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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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