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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

Yasuhiko Kamiyama  and
Shuichi Tsukuda
Yasuhiko Kamiyama
Affiliation:
Department of Mathematics University of the Ryukyus Nishihara-Cho Okinawa 903-0213 Japan, e-mail: [email protected]
Shuichi Tsukuda
Affiliation:
Department of Mathematics University of the Ryukyus Nishihara-Cho Okinawa 903-0213 Japan, e-mail: [email protected]
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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.

Keywords

14D2032C35polygon spacecomplex structure
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 3 , 01 September 2002 , pp. 417 - 421
Copyright
Copyright © Canadian Mathematical Society 2002

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