no. 5
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 von1 ; Ebeling, Wolfgang1 ; Gómez-Mont, Xavier2
1 Leibniz Universität Hannover Institut für Algebraische Geometrie Postfach 6009 30060 Hannover (Germany)
2 CIMAT A.P. 402 Guanajuato, 36000 (México)
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.

DOI : 10.5802/aif.2398
Classification : 32S65, 14B05, 14Q10, 13D02, 13D25, 13H15, 32S25, 58K45
Keywords: Index, Vector Field, Complete Intersections, Complex, Homology of Complexes, Double Complexes, Homological Index, Buchsbaum-Eisenbud Theory
Mot clés : indice, champ de vecteur, intersection complète, complexe, homologie de complexes, double complexes, indice homologique, théorie Buchsbaum-Eisenbud

Bothmer, H.-Ch. Graf von 1 ; Ebeling, Wolfgang 1 ; Gómez-Mont, Xavier 2

1 Leibniz Universität Hannover Institut für Algebraische Geometrie Postfach 6009 30060 Hannover (Germany)
2 CIMAT A.P. 402 Guanajuato, 36000 (México) 
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Bothmer, H.-Ch. Graf von; Ebeling, Wolfgang; Gómez-Mont, Xavier. An Algebraic Formula for the Index of a Vector Field on an Isolated Complete Intersection Singularity. Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1761-1783. doi : 10.5802/aif.2398. https://aif.centre-mersenne.org/articles/10.5802/aif.2398/

[1] Eisenbud, D. Commutative Algebra, with a view toward Algebraic Geometry, Graduate Texts in Math., Springer Verlag, 1995 | MR | Zbl

[2] Gómez-Mont, X. An algebraic formula for the index of a vector field on a hypersurface with an isolated singularity, J. Alg. Geom., Volume 7 (1998), pp. 731-752 | MR | Zbl

[3] Gómez-Mont, X.; Giraldo, L. A law of conservation of number for local Euler characteristics, Contemp. Math., Volume 311 (2002), pp. 251-259 | MR | Zbl

[4] Gómez-Mont, X.; Seade, J.; Verjovsky, A. The index of a holomorphic flow with an isolated singularity, Math. Ann., Volume 291 (1991), pp. 737-751 | DOI | MR | Zbl

[5] Graf von Bothmer, H. C.; Ebeling, W.; Gómez-Mont, X. (A script for calculating the index of a non homogeneous vector field on a complete intersection of two singular quadrics. Available either at http://www.iag.uni-hannover.de/~bothmer/gobelin or at the end of the LaTeX-file of An Algebraic Formula for the Index of a Vector Field on an Isolated Complete Intersection Singularity at http://arXiv.org/abs/math.AG/0601640)

[6] Grayson, D. R.; Stillman, M. E. Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/

[7] Greuel, G. M. Der Gauss-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten, Math. Ann., Volume 214 (1975), pp. 235-266 | DOI | MR | Zbl

[8] Greuel, G. M. Dualität in der lokalen Kohomologie isolierter Singularitäten, Math. Ann., Volume 250 (1980), pp. 157-173 | MR | Zbl

[9] Greuel, G. M.; Pfister, G.; Schönemann, H. Singular, a Computer Algebra System for polynomial computations, Available at http://www.www.singular.uni-kl.de/

[10] Klehn, O. Real and complex indices of vector fields on complete intersection curves with isolated singularity, Compos. Math., Volume 141 (2005), pp. 525-540 | DOI | MR | Zbl

[11] Seade, J. A.; Suwa, T. A residue formula for the index of a holomorphic flow, Math. Ann., Volume 304 (1996), pp. 621-634 | DOI | MR | Zbl

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