Index of Singularities of Real Vector Fields on Singular Hypersurfaces

Abstract : Gómez-Mont, Seade and Verjovsky introduced an index, now called GSV-index, generalizing the Poincaré-Hopf index to complex vector fields tangent to singular hypersurfaces. The GSV-index extends to the real case. This is a survey paper on the joint research with Gómez-Mont and Giraldo about calculating the GSV-index $\Ind_{V_\pm,0}(X)$ of a real vector field $X$ tangent to a singular hypersurface $V=f^{-1}(0)$. The index $\Ind_{V_{\pm,0}}(X)$ is calculated as a combination of several terms. Each term is given as a signature of some bilinear form on a local algebra associated to $f$ and $X$. Main ingredients in the proof are Gómez-Mont's formula for calculating the GSV-index on \emph{ singular complex} hypersurfaces and the formula of Eisenbud, Levine and Khimshiashvili for calculating the Poincaré-Hopf index of a singularity of a \emph{real} vector field in $\R^{n+1}$
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Submitted on : Wednesday, January 9, 2013 - 9:56:40 AM
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  • HAL Id : hal-00771625, version 1
  • ARXIV : 1301.1781

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Pavao Mardesic. Index of Singularities of Real Vector Fields on Singular Hypersurfaces. 2012. ⟨hal-00771625⟩

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