[Paper Review] Higher codimensional foliations and Kupka singularities

Maurício Corrêa, Omegar Calvo-Andrade+1 more|arXiv (Cornell University)|Aug 29, 2014

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper studies holomorphic foliations of dimension $k > 1$ and codimension $\geq 1$ on $\mathbb{P}^n$ with a compact connected Kupka singular set component. It proves that if the transversal type is linear with positive eigenvalues, the foliation must be a rational fibration $\Phi: \mathbb{P}^n \dasharrow \mathbb{P}^{n-k}$. This result extends to cases where the Kupka set is a complete intersection or has radial transversal type, and is applied to derive a normal form for non-integrable codimension-one distributions.

ABSTRACT

We consider holomorphic foliations of dimension $k>1$ and codimension $\geq 1$ in the projective space $\mathbb{P}^n$, with a compact connected component of the Kupka set. We prove that, if the transversal type is linear with positive integers eigenvalues, then the foliation consist on the fibers of a rational fibration. As a corollary, if $\mathcal{F}$ is a foliation such that $dim(\mathcal{F})\geq cod(\mathcal{F})+2$ and has transversal type diagonal with different eigenvalues, then the Kupka component $K$ is a complete intersection and we get the same conclusion. The same conclusion holds if the Kupka set is a complete intersection and has radial transversal type. Finally, as an application, we find a normal form for non integrable codimension one distributions on $\mathbb{P}^{n}$.

Motivation & Objective

• To characterize holomorphic foliations of dimension $k > 1$ and codimension $\geq 1$ on $\mathbb{P}^n$ with a compact connected Kupka singular set component.
• To determine conditions under which such foliations arise as rational fibrations.
• To extend the characterization to cases where the Kupka set is a complete intersection or has radial transversal type.
• To apply the results to derive a normal form for non-integrable codimension-one distributions on $\mathbb{P}^n$.

Proposed method

• Analyzes the Kupka set of holomorphic foliations using local normal forms and transversal type classification.
• Applies the Kupka condition to show that the foliation is locally defined by a closed form with linear transverse structure.
• Uses Baum–Bott theory and cohomological techniques to relate the geometry of the Kupka set to global fibrations.
• Employs the Poincaré lemma and Darboux-type theorems to construct local coordinates where the foliation takes a standard form.
• Applies results on global decomposability of $d\omega$ and radial vector field contraction to deduce homogeneous polynomial structure.
• Uses the fact that $\text{cod}(\text{Sing}((d\omega)^r)) \geq 3$ to conclude global decomposition of $d\omega$ as a sum of wedge products of differentials.

Experimental results

Research questions

• RQ1 Under what conditions does a holomorphic foliation on $\mathbb{P}^n$ with a compact connected Kupka component and linear transverse type with positive eigenvalues arise as a rational fibration?
• RQ2 When is the Kupka set of a foliation a complete intersection, and what global structure does this imply?
• RQ3 Can the radial transversal type of the Kupka set force the foliation to be a rational fibration?
• RQ4 What normal form can be achieved for non-integrable codimension-one distributions on $\mathbb{P}^n$ with a Kupka component?
• RQ5 How does the global decomposability of $d\omega$ relate to the homogeneous structure of the defining 1-form?

Key findings

• If a holomorphic foliation on $\mathbb{P}^n$ has a compact connected Kupka component with linear transverse type and positive eigenvalues, then it is a rational fibration $\Phi: \mathbb{P}^n \dasharrow \mathbb{P}^{n-k}$.
• When $\dim(\mathcal{F}) \geq \text{cod}(\mathcal{F}) + 2$ and the transverse type is diagonal with distinct eigenvalues, the Kupka component $K$ is a complete intersection and the foliation is a rational fibration.
• If the Kupka set is a complete intersection and has radial transversal type, the foliation is again a rational fibration.
• For non-integrable codimension-one distributions of class $r$ on $\mathbb{P}^n$ with $\text{cod}(\text{Sing}((d\omega)^r)) \geq 3$, the 1-form $\omega$ is locally pullback of a standard form via a submersion to $\mathbb{C}^{2r}$.
• The 1-form $\omega$ is globally of the form $\sum_{i=0}^{r-1} (f_i df_{i+r} - f_{i+r} df_i)$ for homogeneous polynomials $f_i$ of equal degree.
• The global structure of $d\omega$ is $\sum_{i=0}^{r-1} df_i \wedge df_{i+r}$, implying a homogeneous and decomposable structure.