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Topological invariants and equidesingularization for holomorphic vector fields

1984; Lehigh University; Volume: 20; Issue: 1 Linguagem: Inglês

10.4310/jdg/1214438995

1945-743X

César Camacho, Alcides Lins Neto, Paulo Sad,

Homotopy and Cohomology in Algebraic Topology

The integrals of a holomorphic vector field Z defined in an open subset °U of C" are complex curves parametrized locally as the solutions of the differential equation They define a complex one-dimensional foliation J^z of °U with singularities at the zeros of Z.The purpose of this paper is to exhibit several topological invariants of these foliations near a singular point.Let Θ n be the ring of germs of holomorphic functions defined in some neighborhood of 0 G C w and let /(Z^ ,Z M ) be the ideal generated by the germs at OeC of the coordinate functions of Z.We define the Mίlnor number μ of the vector field Z at 0 e C n as μ = dim c ^//(Z 1 , ,Zj.This number is finite if and only if 0 e C n is an isolated singularity of Z, a hypothesis which we will assume from now on.In this case μ coincides with the topological degree of the Gauss mapping induced by Z, considered as a real vector field, in a small (In -l)-sphere around 0 e C".In Theorem A we show that: the Milnor number ofZ is a topological invariant of ^z provided that n > 2.Consider now a polydisc Kf centered at OGC" and let /: B -> C*, /(0) = 0, be an irreducible analytic function.Then V = f~ι(0) is an analytic sub variety and we say V is invariant by Z if for any p e V we have df(p) Z(p) = 0. Suppose k = n -1.Then dim c V = 1 and V -{0} is a leaf of J^z.Moreover, if B is small enough, then D = B Π V is homeomorphic to a 2-disc via a Puiseaux's parametrization.Then the restriction of Z to D can be considered as a real vector field X defined in a 2-disc.The multiplicity of Z