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Preprint Number 764

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764. Thomas Scanlon
Algebraic differential equations from covering maps
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Submission date: 22 August 2014

Abstract:

Let Y be a complex algebraic variety, G \curvearrowright Y an action of an algebraic group on Y, U \subseteq Y(C) a complex submanifold, Γ < G(C) a discrete, Zariski dense subgroup of G( C) which preserves U, and π:U \to X(C) an analytic covering map of the complex algebraic variety X expressing X(C) as Γ \backslash U. We note that the theory of elimination of imaginaries in differentially closed fields produces a generalized Schwarzian derivative \widetilde{χ}: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 := \widetilde{χ} \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(C) with π(γ \cdot π^{-1}(a)) = b.

Mathematics Subject Classification: 03C60, 03C64, 12H05, 14G35

Keywords and phrases:

Full text arXiv 1408.5177: pdf, ps.


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