On the holomorphic foliations admitting a common invariant algebraic set

TL;DR

This paper investigates holomorphic foliations with a shared invariant algebraic set, providing conditions for their generating modules to be free, especially characterizing when such modules are generated by Hamiltonian vector fields in two variables.

Contribution

It offers new criteria for the freeness of modules of vector fields admitting a common invariant algebraic set, including a characterization involving Jacobian ideals in two-variable cases.

Findings

The module of vector fields is free under certain conditions.

In two variables, the module is generated by Hamiltonian vector fields if and only if the polynomial is in its Jacobian ideal.

Elementary methods are used for the proofs.

Abstract

In this paper, we study the holomorphic foliations admitting a common invariant algebraic set $C$ defined by a polynomial $f$ in $ \mathbb{K}[x_1,x_2,...,x_n]$ over any characteristic $0$ subfield $\mathbb{K}\subseteq\mathbb{C}$. For the $\mathbb{K}[x_1,x_2,...,x_n]$-module $V_f$ of vector fields generating foliations that admit $C$ as an invariant set, we provide several conditions under which the module $V_f$ can be freely generated by a minimal generating set. In particular, when $n=2$ and $f$ is a weakly tame polynomial, we show that the $\mathbb{K}[x,y]$-module $V_f$ is freely generated by two polynomial vector fields, one of which is the Hamiltonian vector field induced by $f$, if and only if, $f$ belongs to the Jacobian ideal $\langle f_x, f_y\rangle$ in $\mathbb{K}[x,y]$. Our proof employs a purely elementary method.