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# On Deformations of the Complex Structure on the Moduli Space of Spatial Polygons

Published:2002-09-01
Printed: Sep 2002
• Yasuhiko Kamiyama
• Shuichi Tsukuda
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## Abstract

For an integer $n \geq 3$, 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 $TM_n$. In this paper, we prove $H^q (M_n,\Theta_{M_n})=0$ for all $q \geq 0$ 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.
 Keywords: polygon space, complex structure
 MSC Classifications: 14D20 - Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 32C35 - Analytic sheaves and cohomology groups [See also 14Fxx, 18F20, 55N30]

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