Equivalence of Neighborhoods of Embedded Compact Complex Manifolds and Higher Codimension Foliations

TL;DR

This paper investigates conditions under which neighborhoods of embedded compact complex manifolds are biholomorphically equivalent to their normal bundle neighborhoods and extends foliation theory to higher codimensions, advancing the classification of complex structures.

Contribution

It provides new criteria for neighborhood equivalence to normal bundles and extends Ueda's foliation theory to higher codimension cases, broadening the understanding of complex manifold neighborhoods.

Findings

Neighborhoods are biholomorphic to normal bundle neighborhoods under certain conditions.

Existence of holomorphic foliations with the embedded manifold as a leaf in higher codimension.

Extension of Ueda's theory to complex manifolds of higher codimension.

Abstract

We consider an embedded $n$-dimensional compact complex manifold in $n+d$ dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert's formal principle program. We will give conditions ensuring that a neighborhood of $C_n$ in $M_{n+d}$ is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold's result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in $M_{n+d }$ having $C_n$ as a compact leaf, extending Ueda's theory to the high codimension case. Both problems appear as a kind linearization problem involving small divisors condition arising from solutions to their cohomological equations.