Complex submanifolds of almost complex Euclidean spaces

TL;DR

This paper characterizes which compact Riemann surfaces can be realized as pseudo-holomorphic curves in almost complex Euclidean spaces and constructs explicit examples of almost complex structures that are not tamed by symplectic forms.

Contribution

It provides a complete characterization of elliptic curves as pseudo-holomorphic curves in 4 and shows how to embed complex tori and Riemann surfaces into almost complex Euclidean spaces.

Findings

Elliptic curves can be realized as pseudo-holomorphic curves in 4 with suitable almost complex structures.

Any complex 2n-torus can be holomorphically embedded in ^{4n} with an appropriate almost complex structure.

Explicit examples of almost complex structures in ^{2n} that cannot be tamed by any symplectic form.

Abstract

We prove that a compact Riemann surface can be realized as a pseudo-holomorphic curve of $(\mathbb{R}^4,J)$, for some almost complex structure $J$ if and only if it is an elliptic curve. Furthermore we show that any (almost) complex $2n$-torus can be holomorphically embedded in $(\mathbb{R}^{4n},J)$ for a suitable almost complex structure $J$. This allows us to embed any compact Riemann surface in some almost complex Euclidean space and to show many explicit examples of almost complex structure in $\mathbb{R}^{2n}$ which can not be tamed by any symplectic form.