  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 $N_S$ 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.
