AbstractFor an integer n ≥ 3, let Mn be the moduli space of spatial polygons with n edges. We consider the case of odd n. Then Mn is a Fano manifold of complex dimension n − 3. Let ΘMn be the sheaf of germs of holomorphic sections of the tangent bundle TMn. In this paper, we prove Hq(Mn, ΘMn) = 0 for all q ≥ 0 and all odd n. In particular, we see that the moduli space of deformations of the complex structure on Mn consists of a point. Thus the complex structure on Mn is locally rigid.