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, p. 1761-1783

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

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

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
     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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {5},
     year = {2008},
     pages = {1761-1783},
     doi = {10.5802/aif.2398},
     mrnumber = {2445833},
     zbl = {1168.32023},
     language = {en},
     url = {}
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.

[1] Eisenbud, D. Commutative Algebra, with a view toward Algebraic Geometry, Springer Verlag, Graduate Texts in Math. (1995) | MR 1322960 | Zbl 0819.13001

[2] Gómez-Mont, X. An algebraic formula for the index of a vector field on a hypersurface with an isolated singularity, J. Alg. Geom., Tome 7 (1998), pp. 731-752 | MR 1642757 | Zbl 0956.32029

[3] Gómez-Mont, X.; Giraldo, L. A law of conservation of number for local Euler characteristics, Contemp. Math., Tome 311 (2002), pp. 251-259 | MR 1940173 | Zbl 1051.32010

[4] Gómez-Mont, X.; Seade, J.; Verjovsky, A. The index of a holomorphic flow with an isolated singularity, Math. Ann., Tome 291 (1991), pp. 737-751 | Article | MR 1135541 | Zbl 0725.32012

[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 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

[6] Grayson, D. R.; Stillman, M. E. Macaulay2, a software system for research in algebraic geometry, Available at

[7] Greuel, G. M. Der Gauss-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten, Math. Ann., Tome 214 (1975), pp. 235-266 | Article | MR 396554 | Zbl 0285.14002

[8] Greuel, G. M. Dualität in der lokalen Kohomologie isolierter Singularitäten, Math. Ann., Tome 250 (1980), pp. 157-173 | MR 582515 | Zbl 0417.14003

[9] Greuel, G. M.; Pfister, G.; Schönemann, H. Singular, a Computer Algebra System for polynomial computations, Available at

[10] Klehn, O. Real and complex indices of vector fields on complete intersection curves with isolated singularity, Compos. Math., Tome 141 (2005), pp. 525-540 | Article | MR 2134279 | Zbl 1077.32018

[11] Seade, J. A.; Suwa, T. A residue formula for the index of a holomorphic flow, Math. Ann., Tome 304 (1996), pp. 621-634 | Article | MR 1380446 | Zbl 0853.32040