Recent progress on the geometric Bombieri--Lang conjecture

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.