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.