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Contractibility of the space of rational maps. (English) Zbl 1263.14013
For $$X$$ a smooth, connected, complete curve, $$G$$ a reductive group over an algebraically closed field $$k$$ of characteristic zero, let Bun$$_G$$ be the moduli stack of $$G$$-bundles on $$X$$, Gr the moduli space of $$G$$-bundles on $$X$$ with a rational trivialization. One can view Bun$$_G$$ as the quotient of Gr by the group of rational maps from $$X$$ to $$G$$, Maps$$(X,G)^{\text{rat}}$$. This paper studies this latter object. In particular, the author proves that Maps$$(X,G)^{\text{rat}}$$ is homologically contractible, that is we have an isomorphism H$$_{\bullet}(\text{Maps}(X,G)^{\text{rat}}_{\text{RanX}}) \rightarrow k$$ given by a trace map. Instrumental in this formalism is the Ran space Ran$$X$$ of $$X$$. One consequence of the contractibility of the fibers of the uniformization map $$\pi: \text{Gr} \rightarrow \text{Bun}_G$$ is that both spaces have isomorphic cohomology, which allows the author to rederive the Atiyah-Bott formula for the cohomology of Bun$$_G$$.

MSC:
 14D24 Geometric Langlands program (algebro-geometric aspects) 14H10 Families, moduli of curves (algebraic)
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References:
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