Principalization on logarithmically foliated orbifolds

Abstract

In characteristic zero, we construct principalization of ideals on smooth orbifolds endowed with a normal crossings divisor and a foliation. We then illustrate how the method can be used in the general study of foliations via two applications. First, we provide a resolution of singularities of Darboux totally integrable foliations in arbitrary dimensions -- including rational and meromorphic Darboux foliations. Second, we show how to transform a generically transverse section into a transverse section.

We have introduced the notion of a thick class of foliations. We have also established a criterion for reducing the singularities of the pullback of a foliation in a thick class C to another foliation within the same class