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The rigidity of dimension estimate for holomorphic functions on Kähler manifolds

Jianchun Chu, Zihang Hao

TL;DR

Addresses the rigidity of dimension estimates for holomorphic functions with polynomial growth on complete $n$-dimensional Kähler manifolds with non-negative holomorphic bisectional curvature. The authors develop a framework based on the Poincaré–Siegel map, lifting to the universal cover, and a splitting theorem to analyze the structure of $\mathcal{O}_d(M)$, proving a sharp gap: the maximum possible dimension is $N_1=\binom{n+d}{d}$ and the second-largest is $N_2=N_1-\binom{n+d-2}{d-1}$; if $\dim\mathcal{O}_d(M)=N_2$ with $d\ge2$, then $M$ is biholomorphic to $\mathbb{C}^n$. The paper also provides explicit metric examples achieving the gap, illustrating optimality, and discusses potential generalizations under weaker curvature notions such as non-negative holomorphic sectional curvature.

Abstract

In this paper, we obtain the optimal rigidity of dimension estimate for holomorphic functions with polynomial growth on Kähler manifolds with non-negative holomorphic bisectional curvature. There is a specific gap between the largest and the second largest dimension. We also show that the manifold attains the second largest dimension is biholomorphic to the complex Euclidean space.

The rigidity of dimension estimate for holomorphic functions on Kähler manifolds

TL;DR

Addresses the rigidity of dimension estimates for holomorphic functions with polynomial growth on complete -dimensional Kähler manifolds with non-negative holomorphic bisectional curvature. The authors develop a framework based on the Poincaré–Siegel map, lifting to the universal cover, and a splitting theorem to analyze the structure of , proving a sharp gap: the maximum possible dimension is and the second-largest is ; if with , then is biholomorphic to . The paper also provides explicit metric examples achieving the gap, illustrating optimality, and discusses potential generalizations under weaker curvature notions such as non-negative holomorphic sectional curvature.

Abstract

In this paper, we obtain the optimal rigidity of dimension estimate for holomorphic functions with polynomial growth on Kähler manifolds with non-negative holomorphic bisectional curvature. There is a specific gap between the largest and the second largest dimension. We also show that the manifold attains the second largest dimension is biholomorphic to the complex Euclidean space.
Paper Structure (15 sections, 17 theorems, 119 equations)

This paper contains 15 sections, 17 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Previous results
  5. Lifted function
  6. Classical results on $\mathbb{C}^{n}$
  7. The splitting theorem
  8. Simply-connectedness
  9. The first and second rigidity
  10. The first rigidity
  11. The second rigidity
  12. Examples
  13. Further questions
  14. Generalization of the second rigidity
  15. Holomorphic sectional curvature

Key Result

Theorem 1.5

Let $(M,\omega)$ be a complete $n$-dimensional Kähler manifold with non-negative holomorphic bisectional curvature. For any integer $d\geqslant1$, and the equality holds if and only if $(M,\omega)$ is biholomorphically isometric to $(\mathbb{C}^{n},\omega_{\mathbb{C}^{n}})$.

Theorems & Definitions (38)

  • Definition 1.1
  • Conjecture 1.2: Frankel's Conjecture Frankel61
  • Conjecture 1.3: Yau's Uniformization Conjecture Yau89
  • Definition 1.4
  • Theorem 1.5: Ni Ni04, Chen-Fu-Yin-Zhu CFYZ06, Liu Liu16
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Definition 2.1: Vanishing order
  • Definition 2.2: Degree
  • ...and 28 more