On neighborhoods of embedded toroidal and Hopf manifolds and their foliations
Laurent Stolovitch, Xiaojun Wu
TL;DR
This work advances the understanding of how neighborhoods of certain embedded complex manifolds behave under biholomorphic equivalence. It develops a unified method based on vertical/horizontal cohomology, Diophantine conditions, and Reinhardt-domain techniques to obtain full linearization and Ueda-type foliations for non-compact toroidal embeddings, specifically connected abelian complex Lie groups, and separately treats Hopf manifolds embedded as hypersurfaces. The main contributions are (i) full linearization results for toroidal embeddings under Diophantine hypotheses and a vertical foliation when the normal bundle is vertically Diophantine, (ii) an extension to Hopf manifolds showing that generic-type Hopf hypersurfaces with non-Hermitian-flat flat normal bundles satisfy $(C,X)\\cong (C,N_{C/X})$, and (iii) the construction of Stein nested coverings and cohomology-vanishing arguments that support these linearization results. These results broaden the class of ambient manifolds for which germ-level linearization holds and provide tools for constructing foliations with embedded leaves in complex geometry.
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.