Reduction of the singularities of codimension one singular foliations in dimension three. (English) Zbl 1088.32019

Cano, Felipe

Ann. Math. (2) 160, No. 3, 907-1011 (2005).

The author solves the problem of the reduction of singularities for codimension one foliations when the ambient space has dimension three. The main result is:
Theorem: (reduction to simple singularities) Let \(X\) be a three-dimensional germ of nonsingular complex manifold around a compact analytic set. Let \(F\) be a holomorphic singular foliation of codimension one and \(D\) a divisor on \(X\) with normal crossings.
Then there exists a morphism \(\pi: X'\to X\) which is a composition of a finite sequence of blowing-ups with nonsingular centers such that:
1) Each center is invariant for the strict transform of \(F\) and has normal crossings with the total transform of \(D\).
2) The strict transform \(F'\) of \(F\) in \(X'\) has normal crossings with the total transform \(D'\) of \(D\) and it has at most “simple singularities adapted to \(D\)”.

Reviewer: Viorel Vâjâitu (Bucureşti)

MSC:

32S65	Singularities of holomorphic vector fields and foliations
37F75	Dynamical aspects of holomorphic foliations and vector fields
32S45	Modifications; resolution of singularities (complex-analytic aspects)
57R30	Foliations in differential topology; geometric theory

Keywords:

singular foliations; reduction of foliations; simple singularities; blowing-up; normal crossings divisor; integral hypersurfaces; (non)-dicritical foliations