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Deformations of hypercomplex structures related to Heisenberg groups
Differential Geometry · Submitted 28 Nov 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $\bR^3$. | background 0.85 | 0.14 | 0.02 | asserted 0.88 |
| s1 | A left invariant hypercomplex structure on $H_{4m+1}\times \bR^3$ descends onto the compact quotient $X$. | result 0.50 | 0.44 | 0.04 | asserted 1.00 |
| s2 | The space $X$ is a hyperholomorphic fibration of 4-tori over a $4m$-torus. | result 0.84 | 0.55 | 0.06 | asserted 1.00 |
| s3 | We calculate the parameter space and obstructions to deformations of this hypercomplex structure on $X$. | contribution 0.88 | 0.96 | 0.09 | asserted 0.99 |
| s4 | 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. | result 0.99 | 0.95 | 0.76 | asserted 1.00 |
| s5 | This is in contrast to the deformations on the $4m$-torus. | interpretation 0.94 | 0.38 | 0.08 | asserted 0.95 |
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The paper
Original abstract
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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