Neighborhoods of Analytic Varieties in Complex Manifolds

The systematic study of neighborhoods of analytic varieties was started by H. Grauert in his celebrated article 1962. In that article he considers a manifold X and a negatively embedded submanifold A subset X. He introduces n-neighborhood, n in N, of A and studies when an isomorphism of two n-neighborhoods can be extended to an isomorphism of (n+1)-neighborhoods. He observes that obstructions to this extension problem lie in the first cohomology group of certain sheaves involving the normal bundle of A in X. Using a version of Kodaira vanishing theorem he shows that for a large n these cohomology groups vanish and so he concludes that the germ of a negatively embedded manifold A depends only on a finite neighborhood of it. These methods are generalized to a germ of an arbitrary negatively embedded divisor A by Hironaka, Rossi and Laufer. In the case where A is a Riemann surface embedded in a two dimensional manifold, by using Serre duality we can say exactly which finite neighborhood of A determines the embedding. This isa first draft of an expository article about negatively embedded varieties. The text is mainly based on Grauert's paper, but we have used also the contributions of subsequent authors; for instance the Artin's criterion on the existence of convergent solutions is used instead of Grauert's geometrical methods. Our principal aim is to extend this study to the germ of foliated neighborhoods and singularities. In the first steps we will consider the most simple foliations which are transversal foliations. Next, foliations with tangencies and Poincare type singularities will be considered.