The non-Archimedean Green--Griffiths--Lang--Vojta conjecture for commutative algebraic groups with unipotent rank 1
Jackson S. Morrow, Paul Vojta
TL;DR
The paper proves that for commutative algebraic groups $G$ with unipotent rank $1$ and toric rank $0$, the Lang-like exceptional locus of a closed subvariety $X subseteq G$ coincides with the Kawamata locus, and identifies a not-fibered-by-subgroups condition ensuring these loci are proper subschemes. It then establishes a strong non-Archimedean Green--Griffiths--Lang--Vojta statement: any non-Archimedean analytic map from $ ext{G}_{m}^{an}$ to $G^{an}$ has an analytic Zariski closure that is a translate of an algebraic subgroup, which implies $X^{an}$ is Brody hyperbolic modulo the analytification of the Lang-like locus. The work relies on a detailed analysis of the abelian and unipotent parts via equivariant completions, extensions as analytic torsors, and reductions to abelian quotients, to control the exceptional loci and hyperbolicity. Collectively, the results connect algebraic, complex-analytic, and non-Archimedean perspectives on hyperbolicity and exceptional loci in a new class of commutative groups, with corollaries relating pseudo-grouplessness and log-general type.
Abstract
Let $k$ be algebraically closed field of characteristic zero, let $G$ be a commutative algebraic group over $k$ such that the linear part of $G$ is isomorphic to $\mathbb{G}_a$, and let $X$ be a closed subvariety of $G$. We show that the Kawamata locus of $X$ is equal to a Lang-like exceptional locus of $X$, and furthermore, we identify a condition on $X$ that implies that these loci are proper subschemes of $X$. We also prove the strong form of the non-Archimedean Green--Griffiths--Lang--Vojta conjecture for closed subvarieties of commutative algebraic groups where the linear part is isomorphic to $\mathbb{G}_a \times \mathbb{G}_m^t$.