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The Quarterly Journal of Mathematics Advance Access published online on April 2, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hap014
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© 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

COMPLEX SUBMANIFOLDS OF ALMOST COMPLEX EUCLIDEAN SPACES

Antonio J. Di Scala {dagger}

Dipartimento di Matematica, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129 Torino, Italy

Luigi Vezzoni{ddagger}

Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy

{ddagger} Corresponding author. E-mail: luigi.vezzoni{at}unito.it

Received 16 December 2008; revised 23 February 2009
  Abstract

We prove that a compact Riemann surface can be realized as a pseudo-holomorphic curve of (R4, 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 (R4n, 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 structures in R2n, which cannot be tamed by any symplectic form.

{dagger} E-mail: antonio.discala{at}polito.it


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