Sharp dimension estimates of holomorphic functions and rigidity

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