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

Xavier Gómez-MontPavao Mardešić — 1997

Annales de l'institut Fourier

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...

An Algebraic Formula for the Index of a Vector Field on an Isolated Complete Intersection Singularity

H.-Ch. Graf von BothmerWolfgang EbelingXavier Gómez-Mont — 2008

Annales de l’institut Fourier

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.

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