Sharp Dimension Estimates of Holomorphic Functions and Rigidity

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