Nilpotent Holomorphic Foliations on $(\mathbb{C}^n,\mathbf{0})$

TL;DR

The paper analyzes nilpotent holomorphic foliations on $(\mathbb{C}^{n+1},0)$ defined by integrable 1-forms with linear part $z\,dz$, showing that every such foliation is a pullback of Takens' normal form and, in higher dimensions, yields cuspidal invariant hypersurfaces. It develops a generalized hypersurface framework, positing non-dicritical foliations whose generic transversal pullbacks are of generalized curve type, and proves a weighted-order condition $\mathrm{ord}_{s,2}(g)\ge s-2$ when the separatrix is quasi-ordinary cuspidal; in dimension $n=2$ this condition is also sufficient. The Newton polyhedron is then used to characterize generalized hypersurface type via the equality $\mathcal{N}(\mathcal{F}) = \mathcal{N}(F)$ of the foliation and its separatrix, providing a coordinate-invariant, combinatorial criterion that extends prior 2D results to higher dimensions. Together, these results yield a direct, algebraic route to classify a family of generalized surface-type foliations without resorting to resolution of singularities, with explicit connections between the foliation data and its separatrices.

Abstract

In this paper, we study nilpotent holomorphic foliations in complex dimension $n+1$, at the origin, defined by germs of integrable 1-forms whose linear part is given by \(zdz\). These foliations generalize the classical nilpotent foliations in dimension two. We show that every nilpotent foliation in higher dimensions can be described as the pullback of Takens' normal form, which naturally leads to the existence of cuspidal hypersurfaces as invariant sets. We focus on the case where the separatrix is a quasi-ordinary cuspidal hypersurface, and we provide a characterization of those foliations that are of generalized hypersurface type. Furthermore, we recall the Newton polyhedron of a foliation and prove that, for foliations with a quasi-ordinary cuspidal separatrix, being of generalized hypersurface type is equivalent to the coincidence of the Newton polyhedra of the foliation and its separatrix. This provides a new criterion that extends previous results studied by Fernández-Sánchez and J. Mozo (2006), and later partially generalized by the second-named author.

Theorems & Definitions (14)

• Lemma 2.1
• Definition 2.2
• Lemma 2.3
• Definition 3.1
• Theorem 3.2: PF-JM-HN-GeneralizedHypersurfaces
• Proposition 3.3
• proof
• Corollary 3.4
• Definition 4.1
• Remark 4.2