Index of Singularities of Real Vector Fields on Singular Hypersurfaces

TL;DR

This paper surveys methods for calculating the GSV-index of real vector fields tangent to singular hypersurfaces, extending complex case techniques to the real setting using algebraic signatures.

Contribution

It presents a unified approach to compute the GSV-index for real hypersurfaces, combining formulas from complex hypersurface theory and real vector field indices.

Findings

GSV-index can be expressed as signatures of bilinear forms.

The approach extends complex hypersurface index calculations to real cases.

Provides explicit formulas for the GSV-index in terms of local algebra signatures.

Abstract

G\'omez-Mont, Seade and Verjovsky introduced an index, now called GSV-index, generalizing the Poincar\'e-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 G\'omez-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 G\'omez-Mont's formula for calculating the GSV-index on singular complex hypersurfaces and the formula of Eisenbud, Levine and Khimshiashvili for calculating the Poincar\'e-Hopf index of a singularity of a real vector field in $\R^{n+1}$