Foliations on projective spaces associated to the affine Lie Algebra

TL;DR

This paper constructs and classifies certain irreducible components of holomorphic foliations on projective spaces linked to affine Lie algebra representations, focusing on Kupka components in dimensions 3 and 4.

Contribution

It introduces a novel construction of irreducible components of foliations associated with affine Lie algebra representations and classifies Kupka components by degree.

Findings

Classification of Kupka components in terms of degree

Construction of irreducible components linked to affine Lie algebra

Description of foliations on $ ext{P}^3$ and $ ext{P}^4$

Abstract

In this work, we construct some irreducible components of the space of two-dimensional holomorphic foliations on $\mathbb{P}^n$ associated to some algebraic representations of the affine Lie algebra $\mathfrak{aff}(\mathbb{C})$. We give a description of the generalized Kupka components, obtaining a classification of them in terms of the degree of the foliations, in both cases $n=3$ and $n=4$.