Let be a germ of a complete intersection variety in , , having an isolated singularity at and be the germ of a holomorphic vector field having an isolated zero at and tangent to . We show that in this case the homological index and the GSV-index coincide. In the case when the zero of is also isolated in the ambient space we give a formula for the homological index in terms of local linear algebra.
Soit un germe d’intersection complète dans , , avec singularité isolée en et soit un germe de champs de vecteurs holomorphes en tangents à et qui a une singularité isolée dans en . Nous montrons que dans ce cas l’indice homologique et l’indice GSV coïncident. Dans le cas où le zéro de est aussi isolé dans l’espace ambiant , nous donnons une formule pour l’indice homologique en terme de l’algèbre linéaire locale.
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
@article{AIF_2008__58_5_1761_0, author = {Bothmer, H.-Ch. Graf von and Ebeling, Wolfgang and G\'omez-Mont, Xavier}, title = {An {Algebraic} {Formula} for the {Index} {of~a~Vector} {Field} on an {Isolated} {Complete} {Intersection} {Singularity}}, journal = {Annales de l'Institut Fourier}, pages = {1761--1783}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {58}, number = {5}, year = {2008}, doi = {10.5802/aif.2398}, mrnumber = {2445833}, zbl = {1168.32023}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2398/} }
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 SP - 1761 EP - 1783 VL - 58 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2398/ DO - 10.5802/aif.2398 LA - en ID - AIF_2008__58_5_1761_0 ER -
%0 Journal Article %A Bothmer, H.-Ch. Graf von %A Ebeling, Wolfgang %A Gómez-Mont, Xavier %T An Algebraic Formula for the Index of a Vector Field on an Isolated Complete Intersection Singularity %J Annales de l'Institut Fourier %D 2008 %P 1761-1783 %V 58 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2398/ %R 10.5802/aif.2398 %G en %F AIF_2008__58_5_1761_0
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, Volume 58 (2008) no. 5, pp. 1761-1783. doi : 10.5802/aif.2398. https://aif.centre-mersenne.org/articles/10.5802/aif.2398/
[1] Commutative Algebra, with a view toward Algebraic Geometry, Graduate Texts in Math., Springer Verlag, 1995 | MR | Zbl
[2] 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] A law of conservation of number for local Euler characteristics, Contemp. Math., Volume 311 (2002), pp. 251-259 | MR | Zbl
[4] The index of a holomorphic flow with an isolated singularity, Math. Ann., Volume 291 (1991), pp. 737-751 | DOI | MR | Zbl
[5] 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)
(A script for calculating the index of a non homogeneous vector field on a complete intersection of two singular quadrics. Available either at[6] Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/
[7] Der Gauss-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten, Math. Ann., Volume 214 (1975), pp. 235-266 | DOI | MR | Zbl
[8] Dualität in der lokalen Kohomologie isolierter Singularitäten, Math. Ann., Volume 250 (1980), pp. 157-173 | MR | Zbl
[9] Singular, a Computer Algebra System for polynomial computations, Available at http://www.www.singular.uni-kl.de/
[10] 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] A residue formula for the index of a holomorphic flow, Math. Ann., Volume 304 (1996), pp. 621-634 | DOI | MR | Zbl
Cited by Sources: