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On Deformations of the Complex Structure on the Moduli Space of Spatial Polygons
Published online by Cambridge University Press: 20 November 2018
Abstract
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For an integer 
$n\,\ge \,3$, let 
${{M}_{n}}$ be the moduli space of spatial polygons with 
$n$ edges. We consider the case of odd $n$. Then ${{M}_{n}}$ is a Fano manifold of complex dimension 
$n\,-\,3$. Let 
${{\Theta }_{{{M}_{n}}}}$ be the sheaf of germs of holomorphic sections of the tangent bundle 
$T{{M}_{n}}$. In this paper, we prove 
${{H}^{q}}\left( {{M}_{n}},\,{{\Theta }_{{{M}_{n}}}} \right)\,=\,0$ for all 
$q\,\ge \,0$ and all odd $n$. In particular, we see that the moduli space of deformations of the complex structure on ${{M}_{n}}$ consists of a point. Thus the complex structure on ${{M}_{n}}$ is locally rigid.
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- Copyright © Canadian Mathematical Society 2002
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