Indexes of vector fields for mixed functions

2023/05/26 by José Luis Cisneros‐Molina, Cisneros-Molina, José Luis, Agustín Romano-Velázquez +1
Mathematics · #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2305.16719

Abstract

A mixed function is a real analytic map f\colon ℂⁿ → ℂ in the complex variables z₁,…,z_n and their conjugates z₁,…,z_n. In this article we define an integer valued index for vector fields v with isolated singularity at 0 on real analytic varieties V_f:=f^-1(0) defined by mixed functions f with isolated critical point at 0. We call this index the mixed GSV-index and it generalizes the classical GSV-index defined by Gomez-Mont, Seade and Verjovsky, i.e., if the function f is holomorphic, then the mixed GSV-index coincides with the GSV-index. Furthermore, the mixed GSV-index is a lifting to ℤ of the ℤ₂-valued real GSV-index defined by Aguilar, Seade and Verjovsky. As applications we prove that the mixed GSV-index is equal to the Poincaré-Hopf index of v on a Milnor fiber. If f also satisfies the strong Milnor condition, i.e., for every ε>0 (small enough) the map (f)/(‖f‖)\colon \mathbbS_ε∖ L_f → \mathbbS¹ is a fiber bundle, we prove that the mixed GSV-index is equal to the curvatura integra of f defined by Cisneros-Molina, Grulha and Seade based on the curvatura integra defined by Kervaire.

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