Irreducible components of the space of foliations associated to the affine Lie algebra

Abstract

In this paper, we give the explicit construction of certain components of the space of holomorphic foliations of codimension one, in complex projective spaces. These components are associated to some algebraic representations of the affine Lie algebra $\mathfrak{aff}(\mathbb{C})$. Some of them, the so-called exceptional or Klein–Lie components, are rigid in the sense that all generic foliations in the component are equivalent (Example 1). In particular, we obtain rigid foliations of all degrees. Some generalizations and open problems are given at the end of §1.