi-complete vector field relatively to a (relatively compact) open set U was introduced in [10].

Definition 1

Let X be a holomorphic vector field defined on a complex manifold M and UM an open subset of M. We say that X is semi-complete relatively to U if there exists a holomorphic application Φ:ΩC×UU,where Ω is an open set containing {0}×U such that

  • (a)

    Φ(0,x)=x,xM,

  • (b)

    X(x)=ddTT=0Φ(T,x),

  • (c)

    Φ(T1+T2,x)=Φ(T2,Φ(T1,x)), when the two members are defined,

  • (d)

    (Ti,x)Ω and (Ti,x)ΩΦ(Ti,x)U.

We call Φ

Acknowledgements

I would like to thank Professor J. Basto-Gonçalves for the discussions on the subject.

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Financial support from Fundação para a Ciência e Tecnologia (FCT) through Centro de Matemática da Universidade do Porto, from PRODEPIII and Fundação Calouste Gulbenkian.

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