  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 $\pi(\gamma \cdot \pi^{-1}(a)) = b$.
