Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nevanlinna Theory of Algebroid Functions on Complete Kähler Manifolds

Xianjing Dong

TL;DR

This work extends classical Nevanlinna value distribution theory from one complex variable to algebroid functions on complete non-compact Kähler manifolds with curvature constraints, introducing a ν-leaf uniformization to realize algebroid functions as meromorphic objects on a higher-dimensional covering. It develops a full Nevanlinna framework on such manifolds, including First and Second Main Theorems, Defect relations, and Picard-type results, by leveraging Green function techniques, Ricci and curvature bounds, and a detailed analysis of branch structures. The paper further advances algebroid theory by establishing First/Second Main Theorems for algebroid functions, estimating branch divisors, and proving propagation of algebraic dependence and five-value unicity theorems, with separate treatments under non-negative Ricci curvature and non-positive sectional curvature. Overall, the results substantially generalize value distribution and unicity theory to geometric settings, enabling applications to complex geometry and value distribution on Kähler manifolds with curvature constraints.

Abstract

In this paper, we generalize the classical Nevanlinna theory of algebroid functions from $\mathbb C$ to a complete Kähler manifold with either non-negative Ricci curvature or non-positive sectional curvature. As its applications, we establish some Picard type theorems and five-value type theorems for algebroid functions under certain conditions.

Nevanlinna Theory of Algebroid Functions on Complete Kähler Manifolds

TL;DR

This work extends classical Nevanlinna value distribution theory from one complex variable to algebroid functions on complete non-compact Kähler manifolds with curvature constraints, introducing a ν-leaf uniformization to realize algebroid functions as meromorphic objects on a higher-dimensional covering. It develops a full Nevanlinna framework on such manifolds, including First and Second Main Theorems, Defect relations, and Picard-type results, by leveraging Green function techniques, Ricci and curvature bounds, and a detailed analysis of branch structures. The paper further advances algebroid theory by establishing First/Second Main Theorems for algebroid functions, estimating branch divisors, and proving propagation of algebraic dependence and five-value unicity theorems, with separate treatments under non-negative Ricci curvature and non-positive sectional curvature. Overall, the results substantially generalize value distribution and unicity theory to geometric settings, enabling applications to complex geometry and value distribution on Kähler manifolds with curvature constraints.

Abstract

In this paper, we generalize the classical Nevanlinna theory of algebroid functions from to a complete Kähler manifold with either non-negative Ricci curvature or non-positive sectional curvature. As its applications, we establish some Picard type theorems and five-value type theorems for algebroid functions under certain conditions.
Paper Structure (33 sections, 64 theorems, 362 equations)

This paper contains 33 sections, 64 theorems, 362 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main Results
  4. Algebroid Functions on Hermitian Manifolds
  5. Notion of Algebroid Functions
  6. Meromorphic function elements
  7. Analytic continuation of meromorphic function elements
  8. Properties of function elements near a multiple point
  9. Multi-leaf Complex Manifolds by Algebroid Functions
  10. Zero and Pole Divisors of Algebroid Functions
  11. Uniformization of Algebroid Functions without Construction
  12. Algebroid Reduction
  13. Uniformization
  14. Nevanlinna Theory of Meromorphic Functions
  15. First Main Theorem
  16. ...and 18 more sections

Key Result

Theorem 2.3

Let $(x_0, w_0)\in M\times\mathbb C$ be a pair such that Then, there exists a unique regular function element $(x_0, u)$ subordinate to equation $(eq)$ satisfying $w_0=u(x_0).$

Theorems & Definitions (112)

  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Definition 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Theorem 2.7
  • proof
  • Definition 2.8
  • ...and 102 more