Algebraic differential equations from covering maps

Abstract

Let $Y$ be a complex algebraic variety, $G \curvearrowright Y$ an action of an algebraic group on $Y$, $U \subseteq Y({\mathbb C})$ a complex submanifold, $Γ< G({\mathbb C})$ a discrete, Zariski dense subgroup of $G({\mathbb C})$ which preserves $U$, and $π:U \to X({\mathbb C})$ an analytic covering map of the complex algebraic variety $X$ expressing $X({\mathbb C})$ as $Γ\backslash U$. We note that the theory of elimination of imaginaries in differentially closed fields produces a generalized Schwarzian derivative $\widetildeχ:Y \to Z$ (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 \emph{prima facie} differentially analytic relation $χ:= \widetildeχ \circ π^{-1}$ 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 $a, b \in X(K)$ then $χ(a) = χ(b)$ if and only if after suitable restriction there is some $γ\in G({\mathbb C})$ with $π(γ\cdot π^{-1}(a)) = b$.