Deformations Feuilletees Des Varietes De Hopf

TL;DR

This paper studies special foliations with complex leaves, showing how the complex structure of a compact leaf determines the structure of non-compact leaves and vice versa, with applications to Hopf manifolds.

Contribution

It establishes a reciprocal relationship between the complex structures of compact and non-compact leaves in certain foliations and applies this to deformations of Hopf manifolds.

Findings

Complex structure along non-compact leaves is fixed by the compact leaf's structure.

The complex structure on one non-compact leaf determines all others.

Results provide insights into foliated deformations of Hopf manifolds.

Abstract

In this article, we focus on a very special class of foliations with complex leaves whose diffeomorphism type is fixed. They have a unique compact leaf and the noncompact leaves all accumulate onto it. We show that the complex structure along the non-compact leaves is fixed by the complex structure of the compact leaf. Reciprocally, we prove that the complex structure along a non-compact leaf determines the complex structure along the other leaves. We apply these results to the study of foliated deformations of Hopf manifolds, a foliated analogue to the notion of deformation in the large.