2-dimensional Lie algebras and separatrices for vector fields on (C^3,0)

TL;DR

This paper proves that certain holomorphic vector fields on complex 3-space have invariant separatrices if they are associated with a rank 2 Lie algebra, and explores implications for vector fields on compact complex 3-manifolds.

Contribution

It establishes the existence of separatrices for vector fields embedded in a rank 2 Lie algebra representation and applies this to properties of vector fields on compact complex manifolds.

Findings

Holomorphic vector fields with rank 2 Lie algebra embeddings have separatrices.

Second jet of vector fields on compact 3-manifolds cannot vanish at isolated singularities.

Results connect Lie algebra representations to geometric properties of vector fields.

Abstract

We show that holomorphic vector fields on (C^3,0) have separatrices provided that they are embedded in a rank 2 representation of a two-dimensional Lie algebra. In turn, this result enables us to show that the second jet of a holomorphic vector field defined on a compact complex manifold M of dimension 3 cannot vanish at an isolated singular point provided that M carries more than a single holomorphic vector field.