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Some remarks on indices of holomorphic vector fields. (English) Zbl 0912.32024

Let \(v\) be a holomorphic vector field defined on a neighbourhood of \(0\in \mathbb{C}^2\) and having an isolated singularity there. The author compares three residue-type indices for the singularity \((v,0)\). The first is the Baum-Bott index \(BB(v,0)\) defined in [P. F. Baum and R. Bott, Essays Topol. Relat. Top., Mém. dédiés à Georges de Rham, 29-47 (1970; Zbl 0193.52201)]. The second is an index \(CS(v,S,0)\) defined in terms of a separatrix \(S\) for \(v\) at 0 by C. Camacho and P. Sad [Ann. Math., II. Ser. 115, 579-595 (1982; Zbl 0503.32007)] which represents in a sense the intersection index of the trajectories of \(v\) with \(S\). More recently, X. Gomez-Mont, J. Seade and A. Verjovsky [Math. Ann. 291, No. 4, 737-751 (1991; Zbl 0739.32031)] have defined a third index \(GSV(v,S,0)\) which is a sort of Poincaré-Hopf index for the restriction of \(v\) to \(S\).
The main theorem of this paper states that, when \((v,0)\) is a generalised curve, that is a nondicritical singularity whose resolution does not contain saddle-nodes, and \(S\) is the union of all separatrices of \(v\) at 0, then \(BB(v,0)=CS(v,S,0)\), while \(GSV (v,S,0)=0\). The proof is obtained by desingularisation and analysis of the variation of the indices under blow-ups.

MSC:

32S65 Singularities of holomorphic vector fields and foliations
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