An equivalence between pseudo-holomorphic embeddings into almost-complex Euclidean space and CR regular embeddings into complex space

We show that a pseudo-holomorphic embedding of an almost-complex 2n-manifold into almost-complex (2n+2)-Euclidean space exists if and only if there is a CR regular embedding of the 2n-manifold into complex (n+1)-space. We remark that the fundamental group does not place any restriction on the existence of either kind of embedding when n is at least three. We give necessary and sufficient conditions in terms of characteristic classes for a closed almost-complex 6-manifold to admit a pseudo-holomorphic embedding into R8 equipped with an almost-complex structure that need not be integrable.