On embeddings of almost complex manifolds in almost complex Euclidean spaces

TL;DR

This paper establishes conditions for embedding almost complex manifolds into almost complex Euclidean spaces, providing new bounds and criteria based on Segre classes, with specific results for 4-manifolds.

Contribution

It introduces new embedding bounds for compact almost complex manifolds into Euclidean spaces with almost complex structures, using Segre class conditions.

Findings

Any compact almost complex manifold of dimension 2m embeds in 4m+2 space.

Necessary and sufficient conditions for embeddings in 4m space based on Segre classes.

Discussion of embeddings for 4-manifolds in R^6.

Abstract

We prove that any compact almost complex manifold $(M, J)$ of real dimension $2m$ admits a pseudo-holomorphic embedding in a Euclidean space of dimension $4m + 2$, endowed with a suitable non-standard almost complex structure. Moreover, we give a necessary and sufficient condition, expressed in terms of the Segre class of $(M, J)$, for the existence of an embedding or an immersion in an almost complex Euclidean $4m$-space. We also discuss the pseudo-holomorphic embeddings of an almost complex 4-manifold in $R^6$.