The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant

Xavier Gómez-Mont; Pavao Mardešić

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 5, page 1523-1539
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Gómez-Mont, Xavier, and 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 47.5 (1997): 1523-1539. <http://eudml.org/doc/75272>.

@article{Gómez1997,
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.},
author = {Gómez-Mont, Xavier, Mardešić, Pavao},
journal = {Annales de l'institut Fourier},
keywords = {index of a vector field; singularity of a vector field; hypersurface singularities; relative Jacobian determinant},
language = {eng},
number = {5},
pages = {1523-1539},
publisher = {Association des Annales de l'Institut Fourier},
title = {The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant},
url = {http://eudml.org/doc/75272},
volume = {47},
year = {1997},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 5
SP - 1523
EP - 1539
AB - 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.
LA - eng
KW - index of a vector field; singularity of a vector field; hypersurface singularities; relative Jacobian determinant
UR - http://eudml.org/doc/75272
ER -

References

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  1. [1] V. ARNOLD, S. GUSEIN-ZADE & V. VARCHENKO, Singularities of Differentiable Maps, I, Birkhauser, 1985. 
  2. [2] Ch. BONATTI & GÓMEZ-MONT, The index of a holomorphic vector field on a singular variety I, Astérisque, 222 (1994), 9-35. Zbl0810.32017
  3. [3] A. DOUADY, Flatness and Privilige, L'Enseignement Mathématique, 14 (1968), 47-74. Zbl0183.35102MR38 #4716
  4. [4] D. EISENBUD & H. LEVINE, An algebraic formula for the degree of a C∞ map germ, Ann. Math., 106 (1977), 19-38. Zbl0398.57020
  5. [5] X. GÓMEZ-MONT, An Algebraic formula for the index of a vector field on a hypersurface with an isolated singularity, preprint. Zbl0956.32029
  6. [6] X. GÓMEZ-MONT, P. MARDEŠI&#0106;, The index of a vector field tangent to an odd dimensional hypersurface and the signature of the relative Hessian, preprint. 
  7. [7] X. GÓMEZ-MONT, J. SEADE & A. VERJOVSKY, The index of a holomorphic flow with an isolated singularity, Math. Ann., 291 (1991), 737-751. Zbl0725.32012MR93d:32066
  8. [8] GRAUERT & H. REMMERT, Coherent Analytic Sheaves, Grundlehren 265, Springer Verlag, 1984. Zbl0537.32001MR86a:32001
  9. [9] Ph. GRIFFITHS, J. HARRIS, Principles of Algebraic Geometry, J. Wiley, 1978. Zbl0408.14001
  10. [10] KHIMISHIASHVILI, On the local degree of a smooth map, Trudi Tbilisi Math. Inst., (1980), 105-124. Zbl0526.58010

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