For an integer n greater than or equal to 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 circle minusM(n) be the sheaf of germs of holomorphic sections of the tangent bundle TMn. In this paper, we prove H-q(M-n, circle minus(Mn)) = 0 for all q greater than or equal to 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.