Deformations of hypercomplex structures related to Heisenberg groups

Details Deformations of hypercomplex structures related to Heisenberg groups Deformations of hypercomplex structures related to Heisenberg groups

2006-11-28

arxiv.org/abs/math/0611880v1

Mathematics

Differential Geometry

32 pages

Scientific paper

Let $X$ be a compact quotient of the product of the real Heisenberg group $H_{4m+1}$ of dimension $4m+1$ and the 3-dimensional real Euclidean space $\bR^3$. A left invariant hypercomplex structure on $H_{4m+1}\times \bR^3$ 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.

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