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.