Index of Singularities of Real Vector Fields on Singular Hypersurfaces

2013/01/09 by Pavao Mardešić, Pavao Mardesic, Mardesic, Pavao
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.AG #math.DS

paper · pdf · doi:10.48550/arxiv.1301.1781

School and Conference on Algebraic Methods in Geometry In Honour of Xavier Gomez-Mont, Mexico 28/08/2011-02/09/2011

arxiv created 2013/01/09 · openalex publication_date 2013/01/09 · arxiv updated 2013/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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_±,0}(X) of a real vector field X tangent to a singular hypersurface V=f^-1(0). The index \Ind_V_±,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 singular complex hypersurfaces and the formula of Eisenbud, Levine and Khimshiashvili for calculating the Poincaré-Hopf index of a singularity of a real vector field in \Rⁿ⁺¹

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