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

TL;DR

This paper establishes an equivalence between pseudo-holomorphic embeddings into almost-complex Euclidean spaces and CR regular embeddings into complex spaces, providing new insights into embedding conditions based on characteristic classes.

Contribution

It proves the equivalence between pseudo-holomorphic and CR regular embeddings and characterizes when certain 6-manifolds can embed into 8-dimensional Euclidean space with an almost-complex structure.

Findings

Equivalence between pseudo-holomorphic and CR regular embeddings.

No fundamental group restrictions for n ≥ 3.

Characteristic class conditions for 6-manifolds to embed into R^8.

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