Foliated neighborhoods of exceptional submanifolds

TL;DR

This paper investigates the existence and classification of regular foliations transverse to strongly exceptional submanifolds in complex manifolds, providing cohomological conditions and extending classical linearization results.

Contribution

It introduces cohomological criteria for the existence of such foliations and generalizes Poincaré's linearization theorem for singular foliations at submanifolds.

Findings

Cohomological conditions for foliation existence

Classification of transverse foliations

Refinement of Grauert's embedding theorem

Abstract

The present article is a study of germs of regular foliations transverse to an embedded strongly exceptional submanifold of a complex manifold. Cohomological conditions are given on this embedding for the existence of these foliations and their classification is established. One dimensional foliations singular at the submanifold and a generalization of a linearization theorem of Poincar\'e for these foliations, are used in this study. As a consequence of our approach, we obtain a refinement of the embedding theorem of Grauert.