The transverse dynamics of a holomorphic flow
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.
transverse-holomorphically trivial deformation
scientific article; zbMATH DE number 4042288