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