C^*- Actions on Stein analytic spaces with isolated singularities

1984

A generalized Nash Blow-up M ′ with respect to coherent subsheaves of locally free sheaves is defined for complex spaces. It is shown that M ′ is locally isomorphic to a monoidal transformation and hence is analytic. Examples of M ′ are given. Applications are given to Serre’s extension problem and reductive group actions. A C∗-action on Grassmannians is defined, fixed point sets and Bialynicki-Birula decomposition are described. This action is generalized to Grassmann bundles. The Grassmann graph construction is defined for the analytic case and it is shown that for a compact Kaehler manifold the cycle at infinity is an analytic cycle. A calculation involving the localized classes of graph construction is given. Nash residue for singular holomorphic foliations is defined and it is shown that the residue of Baum-Bott and the Nash residue differ by a term that comes from the Grassmann graph construction of the singular foliation. As an application conclusions are drawn about the rati...

Communications in Analysis and Geometry, 1999

Let T be a nondicritical codimension one holomorphic foliation on the fc-dimensional complex projective space M = CP(fc), k > 3. Assume that the closure K^) of the Kupka singular set is a complete intersection projective variety and that the points in K(!F)-K^) exhibit slightly generic holomorphic integrating factors. Then the Kupka components have normalizable transverse type and therefore the foliation F is given by a closed rational 1form on CP(fc). In general (for M a complex manifold and K^) not necessarily complete intersection), we prove the existence of an affine or projective transverse structure in a neighborhood of the Kupka singular set. 1. Introduction. 1.1. Singular holomorphic foliations. According to Probenius Theorem [23] a (nonsingular) codimension one holomorphic foliation T on a complex manifold M is given by an open covering M = (J Uj of M such that in each Uj we have a nonsingular holomorphic 1-form Uj that satisfies the integrability condition ujj A dujj = 0, and such that for each Uj fl [/; ^ 0 we have UJ^JJ nU = dij-^jlu^jj. f or some holomorphic nonvanishing function gij in Ui D Uj. If M is a complex projective space M = CP(fc) then it is known that there are no such foliations without singularities. On the other hand we may consider singular holomorphic foliations defined as follows: A singular codimension one holomorphic foliation on M is defined by an open cover M = [jUj and integrable holo-jeJ morphic 1-forms as above, but that may have singularities. The singular 1 Scardua was supported by CNPq-Brasil and Min Affaires Etrangeres-France.

arXiv: Dynamical Systems, 2016

We consider germs of holomorphic vector fields with an isolated singularity at the origin $0\in\mathbb{C}^2$. We introduce a notion of stability, similar to "Lyapunov stability". For such a germ, called $L$-stable singularity, either the corresponding foliation admits a holomorphic first integral, or it is a real logarithmic foliation singularity. A notion of $L$-stability is also naturally introduced for a leaf of a foliation. In the complex codimension one case, for holomorphic foliations, the holonomy groups of $L$-stable leaves are proved to be abelian, of a suitable type. This implies the existence of local closed meromorphic one-forms defining the foliation, in a neighborhood of $L$-stable leaves.

Journal of Differential Geometry, 1984

2012

We state some generalizations of a theorem due to G. Darboux, which originally states that a polynomial vector field in the complex plane exhibits a rational first integral and has all its orbits algebraic provided that it exhibits infinitely many algebraic orbits. In this paper, we give an interpretation of this result in terms of the classical Reeb stability theorems, for compact leaves of (non-singular) smooth foliations. Then we give versions of Darboux's theorem, assuring, for a (nonsingular) holomorphic foliation of any codimension, the existence of an open set of compact leaves provided that the measure of the set of compact leaves is not zero. As for the case of polynomial vector fields in the complex affine space of dimenion m ≥ 2, we prove suitable versions of the above results, based also on the very special geometry of the complex projective space of dimension m, and on the nature of the singularities of such vector fields we consider.

Archiv Der Mathematik, 2007

We study holomorphic flows on Stein manifolds. We prove that a holomorphic flow with isolated singularities and a dicritical singularity of the form $\sum^{n}_{j=1}\lambda_{j}z_{j}\frac{\partial}{\partial z_{j}}+\ldots, \lambda_{j}\in \mathbb{Q}_{+},\forall j \in \{1,\ldots,n\}$ on a Stein manifold $M^n, n \geq 2$ with ${\mathop{H}\limits^{\vee}}{^{2}}(M^{n}, {{{\mathbb{Z}}}})=0$ , is globally analytically linearizable; in particular M is biholomorphic to ${\mathbb{C}}^{n}$ . A complete stability result for periodic orbits is also obtained.

2014

In this paper we give complete analytic invariants for germs of holomorphic foliations in (C 2 , 0) that become regular after a single blow-up. Some of them describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in finite dimensional complex vector space. Such singularities admit separatrices tangent to any direction at the origin. When enough separatrices coincide with their tangent directions (a condition that can always be attained if the mutiplicity of the germ at the origin is at most four) we are able to describe and realize all the analytical invariants geometrically and provide analytic normal forms. As a consequence we prove that any two such germs sharing the same transverse invariants are conjugated by a very particular type of birational transformations. We also provide the first explicit examples of universal equisingular unfoldings of foliations that cannot be produced by unfolding functions. With these at hand we are able to explicitely parametrize families of analytically distinct foliations that share the same transverse invariants. Contents * The first author is supported by CNPq/Faperj; the second author supported by the ANR under the project ANR-13-JS01-0002-01. Both authors are supported by the Math-Amsud project DFH. Moduli spaces of dicritical foliations 3 Unfoldings in D h. 24

Annals of Mathematics, 2004

2017

We study in this paper several properties concerning singularities of foliations in \begin{document}$ {\left( {{\mathbb{C}}^{3}}\rm{,}\bf{0} \right)}$\end{document} that are pull-back of dicritical foliations in \begin{document}$ {\left( {{\mathbb{C}}^{2}}\rm{,}\bf{0} \right)}$\end{document} . Particularly, we will investigate the existence of first integrals (holomorphic and meromorphic) and the dicriticalness of such a foliation. In the study of meromorphic first integrals we follow the same method used by R. Meziani and P. Sad in dimension two. While the foliations we study are pull-back of foliations in \begin{document}$ {\left( {{\mathbb{C}}^{2}}\rm{,}\bf{0} \right)}$\end{document} , the adaptations are not straightforward.

This paper is about the integrability of complex vector fields in dimension three in a neighborhood of a singular point. More precisely, we study the existence of holomorphic first integrals for isolated singularities of holomorphic vector fields in complex dimension three, pursuing the discussion started in \cite{CaSc2009}. Under generic conditions, we prove a topological criteria for the existence of a holomorphic first integral. Our result may be seen as a kind of Reeb stability result for the framework of vector fields singularities in complex dimension three. As a consequence, we prove that, for the class of singularities we consider, the existence of a holomorphic first integral is invariant under topological equivalence.