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Weil uniformization theorem
Contents
Context
Bundles
bundles

Context
Classes of bundles
covering space

retractive space

fiber bundle , fiber ∞-bundle

numerable bundle

principal bundle , principal ∞-bundle

associated bundle , associated ∞-bundle

vector bundle , 2-vector bundle , (∞,1)-vector bundle

real , complex /holomorphic , quaternionic

topological , differentiable , algebraic

with connection

bundle of spectra

natural bundle

equivariant bundle

Universal bundles
Presentations
Examples
Constructions
Complex geometry
Contents
Idea
The uniformization theorem for principal bundles over algebraic curves $X$ (going back to André Weil ) expresses the moduli stack of principal bundles on $X$ as a double quotient stack of the $G$ -valued Laurent series around finitely many points by the product of the $G$ -valued formal power series around these points and the $G$ -valued functions on the complement of theses points.

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

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

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

The theorem is based on the fact that $G$ -bundles on $X$ trivialize on the complement of finitely many points and that this double quotient then expresses the $G$ -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 $K$ a global field the nonabelian generalization of quotients of the idele class group by integral adeles

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

are analogous to the moduli stack of $G$ -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 $G$ -bundles over algebraic curves , 1999 (pdf )
See also

For more references see at moduli stack of bundles .

Last revised on April 4, 2023 at 07:08:57.
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