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Sharp dimension estimates of holomorphic functions and rigidity

Authors:	Bing-Long Chen Xiao-Yong Fu Le Yin Xi-Ping Zhu

Institution:	Department of Mathematics, Zhongshan University, Guangzhou, 510275, People's Republic of China Xiao-Yong Fu ; Department of Mathematics, Zhongshan University, Guangzhou, 510275, People's Republic of China Le Yin ; Department of Mathematics, Zhongshan University, Guangzhou, 510275, People's Republic of China Xi-Ping Zhu ; Department of Mathematics, Zhongshan University, Guangzhou, 510275, People's Republic of China

Abstract:	Let be a complete noncompact Kähler manifold of complex dimension with nonnegative holomorphic bisectional curvature. Denote by $\mathcal{O}_d(M^n)$ the space of holomorphic functions of polynomial growth of degree at most on . In this paper we prove that $\displaystyle dim_{\mathbb{C}}{\mathcal{O}}_d(M^n)\leq dim_{\mathbb{C}}{\mathcal{O}}_{d]}(\mathbb{C}^n),$ for all , with equality for some positive integer if and only if 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.

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