Topological invariants and equidesingularization for holomorphic vector fields. (English) Zbl 0576.32020

Camacho, César; Lins Neto, Alcides; Sad, Paulo

J. Differ. Geom. 20, 143-174 (1984).

Let \(Z=(Z_ 1,...,Z_ n)\) be a holomorphic vectorfield in a neighbourhood U of \(0\in {\mathbb{C}}^ n\) which has 0 as an isolated singularity. \({\mathcal F}_ z\) denotes the complex one-dimensional foliation with singularity at 0 defined by the integral curves of Z. A special case which should be kept in mind is the gradient vectorfield of a holomorphic function with isolated singularity at the origin.
The aim of the article is to define certain topological invariants of isolated singularities of holomorphic functions in the more general situation described above and to show that these invariants are indeed topological invariants of the foliation. E.g. if \(n\geq 2\), the Milnor number \(\mu (Z,0)=\dim_{{\mathbb{C}}} {\mathcal O}_{{\mathbb{C}}^ n,0}/(Z_ 1,...,Z_ n)\) is a topological invariant of \({\mathcal F}_ z\) as well as the index at 0 of the vectorfield \(Z|_ D\) where D is the intersection of a small ball with an irreducible complex curve V such that V-\(\{\) \(0\}\) is a leaf of \({\mathcal F}_ z.\)
In the second part the authors consider vectorfields Z in \({\mathbb{C}}^ 2\). They show that after finitely many quadratic transformations at singular points, the foliation \({\mathcal F}_ z\) is transformed into a foliation \(\tilde {\mathcal F}_ z\) with finitely many singularities of a very special kind which the authors call ”simple”. These simple singularities persist after further quadratic transformations and \(\tilde {\mathcal F}_ z\) is called a desingularization of \({\mathcal F}_ z\). A ”generalized curve” is a vectorfield Z such that all simple singularities of \(\tilde {\mathcal F}_ z\) have nonvanishing eigenvalues (which is the case for the gradient vectorfield of a plane curve singularity). It is shown that two generalized curves have isomorphic desingularizations. From this the authors deduce that the algebraic multiplicity of a generalized curve is a topological invariant, which was of course well known for ”true” plane curve singularities.

Reviewer: G.M.Greuel

MSC:

32Sxx	Complex singularities
32S45	Modifications; resolution of singularities (complex-analytic aspects)
32S30	Deformations of complex singularities; vanishing cycles
32S05	Local complex singularities
14J15	Moduli, classification: analytic theory; relations with modular forms

Keywords:

holomorphic vectorfield; foliation; topological invariants of isolated singularities of holomorphic; functions; Milnor number; desingularization; algebraic multiplicity of a generalized curve; topological invariants of isolated singularities of holomorphic functions