(see also Chern-Weil theory, parameterized homotopy theory)
vector bundle, (∞,1)-vector bundle
topological vector bundle, differentiable vector bundle, algebraic vector bundle
direct sum of vector bundles, tensor product of vector bundles, inner product of vector bundles?, dual vector bundle
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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
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.
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
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.
Review is for instance
See also
For more references see at moduli stack of bundles.
Last revised on April 12, 2018 at 12:24:55. See the history of this page for a list of all contributions to it.