Germs Of Holomorphic Vector Fields Research Articles

Analytic Classification of Foliations Induced by Germs of Holomorphic Vector Fields in ( ℂ n , 0 ) $(\mathbb {C}^{n},0)$ with Non-isolated Singularities

We consider germs of holomorphic vector fields in $(\mathbb {C}^{n},0)$ , n ≥ 3, with non-isolated singularities. We assume that the set of singular points forms a submanifold of codimension 2, and the sum of the nonzero eigenvalues of the linearization of the germs at each singular point is zero. We give the orbital analytic classification of generic germs of such type. It happens that, unlike the formal classification (which is trivial), the analytic one has functional moduli. The same result is obtained in the real-analytic case (the smooth normalization was obtained earlier in Roussarie Asterisque. 1975;30:1–181).

Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension $n \geq 3$. Based on Belitskii's work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensure that the normalizing transformation is holomorphic at the fixed point. We shall show that this sufficient condition is a nilpotent version of Bruno's condition (A). In dimension 2, no condition is required since, according to Str{\'o}zyna- Zoladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton's method and $\frak{sl}\_2(\Bbb C)$-representations.

Forme normale de perturbation de champs de vecteurs quasi-homogènes

In this Note, we study germs of holomorphic vector fields which are suitable perturbations of a quasihomogeneous vector field in a neighborhood of the origin of C n , fixed point of the vector fields. In particular, we define a “diophantine condition” on the quasihomogeneous initial part S which ensures that if such a perturbation of S is formally conjugate to S then it is also holomorphically conjugate to it. To cite this article: E. Lombardi, L. Stolovitch, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Analytic classification of saddle nodes

Isolated degenerate elementary singular points (saddle nodes) of germs of holomorphic vector fields in $(\mathbb {C}^2,0)$ are studied. An analytic classification of them is obtained; it is shown that the analytic classification has two times more moduli (numeric and functional) than the orbital analytic classification. A theorem on sectorial normalization is proved.

Examples of non-conjugated holomorphic vector fields and foliations

We consider germs of holomorphic vector fields near the origin of C 2 with a saddle-node singularity, and the induced singular foliations. In a previous article we described the invariants addressing the analytical classification of these vector fields. They split into three parts: a formal, an orbital and a tangential component. For a fixed formal class, the orbital invariant (associated to the foliation) was obtained by Martinet and Ramis; we give it an integral representation. We then derive examples of non-orbitally conjugated foliations by the use of a “first-step” normal form, whose first-significative jet is an invariant. The tangential invariant also admits an integral representation, hence we derive explicit examples of vector fields, inducing the same foliation, that are not mutually conjugated. In addition, we provide a family of normal forms for vector fields orbitally equivalent to the model of Poincaré–Dulac.

Examples of non-conjugated holomorphic vector fields and foliations

We consider germs of holomorphic vector fields near the origin of C 2 with a saddle-node singularity, and the induced singular foliations. In a previous article we described the invariants addressing the analytical classification of these vector fields. They split into three parts: a formal, an orbital and a tangential component. For a fixed formal class, the orbital invariant (associated to the foliation) was obtained by Martinet and Ramis; we give it an integral representation. We then derive examples of non-orbitally conjugated foliations by the use of a “first-step” normal form, whose first-significative jet is an invariant. The tangential invariant also admits an integral representation, hence we derive explicit examples of vector fields, inducing the same foliation, that are not mutually conjugated. In addition, we provide a family of normal forms for vector fields orbitally equivalent to the model of Poincaré–Dulac.

Complex vector fields having orbits with bounded geometry

Germs of holomorphic vector fields at the origin $0\in \co^{\kern1pt2}$ and polynomial vector fields on $\co^{\kern1pt2}$ are studied. Our aim is to classify these vector fields whose orbits have bounded geometry in a certain sense. Namely, we consider the following situations: (i) the volume of orbits is an integrable function, (ii) the orbits have sub-exponential growth, (iii) the total curvature of orbits is finite. In each case we classify these vector fields under some generic hypothesis on singularities. Applications to questions, concerning polynomial vector fields having closed orbits and complete polynomial vector fields, are given. We also give some applications to the classical theory of compact foliations.

Rigidity Theorem for Degenerated Singular Points of Germs of Holomorphic Vector Fields in the Complex Plane

We consider the class νn of germs of holomorphic vector fields in ({\Bbb C}^2, 0) with vanishing (n − 1)-jet at zero. We prove that the formal equivalence of two generic germs in νn implies their analytic equivalence. This result is analogous to the one obtained in l16r for the case of orbital analytic equivalence.

Vanishing theorems for cohomologies of automorphic vector bundles

Let M be a compact, irreducible, locally hermitian symmetric space of noncompact type. If M is not a Riemann surface, then Calabi and Vesentini have shown in ([4]), that the complex structure on M is infinitesimally rigid, i.e., they show the vanishing of H(M, ΘM), where ΘM is the tangent sheaf of germs of holomorphic vector fields on M. Their method involves the construction of an 'auxiliary expression', which is simplified in two different ways, to obtain a quadratic form involving curvature terms. The desired vanishing is reduced then to proving that the quadratic form is positive definite. One obtains criteria for the vanishing of the cohomology groups H*(M, ΘM), which depend on the curvature properties of M and not on the lattice defining M. Based on their method, Weil showed (see [17]) that an irreducible, cocompact lattice Γ in a real semisimple Lie group G without compact or three dimensional factors is rigid, i.e., any 'nearby' deformations of Γ inside G are equivalent. This amounts to showing the vanishing of ΐί (Γ, Ad), where Ad is the adjoint representation of G on its Lie algebra. Matsushima refined this method to show vanishing of Betti numbers of M below some degree ([10]). Our main result is to give a criterion for the vanishing of cohomologies of vector bundles, generalising the results of Calabi-Vesentini and Matsushima. More generally we give a criterion for the vanishing of d-cohomology (or (g, K)— cohomology) of unitary gmodules with coeffecients in a Kmodule. The terms involved can be calculated explicitly in terms of the dominant weight of the reprsentation defining the automorphic vector bundle and the curva-

A residue formula for the index of a holomorphic flow

Let ( ~ , P ) be a normal, isolated complex analytic singularity of dimension n ~ 1, and consider germs of holomorphic vector fields on ~ at P. One would like to be able to distinguish different germs. If P is a regular point of Y', there are a number of invariants that sometimes allow us to distinguish germs of vector fields at P. However, for P singular, little is known. If X is a holomorphic vector field on ~ , singular only at P, then a natural way for studying the geometry, the topology and the dynamics of X near P, is to go up to a resolution of P and see what happens there, see for instance [HH]. More precisely, let

On fixed points of conformal pseudogroups

We show in this paper Theorem 2 that if (H, H1) is a pseudogroup generated by a finite numberH1 of germs of conformal diffeomorphisms of ℂ defined on a sufficiently small discD, which is not linearizable and such that the linear group (L,H1)={g′(0)/g∈(H,H1)}⊂ℂ* is dense in ℂ*, then the set of fixed points of the pseudogroup (H, H1) is dense inD. This implies the abundance of distinct homotopy classes of loops in leaves of foliations defined in ℂ2 by generic polynomial vector fields as well as for germs of holomorphic vector fields in ℂ2 beginning with generic jets, both of degree at least 2. These homotopy classes may be realized arbitrarily close to the line at infinity or to 0, respectively. This shows the genericity of polynomial vector fields with infinite Petrovsky-Landis genus ([5]).

Normal forms of invariant vector fields under a finite group action

Let G be a finite subgroup of GL(n, C). This subgroup acts on the space of germs of holomorphic vector fields vanishing at the origin in Cn and on the group of germs of holomorphic diffeomorphisms of (Cn, 0). We prove a theorem of invariant conjugacy to a normal form and linearization for the subspace of invariant germs of holomorphic vector fields and we give a description of this type of normal forms in dimension n = 2.

Germs of holomorphic vector fields in C3 without a separatrix

Oblatum 28-XII-1988 & 26-III-1990 & 10-X-1990 & 13-VI-1991 & 30-IX-1991. Supported by Universite de Toulouse, CONACYT-CNRS and CAICYT-87-336 Fulltext Preview

On deformations of holomorphic foliations

Given a non-singular holomorphic foliation ℱ on a compact manifold M we analyze the relationship between the versal spaces K and K tr of deformations of ℱ as a holomorphic foliation and as a transversely holomorphic foliation respectively. With this purpose, we prove the existence of a versal unfolding of ℱ parametrized by an analytic space K f isomorphic to π -1 (0)×Σ where Σ is smooth and π : K→K tr is the forgetful map. The map π is shown to be an epimorphism in two situations: (i) if H 2 (M,Θ ℱ f )=0, where Θ ℱ f is the sheaf of germs of holomorphic vector fields tangent to ℱ, and (ii) if there exists a holomorphic foliation ℱ ⋔ transverse and supplementary to ℱ. When the conditions (i) and (ii) are both fulfilled then K≅K f ×K tr .

ORBITAL ANALYTIC NONEQUIVALENCE OF SADDLE RESONANCE VECTOR FIELDS IN $ (\mathbf{C}^2,\,0)$

This article examines germs of holomorphic vector fields for the form under the assumption that the support of the power series lies either above the bisector of the first quadrant of the integer lattice , or below it. Necessary conditions (imposed on the coefficients of ) are formulated for orbital analytic equivalence of vector fields of the type indicated; these are obtained with the help of approximate calculation of the Ecalle-Voronin functional moduli for the analytic classification of germs of holomorphic mappings which are monodromy transformations of the vector fields considered.Bibliography: 18 titles.

Deformations of holomorphic Seifert fiber spaces

Let M be a holomorphic Seifert fiber space as constructed in Conner and Raymond [3] and let O M be the sheaf of germs of holomorphic vector fields on M. The purpose of this paper is to study the cohomology group H I(M, OM) and to describe the small deformations of M. Since M is given as a quotient of a principal torus bundle, it would be natural to use the cohomology of groups with coefficients in sheaves with operators to describe M and objects associated to it. These groups were introduced in Grothendieck [4] and also in Kodaira [7], Conner [2] and Conner and Raymond [3] by different approaches. In Sect. 1, we review these groups. Using the equivariant version of the Leray spectral sequence obtained in Sect. 1, we give a decomposition of H~(M, OM) into the direct sum of complex vector groups C, E, F and H in Sect. 2. In Sect's. 3 through 6, we study the geometric significance of the groups C, E, F and H. In Sect. 7, we compute these vector groups when the dimension of the base space is one and give some examples. It is possible to deal with r fibrations in a similar manner. It will be done in a forthcoming paper [14]. I would like to thank F. Raymond for many valuable conversations.

CONDITIONS FOR TRIVIALITY OF DEFORMATIONS OF COMPLEX STRUCTURES

Let f : X→S be a characteristic, holomorphic mapping of complex spaces (with nilpotent elements). The paper proves that, if f is a flat mapping and all its fibers are equivalent to one and the same compact complex space X0, then, with respect to this mapping, X is equivalent to a holomorphic fibering over S with fiber X0 and structure group Aut(X0). It is further proved that, if the base S is reduced, the assertion remains true for any holomorphic mapping f, at least in the case when the fiber X0 is an irreducible space. This is a strong generalization of the corresponding result of Fischer and Grauert, in which a similar assertion is proved for the case when X and S are complex manifolds and f is a locally trivial mapping. This paper also proves that, if the compact complex space X0 satisfies the condition H1(Ω, X0) = 0, where Ω is the sheaf of germs of holomorphic vector fields on X0, then any locally trivial deformation of the space X0, with arbitrary parameter space, is trivial. This generalizes Kerner's result, in which the parameter space is assumed to be a manifold. Bibliography: 7 titles.