The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant
Annales de l'Institut Fourier, Volume 47 (1997) no. 5, pp. 1523-1539.

Given a real analytic vector field tangent to a hypersurface V with an algebraically isolated singularity we introduce a relative Jacobian determinant in the finite dimensional algebra B Ann B (h) associated with the singularity of the vector field on V. We show that the relative Jacobian generates a 1-dimensional non-zero minimal ideal. With its help we introduce a non-degenerate bilinear pairing, and its signature measures the size of this point with sign. The signature satisfies a law of conservation of number and for even dimensional hypersurfaces it gives a method to compute the Poincaré-Hopf index of the vector field restricted to the hypersurface.

Pour tout champ de vecteurs analytique réel tangent à une hypersurface V et ayant une singularité algébriquement isolée on introduit le déterminant jacobien relatif dans l’algèbre finiment engendrée B Ann B (h) associée à la singularité du champ de vecteurs sur V. On montre que le déterminant jacobien relatif engendre un idéal unidimensionnel qui est l’idéal minimal non trivial. À l’aide du déterminant jacobien relatif on introduit un produit bilinéaire non dégénéré dont la signature mesure la taille (avec signe) de ce point. La signature satisfait une loi de conservation. Pour les hypersurfaces de dimension paire la signature donne l’indice de Poincaré-Hopf de la restriction du champ de vecteurs à l’hypersurface.

@article{AIF_1997__47_5_1523_0,
     author = {G\'omez-Mont, Xavier and Marde\v{s}i\'c, Pavao},
     title = {The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant},
     journal = {Annales de l'Institut Fourier},
     pages = {1523--1539},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {5},
     year = {1997},
     doi = {10.5802/aif.1608},
     mrnumber = {99g:32062},
     zbl = {0891.32013},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1608/}
}
TY  - JOUR
AU  - Gómez-Mont, Xavier
AU  - Mardešić, Pavao
TI  - The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant
JO  - Annales de l'Institut Fourier
PY  - 1997
SP  - 1523
EP  - 1539
VL  - 47
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.1608/
DO  - 10.5802/aif.1608
LA  - en
ID  - AIF_1997__47_5_1523_0
ER  - 
%0 Journal Article
%A Gómez-Mont, Xavier
%A Mardešić, Pavao
%T The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant
%J Annales de l'Institut Fourier
%D 1997
%P 1523-1539
%V 47
%N 5
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.1608/
%R 10.5802/aif.1608
%G en
%F AIF_1997__47_5_1523_0
Gómez-Mont, Xavier; Mardešić, Pavao. The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant. Annales de l'Institut Fourier, Volume 47 (1997) no. 5, pp. 1523-1539. doi : 10.5802/aif.1608. http://archive.numdam.org/articles/10.5802/aif.1608/

[1] V. Arnold, S. Gusein-Zade & V. Varchenko, Singularities of Differentiable Maps, I, Birkhauser, 1985.

[2] Ch. Bonatti & Gómez-Mont, The index of a holomorphic vector field on a singular variety I, Astérisque, 222 (1994), 9-35. | Zbl

[3] A. Douady, Flatness and Privilige, L'Enseignement Mathématique, 14 (1968), 47-74. | MR | Zbl

[4] D. Eisenbud & H. Levine, An algebraic formula for the degree of a C∞ map germ, Ann. Math., 106 (1977), 19-38. | Zbl

[5] X. Gómez-Mont, An Algebraic formula for the index of a vector field on a hypersurface with an isolated singularity, preprint. | Zbl

[6] X. Gómez-Mont, P. Mardešij, The index of a vector field tangent to an odd dimensional hypersurface and the signature of the relative Hessian, preprint.

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

[8] Grauert & H. Remmert, Coherent Analytic Sheaves, Grundlehren 265, Springer Verlag, 1984. | MR | Zbl

[9] Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, J. Wiley, 1978. | Zbl

[10] Khimishiashvili, On the local degree of a smooth map, Trudi Tbilisi Math. Inst., (1980), 105-124. | Zbl

Cited by Sources: