Sharp Dimension Estimates of Holomorphic Functions and Rigidity

TL;DR

This paper establishes sharp bounds on the dimension of polynomial growth holomorphic functions on certain Kähler manifolds, characterizing when equality holds and providing improved estimates under specific geometric conditions.

Contribution

It proves optimal dimension estimates for holomorphic functions on noncompact Kähler manifolds with nonnegative bisectional curvature, including rigidity results and refined bounds under additional curvature assumptions.

Findings

Dimension of holomorphic functions matches that of complex Euclidean space when equality holds.

Sharp dimension bounds are obtained for manifolds with non-maximal volume growth.

Rigidity characterized by isometry to complex Euclidean space when bounds are attained.

Abstract

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.