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Weil uniformization theorem

Context

Bundles

Complex geometry

Contents

Idea

The uniformization theorem for principal bundles over algebraic curves XX (going back to André Weil) expresses the moduli stack of principal bundles on XX as a double quotient stack of the GG-valued Laurent series around finitely many points by the product of the GG-valued formal power series around these points and the GG-valued functions on the complement of theses points.

If a single point xx is sufficient and if DD denotes the formal disk around that point and X *,D *X^\ast, D^\ast denote the complements of this point, respectively then the theorem says for suitable algebraic group GG that there is an equivalence of stacks

[X *,G]\[D *,G]/[D,G]Bun X(G), [X^\ast, G] \backslash [D^\ast, G] / [D,G] \simeq Bun_X(G) \,,

between the double quotient stack of GG-valued functions (mapping stacks) as shown on the left and the moduli stack of G-principal bundles over XX, as shown on the right.

The theorem is based on the fact that GG-bundles on XX trivialize on the complement of finitely many points and that this double quotient then expresses the GG-Cech cohomology with respect to the cover given by the complement of the points and the formal disks around them.

For details see at moduli stack of bundles – over curves.

Applications

The theorem is a key ingredient in the function field analogy where for KK a global field the nonabelian generalization of quotients of the idele class group by integral adeles

GL n(K)\GL n(𝔸 K)/GL n(𝒪 K) GL_n(K) \backslash GL_n(\mathbb{A}_K) / GL_n(\mathcal{O}_K)

are analogous to the moduli stack of GG-bundles. This motivates notably the geometric Langlands correspondence as a geometric analog of the number-theoretic Langlands correspondence.

References

Review is for instance

  • Christoph Sorger, Lectures on moduli of principal GG-bundles over algebraic curves, 1999 (pdf)

See also

For more references see at moduli stack of bundles.

Last revised on April 12, 2018 at 08:24:55. See the history of this page for a list of all contributions to it.