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Hyperbolicity and GCD for n+1 divisors with non-empty intersection

Julie Tzu-Yueh Wang, Zheng Xiao

Abstract

We study hyperbolicity for quasi-projective varieties where the boundary divisor consists of n+1 numerically parallel effective divisors on a complex projective variety of dimension n, allowing non-empty intersection. Under explicit local conditions on beta constants or intersection multiplicities, we prove that all entire curves are algebraically degenerate. Our approach extends the method of Levin-Huang-Xiao to higher dimensions, establishing a second main theorem for regular sequences of closed subschemes. This also yields a GCD-type estimate in the same geometric setting.

Hyperbolicity and GCD for n+1 divisors with non-empty intersection

Abstract

We study hyperbolicity for quasi-projective varieties where the boundary divisor consists of n+1 numerically parallel effective divisors on a complex projective variety of dimension n, allowing non-empty intersection. Under explicit local conditions on beta constants or intersection multiplicities, we prove that all entire curves are algebraically degenerate. Our approach extends the method of Levin-Huang-Xiao to higher dimensions, establishing a second main theorem for regular sequences of closed subschemes. This also yields a GCD-type estimate in the same geometric setting.

Paper Structure

This paper contains 7 sections, 12 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminary in Nevanlinna Theory
  3. Proof of Theorem \ref{['trungeneral']}
  4. Some Technical Lemmas
  5. Proof of Theorem \ref{['trungeneral']}
  6. Proof of Theorem \ref{['mainTheorem2']}
  7. Proof of Theorem \ref{['MainThmgcd']}

Key Result

Theorem 1.5

Let $X$ be a complex projective variety of dimension $n$. Let $D_1, \ldots,D_{n+1}$ be effective Cartier divisors on $X$ such that there exist positive integers $a_1,\ldots, a_{n+1}$ such that $a_1D_1, \ldots, a_{n+1}D_{n+1}$ are all numerically equivalent to an ample divisor $D$. Assume that any $n Suppose further that for every point $Q\in \bigcap_{i=1}^{n+1}D_i$ and every subset $I \subset \{1

Theorems & Definitions (30)

  • Conjecture 1.1: Green-Griffiths-Lang Conjecture
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 1.7: HLX
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 20 more