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

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.