Algebraic differential equations from covering maps

Abstract

Let Y be a complex algebraic variety, G↷Y an action of an algebraic group on Y, U⊆Y(C) a complex submanifold, Γ<G(C) a discrete, Zariski dense subgroup of G(C) which preserves U, and π:U→X(C) an analytic covering map of the complex algebraic variety X expressing X(C) as Γ\\U. We note that the theory of elimination of imaginaries in differentially closed fields produces a generalized Schwarzian derivative χ˜:Y→Z (where Z is some algebraic variety) expressing the quotient of Y by the action of the constant points of G. Under the additional hypothesis that the restriction of π to some set containing a fundamental domain is definable in an o-minimal expansion of the real field, we show as a consequence of the Peterzil–Starchenko o-minimal GAGA theorem that the prima facie differentially analytic relation χ:=χ˜∘π−1 is a well-defined, differential constructible function. The function χ nearly inverts π in the sense that for any differential field K of meromorphic functions, if a,b∈X(K) then χ(a)=χ(b) if and only if after suitable restriction there is some γ∈G(C) with π(γ⋅π−1(a))=b.