logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Previous article | Next article
H.-Ch. Graf von Bothmer; Wolfgang Ebeling; Xavier Gómez-Mont
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)
Annales de l'institut Fourier, 58 no. 5 (2008), p. 1761-1783, doi: 10.5802/aif.2398
Article PDF | Reviews MR 2445833 | Zbl 1168.32023
Class. Math.: 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

Résumé - 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.

Bibliography

[1] D. Eisenbud, Commutative Algebra, with a view toward Algebraic Geometry, Graduate Texts in Math., Springer Verlag, 1995  MR 1322960 |  Zbl 0819.13001
[2] X. Gómez-Mont, “An algebraic formula for the index of a vector field on a hypersurface with an isolated singularity”, J. Alg. Geom. 7 (1998), p. 731-752  MR 1642757 |  Zbl 0956.32029
[3] X. Gómez-Mont & L. Giraldo, “A law of conservation of number for local Euler characteristics”, Contemp. Math. 311 (2002), p. 251-259  MR 1940173 |  Zbl 1051.32010
[4] X. Gómez-Mont, J. Seade & A. Verjovsky, “The index of a holomorphic flow with an isolated singularity”, Math. Ann. 291 (1991), p. 737-751 Article |  MR 1135541 |  Zbl 0725.32012
[5] H. C. Graf von Bothmer, W. Ebeling & X. Gómez-Mont, 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] D. R. Grayson & M. E. Stillman, “Macaulay2, a software system for research in algebraic geometry”, Available at http://www.math.uiuc.edu/Macaulay2/
[7] G. M. Greuel, “Der Gauss-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten”, Math. Ann. 214 (1975), p. 235-266 Article |  MR 396554 |  Zbl 0285.14002
[8] G. M. Greuel, “Dualität in der lokalen Kohomologie isolierter Singularitäten”, Math. Ann. 250 (1980), p. 157-173  MR 582515 |  Zbl 0417.14003
[9] G. M. Greuel, G. Pfister & H. Schönemann, “Singular, a Computer Algebra System for polynomial computations”, Available at http://www.www.singular.uni-kl.de/
[10] O. Klehn, “Real and complex indices of vector fields on complete intersection curves with isolated singularity”, Compos. Math. 141 (2005), p. 525-540 Article |  MR 2134279 |  Zbl 1077.32018
[11] J. A. Seade & T. Suwa, “A residue formula for the index of a holomorphic flow”, Math. Ann. 304 (1996), p. 621-634 Article |  MR 1380446 |  Zbl 0853.32040
top