Holomorphic vector fields tangent to foliations in dimension three

Abstract

Abstract We consider a singular holomorphic vector field in a neighborhood of 0 ∈ C 3and suppose that there is a singular holomorphic foliation of codimension one (outside its singular set, given by a holomorphic decomposition of this neighborhood into complex surfaces, called leaves) to which it is tangent. This means that, when both objects are non-singular, the orbits of the vector field are contained in the leaves of the foliation. First we consider the desingularizations of both objects, trying to relate their final models. Then we analyse the situation where the vector field is tangent to three independent foliations.

Keywords

• Holomorphic foliations
• holomorphic vector fields
• pencil of foliations
• invariant varieties