On resolution of 1-dimensional foliations on 3-manifolds

TL;DR

This paper advances the resolution of singularities in 1-dimensional holomorphic foliations on 3-manifolds, building on prior work and introducing sharper results for specific classes of singularities.

Contribution

It completes and extends existing resolution theorems for foliations, providing new methods and sharper results for particular singularity classes in three-dimensional complex manifolds.

Findings

Complete resolution theorem for 1-dimensional foliations on 3-manifolds.

Sharper resolution results for special classes of singularities.

New methods combining classical differential equations with foliation theory.

Abstract

This paper is devoted to the resolution of singularities of holomorphic vector fields and of one-dimensional holomorphic foliations in dimension 3 and it has two main objectives. First, from the general perspective of one-dimensional foliations, we build upon the work of Cano-Roche-Spivakovsky and essentially complete it. As a consequence, we obtain a general resolution theorem comparable to the resolution theorem of McQuillan-Panazzolo but proved by means of rather different methods. The second objective of this paper consists of looking at a special class of singularities of foliations containing, in particular, all singularities of complete holomorphic vector fields on complex manifolds of dimension 3. We then prove that for this class of holomorphic foliations, there holds a much sharper resolution theorem. This second result was the initial motivation of this paper and it relies on the combination of the previous resolution theorems for (general) foliations with some classical material on asymptotic expansions for solutions of differential equations.