Sharp dimension estimates of holomorphic functions and rigidity

Let M-n be a complete noncompact Kahler manifold of complex dimension n with nonnegative holomorphic bisectional curvature. Denote by O-d(M-n) the space of holomorphic functions of polynomial growth of degree at most d on M-n. In this paper we prove that dim(C)O(d)(M-n) <= dim(C)O([d])(C-n), for all d > 0, with equality for some positive integer d if and only if M-n is holomorphically isometric to C-n. We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.