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The Quarterly Journal of Mathematics Advance Access published online on December 4, 2007
The Quarterly Journal of Mathematics, doi:10.1093/qmath/ham040
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© 2007. Published by Oxford University Press. All rights reserved For permissions, please email: journals.permissions@oxfordjournals.org

DEFORMATIONS OF HYPERCOMPLEX STRUCTURES ASSOCIATED TO HEISENBERG GROUPS

Gueo Grantcharov

Department of Mathematics, Florida International University, Miami, FL 33199, USA

Henrik Pedersen {dagger}

Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, Odense M, DK-5230, Denmark

Yat Sun Poon{ddagger}

Department of Mathematics, University of California at Riverside, Riverside, CA 92521, USA

{ddagger} Corresponding author. E-mail: ypoon{at}math.ucr.edu

Received 5 January 2007;
  Abstract

Let X be a compact quotient of the product of the real Heisenberg group H4m+1 of dimension 4m + 1 and the three-dimensional real Euclidean space R3. A left-invariant hypercomplex structure on H4m+1 x R3 descends onto the compact quotient X. The space X is a hyperholomorphic fibration of 4-tori over a 4m-torus. We calculate the parameter space and obstructions to deformations of this hypercomplex structure on X. Using our calculations, we show that all small deformations generate invariant hypercomplex structures on X but not all of them arise from deformations of the lattice. This is in contrast to the deformations on the 4m-torus.

{dagger} E-mail: henrik{at}adm.sdu.dk


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