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In this article we develop the theory of local models for the moduli
stacks of global G-shtukas, the function field analogs for Shimura
varieties. Here G is a smooth affine group scheme over a smooth
projective curve. As the first approach, we relate the local ge-
ometry of these moduli stacks to the geometry of Schubert va-
rieties inside global affine Grassmannian, only by means of global
methods. Alternatively, our second approach uses the relation be-
tween the deformation theory of global G-shtukas and associated
local P-shtukas at certain characteristic places. Regarding the anal-
ogy between function fields and number fields, the first (resp. sec-
ond) approach corresponds to Beilinson-Drinfeld-Gaitsgory (resp.
Rapoport-Zink) type local model for (PEL-)Shimura varieties. This
discussion will establish a conceptual relation between the above
approaches. Furthermore, as applications of this theory, we discuss
the flatness of these moduli stacks over their reflex rings, we intro-
duce the Kottwitz-Rapoport stratification on them, and we study
the intersection cohomology of the special fiber.
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