The Green--Griffiths--Lang--Vojta conjecture for commutative algebraic groups with unipotent rank 1

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 $X$. Using this, we prove the strong form of the Green--Griffiths--Lang--Vojta conjecture for closed subvarieties of ${\mathbb{G}}_a\times A$ where $A$ is an abelian variety.

arXiv commentv2: 17 pages, new title, removed non-Archimedean statements as there was mistake, extended some algebraic arguments, and added complex analytic results (namely Theorem D and Corollary E are new). Comments welcome!