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Improvements to the Montel-Carathéodory Theorem for families of $\mathbb{P}^n$-valued holomorphic curves

Gopal Datt

Abstract

In this paper, we establish various sufficient conditions for a family of holomorphic mappings on a domain $D\subseteq\mathbb{C}$ into $\mathbb{P}^n$ to be normal. Our results are improvements to the Montel-Carathéodory Theorem for a family of $\mathbb{P}^n$-valued holomorphic curves.

Improvements to the Montel-Carathéodory Theorem for families of $\mathbb{P}^n$-valued holomorphic curves

Abstract

In this paper, we establish various sufficient conditions for a family of holomorphic mappings on a domain into to be normal. Our results are improvements to the Montel-Carathéodory Theorem for a family of -valued holomorphic curves.

Paper Structure

This paper contains 7 sections, 9 theorems, 66 equations.

Table of Contents

  1. Introduction and main results
  2. Basic notions
  3. Some Examples
  4. Essential lemmas
  5. Results on Nevanlinna theory
  6. The proof of Theorem \ref{['T: Main Xu shared hyperplane']}
  7. The proof of Theorem \ref{['T: vol of divisor']}

Key Result

Theorem 1.2

Let $\mathcal{F}$ be a family of holomorphic curves on a planar domain $D$ into $\mathbb{P}^n$. Let $H_1,\dots, H_{2n+1}$ be $2n+1$ hyperplanes in general position in $\mathbb{P}^n$. Assume that the first $n+1$ of these hyperplanes, i.e., $H_1,\dots, H_{n+1}$, are the coordinate hyperplanes of $\mat Then the family $\mathcal{F}$ is normal.

Theorems & Definitions (19)

  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.1
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Lemma 4.1: AladroKrantz, Thai Trang Huong 03
  • ...and 9 more