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

• [1] Leibniz Universität Hannover Institut für Algebraische Geometrie Postfach 6009 30060 Hannover (Germany)
• [2] CIMAT A.P. 402 Guanajuato, 36000 (México)
• Volume: 58, Issue: 5, page 1761-1783
• ISSN: 0373-0956

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## Abstract

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Let $\left(V,0\right)$ 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.

## How to cite

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Bothmer, H.-Ch. Graf von, Ebeling, Wolfgang, and Gómez-Mont, Xavier. "An Algebraic Formula for the Index of a Vector Field on an Isolated Complete Intersection Singularity." Annales de l’institut Fourier 58.5 (2008): 1761-1783. <http://eudml.org/doc/10362>.

@article{Bothmer2008,
abstract = {Let $(V,0)$ be a germ of a complete intersection variety in $\{\mathbb\{C\}\}^\{n+k\}$, $n&gt;0$, 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.},
affiliation = {Leibniz Universität Hannover Institut für Algebraische Geometrie Postfach 6009 30060 Hannover (Germany); Leibniz Universität Hannover Institut für Algebraische Geometrie Postfach 6009 30060 Hannover (Germany); CIMAT A.P. 402 Guanajuato, 36000 (México)},
author = {Bothmer, H.-Ch. Graf von, Ebeling, Wolfgang, Gómez-Mont, Xavier},
journal = {Annales de l’institut Fourier},
keywords = {Index; Vector Field; Complete Intersections; Complex; Homology of Complexes; Double Complexes; Homological Index; Buchsbaum-Eisenbud Theory; index; GSV-index; vector field; complete intersection; complex; homology of complexes; homological index; Buchsbaum–Eisenbud theory},
language = {eng},
number = {5},
pages = {1761-1783},
publisher = {Association des Annales de l’institut Fourier},
title = {An Algebraic Formula for the Index of a Vector Field on an Isolated Complete Intersection Singularity},
url = {http://eudml.org/doc/10362},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Bothmer, H.-Ch. Graf von
AU - Ebeling, Wolfgang
AU - Gómez-Mont, Xavier
TI - An Algebraic Formula for the Index of a Vector Field on an Isolated Complete Intersection Singularity
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 5
SP - 1761
EP - 1783
AB - Let $(V,0)$ be a germ of a complete intersection variety in ${\mathbb{C}}^{n+k}$, $n&gt;0$, 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.
LA - eng
KW - Index; Vector Field; Complete Intersections; Complex; Homology of Complexes; Double Complexes; Homological Index; Buchsbaum-Eisenbud Theory; index; GSV-index; vector field; complete intersection; complex; homology of complexes; homological index; Buchsbaum–Eisenbud theory
UR - http://eudml.org/doc/10362
ER -

## References

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3. X. Gómez-Mont, L. Giraldo, A law of conservation of number for local Euler characteristics, Contemp. Math. 311 (2002), 251-259 Zbl1051.32010MR1940173
4. X. Gómez-Mont, J. Seade, A. Verjovsky, The index of a holomorphic flow with an isolated singularity, Math. Ann. 291 (1991), 737-751 Zbl0725.32012MR1135541
5. H. C. Graf von Bothmer, W. Ebeling, X. Gómez-Mont
6. D. R. Grayson, M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at
7. G. M. Greuel, Der Gauss-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten, Math. Ann. 214 (1975), 235-266 Zbl0285.14002MR396554
8. G. M. Greuel, Dualität in der lokalen Kohomologie isolierter Singularitäten, Math. Ann. 250 (1980), 157-173 Zbl0417.14003MR582515
9. G. M. Greuel, G. Pfister, H. Schönemann, Singular, a Computer Algebra System for polynomial computations, Available at Zbl0902.14040
10. O. Klehn, Real and complex indices of vector fields on complete intersection curves with isolated singularity, Compos. Math. 141 (2005), 525-540 Zbl1077.32018MR2134279
11. J. A. Seade, T. Suwa, A residue formula for the index of a holomorphic flow, Math. Ann. 304 (1996), 621-634 Zbl0853.32040MR1380446

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