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Xavier Gómez-Mont; Pavao Mardešić
The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant
Annales de l'institut Fourier, 47 no. 5 (1997), p. 1523-1539, doi: 10.5802/aif.1608
Article PDF | Reviews MR 99g:32062 | Zbl 0891.32013

Résumé - Abstract

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 ${{\bf B}\over {\rm Ann}_{\bf B}(h)}$ 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.


[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 0810.32017
[3] A. DOUADY, Flatness and Privilige, L'Enseignement Mathématique, 14 (1968), 47-74.  MR 38 #4716 |  Zbl 0183.35102
[4] D. EISENBUD & H. LEVINE, An algebraic formula for the degree of a C∞ map germ, Ann. Math., 106 (1977), 19-38.  Zbl 0398.57020
[5] X. GÓMEZ-MONT, An Algebraic formula for the index of a vector field on a hypersurface with an isolated singularity, preprint.  Zbl 0956.32029
[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 93d:32066 |  Zbl 0725.32012
[8] GRAUERT & H. REMMERT, Coherent Analytic Sheaves, Grundlehren 265, Springer Verlag, 1984.  MR 86a:32001 |  Zbl 0537.32001
[9] Ph. GRIFFITHS, J. HARRIS, Principles of Algebraic Geometry, J. Wiley, 1978.  Zbl 0408.14001
[10] KHIMISHIASHVILI, On the local degree of a smooth map, Trudi Tbilisi Math. Inst., (1980), 105-124.  Zbl 0526.58010