Indexes of vector fields for mixed functions

Abstract

A mixed function is a real analytic map $f\colon \mathbb{C}^n \to \mathbb{C}$ in the complex variables $z_1,\dots,z_n$ and their conjugates $\bar{z}_1,\dots,\bar{z}_n$. In this article we define an integer valued index for vector fields $v$ with isolated singularity at $\mathbf{0}$ on real analytic varieties $V_f:=f^{-1}(0)$ defined by mixed functions $f$ with isolated critical point at $\mathbf{0}$. 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 $\mathbb{Z}$ of the $\mathbb{Z}_2$-valued real GSV-index defined by Aguilar, Seade and Verjovsky. As applications we prove that the mixed GSV-index is equal to the Poincar\'e-Hopf index of $v$ on a Milnor fiber. If $f$ also satisfies the strong Milnor condition, i.e., for every $\epsilon>0$ (small enough) the map $\frac{f}{\|f\|}\colon \mathbb{S}_\epsilon \setminus L_f \to \mathbb{S}^1$ 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.