Pairs of foliations and Mattei-Moussu's Theorem☆
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 . Another vector field will define the same foliation if, and only if, it takes the form 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 and assume that the holonomies of the separatrices associated to 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 of a saddle with a regular foliation such that one of the separatrices of is a leaf of . 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 obtained by a finite sequence of blowing-up such that the singular points of the lifted pair 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 on a complex surface M is defined by coherent analytic subsheaf of rank 1 such that the quotient sheaf is torsion free. The locus where is not locally free is the singular locus, a discrete set. Locally, sections of take the form where v is a holomorphic vector field with isolated zeroes, and f is any holomorphic function. Any other generating vector field takes the form with g non vanishing. Equivalently, one can locally
Mattei-Moussu's Theorem
Theorem 3.1 Mattei-Moussu Let and be two foliations of the form Assume that they are well defined near and have the same holonomy along this leaf. Then, there exists an analytic diffeomorphism conjugating the foliations: , i.e. .
Proof The first part of the proof follows the paper [11]. Let and be local first integrals of respectively at . Assume that and have the same monodromy along a
Reduction of singularities for a pair of foliations
Consider a pair of holomorphic singular foliations on a complex surface M. Denote by the tangency divisor between the two foliations: locally, if where is a holomorphic one-form with isolated zeroes, then is defined by the ideal (f) given by . We note that is passing through all singular points of each . Recall Seidenberg's Theorem:
Theorem 4.1 Let be a holomorphic singular foliation on a complex surface M. Then there is a proper map Seidenberg [14]
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 of type (6) in the list of Theorem 4.2 and assume that is reduced, i.e. without multiplicity. One can easily check that this implies that it is possible to write with , and with (otherwise cannot be reduced). Conversely, any pair as above with
Declaration of Competing Interest
There is no competing interest.
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