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arXiv · 2403.18654

An upper bound for the GSV-index of a foliation

Arturo Fernández-Pérez·Evelia R. García Barroso·Nancy Saravia-Molina
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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 $3μ_p(\mathcal{F},B)-4τ_p(\mathcal{F},B)+GSV_p(\mathcal{F},B)\leq 0$, where $μ_p(\mathcal{F},B)$ is the multiplicity of $\mathcal{F}$ along $B$, $τ_p(\mathcal{F},B)$ 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 $GSV_p(\mathcal{F},B)$ 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.

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Arturo Fernández-Pérez, Evelia R. García Barroso, Nancy Saravia-Molina. 2025-06-13. An upper bound for the GSV-index of a foliation. https://doi.org/10.1007/s00009-024-02625-0

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