An Algebraic Formula for the Index of a Vector Field on an Isolated Complete Intersection Singularity

[Une formule algébrique pour l’indice d’un champ de vecteurs sur une singularité d’intersection complète isolée]

Bothmer, H.-Ch. Graf von ; Ebeling, Wolfgang ; Gómez-Mont, Xavier

Annales de l'Institut Fourier, Tome 58 (2008), p. 1761-1783 / Harvested from Numdam

Résumé

Soit $(V,0)$ un germe d’intersection complète dans ${\mathbb{C}}^{n+k}$, $n>0$, avec singularité isolée en $0$ et soit $X$ un germe de champs de vecteurs holomorphes en ${\mathbb{C}}^{n+k}$ tangents à $V$ et qui a une singularité isolée dans $V$ en $0$. Nous montrons que dans ce cas l’indice homologique et l’indice GSV coïncident. Dans le cas où le zéro de $X$ est aussi isolé dans l’espace ambiant ${\mathbb{C}}^{n+k}$, nous donnons une formule pour l’indice homologique en terme de l’algèbre linéaire locale.

Abstract

Let $(V,0)$ be a germ of a complete intersection variety in ${\mathbb{C}}^{n+k}$, $n>0$, having an isolated singularity at $0$ and $X$ be the germ of a holomorphic vector field having an isolated zero at $0$ and tangent to $V$. We show that in this case the homological index and the GSV-index coincide. In the case when the zero of $X$ is also isolated in the ambient space ${\mathbb{C}}^{n+k}$ we give a formula for the homological index in terms of local linear algebra.

Détail

Publié le : 2008-01-01

MR 2445833 | Zbl 1168.32023

DOI : https://doi.org/10.5802/aif.2398
Classification:  32S65,  14B05,  14Q10,  13D02,  13D25,  13H15,  32S25,  58K45
Mots clés: indice, champ de vecteur, intersection complète, complexe, homologie de complexes, double complexes, indice homologique, théorie Buchsbaum-Eisenbud

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