On the k-th Milnor and k-th Tjurina Numbers of a Foliation
Marcela Ribeiro, Arturo Fernández-Pérez
TL;DR
This work introduces and analyzes the $k$-th Milnor and $k$-th Tjurina numbers for germs of holomorphic foliations in $\mathbb{C}^2$ with isolated singularities, establishing explicit formulas and connecting them to classical foliation indices. It proves that $\mu^{k}$ is a topological invariant in dimension two (while $\tau^{k}$ is not), provides a sharp lower bound for $\tau^{k}$ of weighted homogeneous plane curves, and constructs counterexamples to a proposed universal bound on the ratio $\mu^{k}/\tau^{k}$. A key achievement is the extension of the Gomez-Mont–Seade–Verjovsky index to the $k$-th level, yielding $\tau^{k}(\mathcal{F},C)-\tau^{k}(C)=GSV(\mathcal{F},C)$ for all $k$, and the development of a Teissier-type relation for $k$-th polar intersection indices. The paper also provides a bound on $\mu^{k}$ via the $k$-th Tjurina number on balanced divisors and, in the quasi-homogeneous, non-dicritical setting, a closed formula linking $\mu^{k}$ and $\tau^{k}$. These results unify algebraic, topological, and polar-geometric aspects of foliations and singularities, with explicit formulas and counterexamples informing future investigations in higher dimensions and broader classes of foliations.
Abstract
In this paper, we introduce the notions of the $k$-th Milnor number and the $k$-th Tjurina number for a germ of holomorphic foliation on the complex plane with an isolated singularity at the origin. We develop a detailed study of these invariants, establishing explicit formulas and relating them to other indices associated with holomorphic foliations. In particular, we obtain an explicit expression for the $k$-th Milnor number of a foliation and, as a consequence, a formula for the $k$-th Milnor number of a holomorphic function. We analyze their topological behavior, proving that the $k$-th Milnor number of a holomorphic function is a topological invariant, whereas the $k$-th Tjurina number is not. In dimension two, we provide a positive answer to a conjecture posed by Hussain, Liu, Yau, and Zuo concerning a sharp lower bound for the $k$-th Tjurina number of a weighted homogeneous polynomial. We also present a counterexample to another conjecture of Hussain, Yau, and Zuo regarding the ratio between these invariants. Moreover, we establish a fundamental relation linking the $k$-th Tjurina numbers of a foliation and of an invariant curve via the Gómez-Mont--Seade--Verjovsky index, and we extend Teissier's Lemma to the setting of $k$-th polar intersection numbers. In addition, we derive an upper bound for the $k$-th Milnor number of a foliation in terms of its $k$-th Tjurina number along balanced divisors of separatrices. Finally, for non-dicritical quasi-homogeneous foliations, we obtain a closed formula for their $k$-th Milnor and Tjurina numbers.