Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Vojta's abc conjecture for entire curves in toric varieties highly ramified over the boundary

Min Ru, Julie Tzu-Yueh Wang

TL;DR

This work proves a Vojta-ABC type inequality for entire curves in ${\mathbb P}^n(\mathbb C)$ under the hypothesis of high ramification along the coordinate hyperplanes, and extends the framework to projective toric varieties to obtain Campana orbifold conclusions for finite coverings. The authors develop a parabolic Nevanlinna theory, including Jensen's formula, logarithmic derivative, and a moving gcd theorem, and then establish an abc theorem on parabolic Riemann surfaces. The main theorem is proved by translating to a parabolic setting, applying gcd- and proximity-type bounds to holomorphic curves represented by units on parabolic open Riemann surfaces, and combining these with a parabolic abc inequality to yield truncation-level results with effective exceptional sets. The toric extension leverages affine covers and Campana orbifold formalism to deduce orbifold-structure hyperbolicity-type consequences for finite covers, culminating in a Campana orbifold conjecture for toric-geometric settings. Overall, the paper provides a rigorous analytic-geometry pipeline from parabolic Nevanlinna theory to abc-type results and toric/campana consequences with explicit, computable data.

Abstract

We prove Vojta's abc conjecture for projective space ${\Bbb P}^n({\Bbb C})$, assuming that the entire curves in ${\Bbb P}^n({\Bbb C})$ are highly ramified over the coordinate hyperplanes. This extends the results of Guo Ji and the second-named author for the case $n=2$ (see \cite{GW22}). We also explore the corresponding results for projective toric varieties. Consequently, we establish a version of Campana's orbifold conjecture for finite coverings of projective toric varieties.

Vojta's abc conjecture for entire curves in toric varieties highly ramified over the boundary

TL;DR

This work proves a Vojta-ABC type inequality for entire curves in under the hypothesis of high ramification along the coordinate hyperplanes, and extends the framework to projective toric varieties to obtain Campana orbifold conclusions for finite coverings. The authors develop a parabolic Nevanlinna theory, including Jensen's formula, logarithmic derivative, and a moving gcd theorem, and then establish an abc theorem on parabolic Riemann surfaces. The main theorem is proved by translating to a parabolic setting, applying gcd- and proximity-type bounds to holomorphic curves represented by units on parabolic open Riemann surfaces, and combining these with a parabolic abc inequality to yield truncation-level results with effective exceptional sets. The toric extension leverages affine covers and Campana orbifold formalism to deduce orbifold-structure hyperbolicity-type consequences for finite covers, culminating in a Campana orbifold conjecture for toric-geometric settings. Overall, the paper provides a rigorous analytic-geometry pipeline from parabolic Nevanlinna theory to abc-type results and toric/campana consequences with explicit, computable data.

Abstract

We prove Vojta's abc conjecture for projective space , assuming that the entire curves in are highly ramified over the coordinate hyperplanes. This extends the results of Guo Ji and the second-named author for the case (see \cite{GW22}). We also explore the corresponding results for projective toric varieties. Consequently, we establish a version of Campana's orbifold conjecture for finite coverings of projective toric varieties.

Paper Structure

This paper contains 12 sections, 25 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Motivation through the rational case
  3. The Rational Case
  4. Motivation for the proof of the Main Theorem
  5. Nevanlinna theory and the GCD theorem on parabolic Riemann surfaces
  6. Parabolic Riemann surfaces and Jensen's formula
  7. The characteristic functions in the parabolic setting
  8. The parabolic Logarithmic Derivative Lemma and H. Cartan's theorem
  9. The moving GCD theorem in the parabolic setting
  10. The abc theorem for parabolic Riemann surfaces
  11. Proof of the main theorem
  12. Toric Varieties

Key Result

Theorem 1

Let $\Delta_0$ be an orbifold divisor on ${\Bbb P}^n({\Bbb C})$ and let $H_0 , H_1 , \dots, H_n$ be the coordinate hyperplanes in ${\Bbb P}^n({\Bbb C})$, such that $\Delta_0$ and $H_0, H_1, \dots, H_n$ are in general position. Let $m_i \in (1, \infty)\cap {\Bbb Q}$, $0\leq i\leq n$, and $\Delta=\D

Theorems & Definitions (45)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5: GNSW
  • Theorem 6
  • Proposition 7: GNSW
  • proof : Proof of Theorem \ref{['rational']}
  • Lemma 8: Lemma 3.2 in CHSX
  • proof
  • ...and 35 more