A differential analogue of the wild automorphism conjecture

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. Some of the analysis works in the more general context of $D$-varieties over differential fields: projective $D$-varieties without proper $D$-subvarieties are homogeneous. But the main theorem does not extend: an example of a $D$-variety structure on the projective line without proper $D$-subvarieties is given.