An upper bound for the GSV-index of a foliation

Abstract

Let $\mathcal{F}$ be a holomorphic foliation at $p\in {\mathbb{C}}^2$, and $B$ be a separatrix of $\mathcal{F}$. We prove the following Dimca-Greuel type inequality $3{\mu }_p(\mathcal{F},B)-4{\tau }_p(\mathcal{F},B)+GS{V}_p(\mathcal{F},B)\le 0$, where ${\mu }_p(\mathcal{F},B)$ is the multiplicity of $\mathcal{F}$ along $B$, ${\tau }_p(\mathcal{F},B)$ is the dimension of the quotient of $\mathbb{C}\{x,y\}$ by the ideal generated by the components of any $1$-form defining $\mathcal{F}$ and any equation of $B$, and $GS{V}_p(\mathcal{F},B)$ is the \textit{Gómez-Mont-Seade-Verjovsky index} of the foliation $\mathcal{F}$ with respect to $B$. As a consequence, we provide a new proof of the $\frac{4}{3}$-Dimca-Greuel conjecture for singularities of irreducible plane curve germs, with foliations ingredients, that differs from those given by Alberich-Carramiñana, Almirón, Blanco, Melle-Hernández and Genzmer-Hernandes, but it is in line with the idea developed by Wang.

10 pages. We have modified the hypothesis of the main theorem and its proof. The title has been changed to match the published version. The previous title was A Dimca-Greuel type inequality for foliations