A differential analogue of the wild automorphism conjecture

TL;DR

This paper proves a differential analogue of a conjecture in algebraic dynamics, showing that certain projective varieties with a specific vector field structure are necessarily abelian varieties, and explores vector fields on these varieties.

Contribution

It establishes a differential analogue of the wild automorphism conjecture, linking invariant vector fields to the structure of abelian varieties in algebraic dynamics.

Findings

Varieties with no proper invariant subvarieties under a global algebraic vector field are abelian varieties.

Examines properties of vector fields on abelian varieties with similar invariance conditions.

Abstract

A differential analogue of the conjecture of Reichstein, Rogalski, and Zhang in algebraic dynamics is here established: if $X$ is a projective variety over an algebraically closed field of characteristic zero which admits a global algebraic vector field $v:X\to TX$ such that $(X,v)$ has no proper invariant subvarieties then $X$ is an abelian variety. Vector fields on abelian varieties with this property are also examined.