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

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 of linearization problems involving small divisors condition arising from solutions to their cohomological equations.