On the isotriviality of projective iterative ∂-varieties

We study algebraic varieties X over a universal iterative differential field (K,∂) (typically of positive characteristic), together with an extension of ∂ to an iterative derivation D of the structure sheaf of X. Our work is motivated by the conjecture that if X is projective then the pair (X,D) is isotrivial, namely isomorphic over K to a pair (Y,D0) where Y is defined over the constants C of (K,∂) and D0 is the lifting to K of the trivial iterative derivation on YC. We prove that up to isomorphism there is at most one such D on X extending ∂, thus answering the question when X is defined over C. Other special cases are also proved, including abelian varieties, and smooth curves.