Sharp Dimension Estimates of Holomorphic Functions and Rigidity

Let $M^n$ be a complete noncompact K$ddot{a}$hler manifold of complex dimension $n$ with nonnegative holomorphic bisectional curvature. Denote by $mathcal{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_{mathbb{C}}{mathcal{O}}_d(M^n)leq dim_{mathbb{C}}{mathcal{O}}_{[d]}(mathbb{C}^n),$$ for all $d>0$, with equality for some positive integer $d$ if and only if $M^n$ is holomorphically isometric to $mathbb{C}^n$. We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.