An embedding is an injective map into a universal, simpler model object. Many embedding theorems are without obstruction, in the sense that every object which you wish to embed can be embedded. Examples of such theorems are Yoneda lemma, algebraic closure of fields, Nash embedding theorem for Riemannian manifolds. unconditional.

I'm interested in embedding theorems with obstruction. Do you have examples of theorems that give an obstruction to embedding? In the case where there is an obstruction, would you consider the obstruction to be local or instrinsic? Most embedding problems are possible locally, but there is often a local-global obstruction.

The example that led me to this question is Kodiara's embedding theorem that gives an obstruction for a complex manifold to be a submanifold of complex projective space. Here the obstruction is that the manifold must carry a positive line bundle. Positivity of curvature is a local criterion.

PS. Sorry, but I really don't know how to tag this question.