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}$${}_d(M^n)$ the space of holomorphic functions of polynomial growth of degree at most $d$ on $M^n$. In this paper we prove that $$di{m}_{\mathbb{C}}{\mathcal{O}}_d(M^n)\le di{m}_{\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.