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In this article we develop the theory of local models for the moduli stacksof global $G$-shtukas, the function field analogs for Shimura varieties. Here$G$ is a smooth affine group scheme over a smooth projective curve. As thefirst approach, we relate the local geometry of these moduli stacks to thegeometry of Schubert varieties inside global affine Grassmannian, only by meansof global methods. Alternatively, our second approach uses the relation betweenthe deformation theory of global $G$-shtukas and associated local $P$-shtukasat certain characteristic places. Regarding the analogy between function fieldsand number fields, the first (resp. second) approach corresponds to theBeilinson-Drinfeld-Gaitsgory (resp. Rapoport-Zink) local model for(PEL-)Shimura varieties. As an application, we prove the flatness of thesemoduli stacks over their reflex rings, for tamely ramified group $G$.Furthermore, we introduce the Kottwitz-Rapoport stratification on these modulistacks and discuss the intersection cohomology of the special fiber.

Local Models For The Moduli Stacks of Global $G$-Shtukas