Recent progress on the geometric Bombieri--Lang conjecture

Junyi Xie

Recent progress on the geometric Bombieri--Lang conjecture

Algebraic Geometry 2026-07-02 v1 Complex Variables Number Theory

Authors: Junyi Xie

Abstract

We survey recent progress on the geometric Bombieri--Lang conjecture over function fields of characteristic zero. We discuss recent work of Xie--Yuan and Guoquan Gao, which together proves the conjecture for varieties admitting finite morphisms to abelian varieties. The guiding idea, developed in joint work with Xinyi Yuan, is that Vojta's dictionary can be made concrete in this setting: from rational points of large height one constructs entire curves on complex fibers.

Cite

@article{arxiv.2607.02165,
  title  = {Recent progress on the geometric Bombieri--Lang conjecture},
  author = {Junyi Xie},
  journal= {arXiv preprint arXiv:2607.02165},
  year   = {2026}
}

Comments

This article is based on a talk given at the International Congress of Chinese Mathematicians