A classical result due to Seidenberg states that every singular holomorphic foliation on a

complex surface can be turned into a foliation possessing only elementary singular points

(i.e. singular points possessing at least one eigenvalue different from zero) by means of a

finite sequence of (one-point) blow-ups. However, in dimension 3, the natural analogue

of Seidenberg theorem no longer holds as shown by Sancho and Sanz.

More recently, two major works have been devoted to this problem. Cano, Roche and

Spivakovsky have worked out a reduction procedure using (standard) blow-ups. The main

disadvantage of their theorem lies, however, in the fact that some of their final models are

quadratic and hence have all eigenvalues equal to zero. On the other hand, McQuillan and

Panazzolo have successfully used weighted blow-ups to obtain a satisfactory desingularization

theorem in the category of stacks, rather than in usual complex manifolds.

A basic question is how far these theorems can be improved if we start with a complete vector

field on a complex manifold of dimension 3, rather than with a general 1-dimensional holomorphic

foliation. In this context of complete vector fields, we will prove a sharp desingularization theorem.

Our proof of the mentioned result will naturally require us to revisit the works of Cano-Roche-Spivakovsky and of McQuillan-Panazzolo on general 1-dimensional foliations. In particular, by

building on the first mentioned work, our discussion will also shed some new light on the

desingularization problem for general 1-dimensional foliation on complex manifolds of dimension 3.