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

TL;DR

This paper proves a non-Archimedean version of the Green--Griffiths--Lang--Vojta conjecture for certain subvarieties of commutative algebraic groups with unipotent rank 1, establishing loci equivalences and conditions for properness.

Contribution

It demonstrates the equality of Kawamata and Lang-like loci and identifies conditions for their properness in commutative algebraic groups with unipotent rank 1.

Findings

Kawamata locus equals Lang-like exceptional locus for these groups.

Conditions are provided under which these loci are proper subschemes.

The strong form of the conjecture is proved for subvarieties with linear part isomorphic to G_a × G_m^t.

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 $\mathbb{G}_a \times \mathbb{G}_m^t$.