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The index of holomorphic vector fields on singular varieties. I. (English) Zbl 0810.32017
Camacho, C. (ed.) et al., Complex analytic methods in dynamical systems. Proceedings of the congress held at Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Brazil, January 1992. Paris: Société Mathématique de France, Astérisque. 222, 9-35 (1994).
Let $$(X,0) \subseteq (\mathbb{C}^ n,0)$$ the germ of an isolated singularity defined by the ideal $$I = (f_ 1, \dots, f_ r)$$. Let $$\delta \in \text{Der}_ \mathbb{C} \mathbb{C} \{X_ 1, \dots, X_ n\}$$ be a vector field with $$\delta (I) \subset I$$, $$\delta = \sum^ n_{i=1} h_ i {\partial \over \partial X_ i}$$.
The $$X$$-multiplicity of $$\delta$$ is defined by $\mu_ X (\delta,0) = \dim_ \mathbb{C} \mathbb{C} \{X_ 1, \dots, X_ n\}/(f_ 1, \dots, f_ r, h_ 1, \dots, h_ n).$ The function $$\mu_ X (-,0)$$: $$\text{Der}_ \mathbb{C} \mathbb{C} \{X_ 1, \dots, X_ n\}/I \to \mathbb{Z} \cup \{\infty\}$$ is studied and related to other invariants (as for instance the index).
For the entire collection see [Zbl 0797.00019].

##### MSC:
 32S65 Singularities of holomorphic vector fields and foliations 32L05 Holomorphic bundles and generalizations
##### Keywords:
singularities; index of vector fields; multiplicity
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