Algebraic differential equations from covering maps

Let Y be a complex algebraic variety, G curved right arrow Y an action of an algebraic group on Y, U subset of Y(C) a complex submanifold, Gamma < G(C) a discrete, Zariski dense subgroup of G(C) which preserves U, and pi : U -> X(C) an analytic covering map of the complex algebraic variety X expressing X(C) as Gamma\U. We note that the theory of elimination of imaginaries in differentially closed fields produces a generalized Schwarzian derivative (chi)over-bar : 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 pi 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 chi := (chi)over-baro pi(-1) is a well-defined, differential constructible function. The function chi nearly inverts pi in the sense that for any differential field K of meromorphic functions, if a, b is an element of X(K) then chi(a) = chi(b) if and only if after suitable restriction there is some gamma is an element of G(C) with pi(gamma . pi(-1)(a)) = b. (C) 2018 Elsevier Inc. All rights reserved.