Embeddings of submanifolds and normal bundles

This paper is devoted to the study of the embeddings of a complex submanifold S inside a larger complex manifold M; in particular, we are interested in comparing the embedding of S in M with the embedding of S as the zero section in the total space of the normal bundle NS of S in M. We explicitely describe some cohomological classes allowing to measure the difference between the two embeddings, in the spirit of the work by Grauert, Griffiths, and Camacho-Movasati-Sad; we are also able to explain the geometrical meaning of the separate vanishing of these classes. Our results holds for any codimension, but even for curves in a surface we generalize previous results due to Laufert and Camacho-Movasati-Sad.