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

Kamiyama, Yasuhiko ; Tsukuda, Shuichi

Canadian mathematical bulletin, Tome 45 (2002) no. 3, pp. 417-421

Résumé

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.

Détail

DOI : 10.4153/CMB-2002-043-8

Mots-clés : 14D20, 32C35, polygon space, complex structure