Transversely Lie holomorphic foliations on projective spaces

TL;DR

This paper characterizes certain holomorphic foliations on projective spaces with Lie group transverse structures, showing they are logarithmic and identifying conditions under which they are induced by linear vector fields.

Contribution

It proves that foliations with generic singularities and Lie group transverse structures are logarithmic, and characterizes those with a single singularity as linear diagonal vector fields.

Findings

Foliations with Lie group transverse structures are logarithmic.

Single singularity foliations are induced by linear diagonal vector fields.

Provides classification criteria for such foliations.

Abstract

We prove that a one-dimensional foliation with generic singularities on a projective space, exhibiting a Lie group transverse structure in the complement of some codimension one algebraic subset is logarithmic, i.e., it is the intersection of codimension one foliations given by closed one-forms with simple poles. If there is only one singularity in a suitable affine space, then the foliation is induced by a linear diagonal vector field.