The persistence conjecture for arithmetic hyperbolicity

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The persistence conjecture for arithmetic hyperbolicity

   Let $k$ be an algebraically closed field of characteristic zero, and let $X$ be a variety over $k$. Persistence Conjecture. If $X$ is arithmetically hyperbolic over $k$, then $X$ is absolutely arithmetically hyperbolic. This conjecture asks whether arithmetic hyperbolicity persists under extension to every algebraically closed field of characteristic zero. It is motivated by Lang's philosophy on rational points over finitely generated fields and is known in the case of $\mathbb{A}^1\setminus\{0,1\}$, but the general statement remains open.

   source: Philipp Licht, “Finiteness theorems for complements of large divisors”, arXiv:2203.11126 (2022).