On neighborhoods of embedded toroidal and Hopf manifolds and their foliations

TL;DR

This paper introduces new examples of embedded complex manifolds with neighborhoods equivalent to their normal bundle, including abelian complex Lie groups and Hopf manifolds, and explores conditions for holomorphic foliations with these manifolds as leaves.

Contribution

It provides novel examples of embedded complex manifolds and establishes conditions for holomorphic foliations involving these manifolds.

Findings

Examples of neighborhoods equivalent to normal bundle neighborhoods for abelian complex Lie groups and Hopf manifolds.

Conditions for the existence of holomorphic foliations with these manifolds as leaves.

New insights into the structure of neighborhoods around embedded complex manifolds.

Abstract

In this article, we give completely new examples of embedded complex manifolds the germ of neighborhood of which is holomorphically equivalent to a germ of neighborhood of the zero section in its normal bundle. The first set of examples is composed of connected abelian complex Lie groups, embedded in some complex manifold $M$. These are non compact manifolds in general. We also give some conditions ensuring the existence a holomorphic foliation having the embedded manifold as leaf. The second set of examples are $n$-dimensional Hopf manifolds, embedded as hypersurfaces.