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

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

Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1761-1783.

Soit $(V,0)$ un germe d’intersection complète dans ${ℂ}^{n+k}$, $n>0$, avec singularité isolée en $0$ et soit $X$ un germe de champs de vecteurs holomorphes en ${ℂ}^{n+k}$ 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 ${ℂ}^{n+k}$, nous donnons une formule pour l’indice homologique en terme de l’algèbre linéaire locale.

Let $(V,0)$ be a germ of a complete intersection variety in ${ℂ}^{n+k}$, $n>0$, 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 ${ℂ}^{n+k}$ we give a formula for the homological index in terms of local linear algebra.

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