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The transverse dynamics of a holomorphic flow. (English) Zbl 0639.32013

Let P(z) be a homogeneous polynomial vector field in \({\mathbb{C}}^{n+1}\). The solutions of the system of complex differential equations \(dz/dT=P(z)\) induce a foliation with singularities \({\mathcal F}={\mathcal F}(P)\) in \({\mathbb{C}}P^ n\). Denote by \(f_ i: U_ i\to {\mathbb{C}}^{n-1}\) the family of submersions with connected fibers and disjoint targets defining \({\mathcal F}\), with transition biholomorphisms \(g_{ij}\) between the targets so that \(f_ i=g_{ij}\cdot f_ j\). Note that those compositions \(g_{i_ 1i_ 2}...g_{i_{r-1}i_ r}\) determine a dynamics of this foliation.
By a deformation of \({\mathcal F}\) in \({\mathbb{C}}P^ n\), we mean a germ of a family of foliations \(\{\) \({\mathcal F}_ s={\mathcal F}(P_ s)\}\) given by homogeneous polynomials \(\{P_ s\}\) with s in a complex analytic space (S,0) and \(P_ 0=P\). The deformation is said topologically trivial if there are a neighborhood S’ of 0 in S and a self-homomorphism \(\phi\) of \(S'\times {\mathbb{C}}P^ n\) over S’ such that \(\phi\) maps the leaves of \({\mathcal F}_ s\) to those of \({\mathcal F}\). If further S is reduced and this \(\phi\) induces biholomorphisms \({\bar \phi}_{ij}\) between the targets of the submersions in \(S'\times {\mathbb{C}}^{n-1}\) defining \({\mathcal F}_ s\), we say that the deformation is transverse-holomorphically trivial. If \(\phi\) is holomorphic by itself, then the deformation is holomorphically trivial.
In this paper, the author proves the following rigidity theorem: 1) There is an open and dense set \(\{\) \(P\}\) of homogeneous polynomial vector fields in \({\mathbb{C}}^{n+1}\), \(n>1\), such that any transverse- holomorphically trivial deformation of \({\mathcal F}(P)\) parametrized by a reduced analytic space is holomorphically trivial. 2) For most homogeneous polynomial vector fields in \({\mathbb{C}}^ 3\) having an algebraic solution (i.e. the closure of some leaf of the induced foliation in \({\mathbb{C}}P^ 2\) is an algebraic curve), any topologically trivial deformation parametrized by a reduced analytic space is holomorphically trivial. Note that the meaning of “most” is also explained in the course of the proof of this theorem.
Reviewer: G.Tian

MSC:

32S30 Deformations of complex singularities; vanishing cycles
57R30 Foliations in differential topology; geometric theory
37C10 Dynamics induced by flows and semiflows
58H15 Deformations of general structures on manifolds
32G07 Deformations of special (e.g., CR) structures
14H05 Algebraic functions and function fields in algebraic geometry
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