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

Abstract

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 ${\mathbb{R}}^8$ equipped with an almost-complex structure that need not be integrable.