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A complete proof of the geometric Bombieri-Lang conjecture for ramified covers of abelian varieties

Guoquan Gao

TL;DR

The paper proves a complete geometric Bombieri–Lang-type statement for ramified covers X of abelian varieties over function fields of characteristic 0. It extends prior results by Xie–Yuan by removing hyperbolicity and trace restrictions, using a strategy of constructing entire curves and a new mixture of algebraic geometry, minimal model theory, and Nevanlinna theory. A key technical engine is a height-boundedness result for generic sequences of K-rational points on X when X→A×_K T_K is finite; this feeds into a contradiction argument via tangency and positivity, aided by a canonical-model reduction and semistable reduction. The outcome is a precise description of the K-rational points outside the special set Sp(X) in terms of k-points of fixed base varieties, yielding Zariski-non-density results and generalizing previous cases to ramified covers with nontrivial K/k-trace. This advances understanding of distribution of rational points on higher-dimensional varieties over function fields and demonstrates the power of combining Nevanlinna theory with modern birational geometry.

Abstract

We prove the geometric Bombieri-Lang conjecture for projective varieties which have finite maps to abelian varieties over function fields of characteristic 0. This generalizes the recent results of Xie-Yuan, which require either the hyperbolicity assumption or the non-isotriviality assumption. The proof is based on their strategy for constructing entire curves, but requires some new ingredients from algebraic geometry and Nevanlinna theory.

A complete proof of the geometric Bombieri-Lang conjecture for ramified covers of abelian varieties

TL;DR

The paper proves a complete geometric Bombieri–Lang-type statement for ramified covers X of abelian varieties over function fields of characteristic 0. It extends prior results by Xie–Yuan by removing hyperbolicity and trace restrictions, using a strategy of constructing entire curves and a new mixture of algebraic geometry, minimal model theory, and Nevanlinna theory. A key technical engine is a height-boundedness result for generic sequences of K-rational points on X when X→A×_K T_K is finite; this feeds into a contradiction argument via tangency and positivity, aided by a canonical-model reduction and semistable reduction. The outcome is a precise description of the K-rational points outside the special set Sp(X) in terms of k-points of fixed base varieties, yielding Zariski-non-density results and generalizing previous cases to ramified covers with nontrivial K/k-trace. This advances understanding of distribution of rational points on higher-dimensional varieties over function fields and demonstrates the power of combining Nevanlinna theory with modern birational geometry.

Abstract

We prove the geometric Bombieri-Lang conjecture for projective varieties which have finite maps to abelian varieties over function fields of characteristic 0. This generalizes the recent results of Xie-Yuan, which require either the hyperbolicity assumption or the non-isotriviality assumption. The proof is based on their strategy for constructing entire curves, but requires some new ingredients from algebraic geometry and Nevanlinna theory.

Paper Structure

This paper contains 13 sections, 27 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Ideas of the proof
  4. Notations and terminology
  5. Locations of limit points
  6. Linear entire curves
  7. Locations of limit points
  8. Boundedness of height on generic sequences
  9. Preliminaries in Nevanlinna theory
  10. A key lemma
  11. A positivity lemma
  12. Proof of Proposition \ref{['generic1']}
  13. Proof of the main theorem

Key Result

Theorem 1.1

Let $K$ be a finitely generated field over a field $k$ of characteristic 0 such that $k$ is algebraically closed in $K$. Let $X$ be a projective variety over $K$ with a finite morphism $f\colon X\to A$ to an abelian variety $A$ over $K$. Let $Z$ be the Zariski closure of $(X\backslash{\rm Sp}(X))(K)

Theorems & Definitions (50)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Theorem 1.3: maximal Albanese dimension
  • Proposition 1.4
  • Remark 1.5
  • Proposition 1.6
  • Lemma 1.7: Lemma \ref{['key']}
  • Lemma 2.1: XY2
  • Proposition 2.2: XY2
  • ...and 40 more