Geodesic
On Deformations of the Complex Structure on the Moduli Space of Spatial Polygons
Canadian mathematical bulletin, Tome 45 (2002) no. 3, pp. 417-421

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2002-043-8
Mots-clés : 14D20, 32C35, polygon space, complex structure 
Kamiyama, Yasuhiko; Tsukuda, Shuichi. On Deformations of the Complex Structure on the Moduli Space of Spatial Polygons. Canadian mathematical bulletin, Tome 45 (2002) no. 3, pp. 417-421. doi: 10.4153/CMB-2002-043-8
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[1] [1] Brion, M., Cohomologie équivariante des points semi-stables. J. Reine Angew.Math. 421 (1991), 421–1991. Google Scholar

[2] [2] Griffiths, P. and Harris, J., Principles of algebraic geometry. Wiley-Interscience, New York, 1978. Google Scholar

[3] [3] Hausmann, J.-C. and Knutson, A., The cohomology ring of polygon spaces. Ann. Inst. Fourier (Grenoble) 48 (1998), 48–1998. Google Scholar

[4] [4] Kamiyama, Y. and Tezuka, M., Symplectic volume of the moduli space of spatial polygons. J. Math. Kyoto Univ. 39 (1999), 39–1999. Google Scholar

[5] [5] Kapovich, M. and Millson, J., The symplectic geometry of polygons in Euclidean space. J. Differential Geom. 44 (1996), 44–1996. Google Scholar

[6] [6] Kirwan, F., Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes 31, Princeton University Press, Princeton, 1984. Google Scholar

[7] [7] Kirwan, F., The cohomology rings of moduli spaces of bundles over Riemann surfaces. J. Amer. Math. Soc. 5 (1992), 5–1992. Google Scholar

[8] [8] Klyachko, A., Spatial polygons and stable configurations of points in the projective line. In: Algebraic geometry and its applications, Yaroslavl', 1992, 67–84, Aspects Math. E25, Vieweg, Braunschweig, 1994. Google Scholar

[9] [9] Kodaira, K., Complex manifolds and deformation of complex structures. Grundlehren der Mathematischen Wissenschaften 283, Springer-Verlag, 1986. Google Scholar

[10] [10] Siu, Y.-T. and Trautmann, G., Gap-sheaves and extension of coherent analytic subsheaves. Lecture Notes in Mathematics 172, Springer-Verlag, 1971. Google Scholar

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