Complex submanifolds of almost complex Euclidean spaces

Details Complex submanifolds of almost complex Euclidean spaces Complex submanifolds of almost complex Euclidean spaces

2009-05-26

arxiv.org/abs/0905.4190v1

Quart. J. Math. 61 (2010), 401-405

Mathematics

Differential Geometry

7 pages. To be appear in Q. J. Math

Scientific paper

10.1093/qmath/hap014

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.

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