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Moduli stack of vector bundles
Concept in algebraic geometry

In algebraic geometry, the moduli stack of rank-n vector bundles Vectn is the stack parametrizing vector bundles (or locally free sheaves) of rank n over some reasonable spaces.

It is a smooth algebraic stack of the negative dimension − n 2 {\displaystyle -n^{2}} . Moreover, viewing a rank-n vector bundle as a principal G L n {\displaystyle GL_{n}} -bundle, Vectn is isomorphic to the classifying stack B G L n = [ pt / G L n ] . {\displaystyle BGL_{n}=[{\text{pt}}/GL_{n}].}

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Definition

For the base category, let C be the category of schemes of finite type over a fixed field k. Then Vect n {\displaystyle \operatorname {Vect} _{n}} is the category where

  1. an object is a pair ( U , E ) {\displaystyle (U,E)} of a scheme U in C and a rank-n vector bundle E over U
  2. a morphism ( U , E ) → ( V , F ) {\displaystyle (U,E)\to (V,F)} consists of f : U → V {\displaystyle f:U\to V} in C and a bundle-isomorphism f ∗ F → ∼ E {\displaystyle f^{*}F{\overset {\sim }{\to }}E} .

Let p : Vect n → C {\displaystyle p:\operatorname {Vect} _{n}\to C} be the forgetful functor. Via p, Vect n {\displaystyle \operatorname {Vect} _{n}} is a prestack over C. That it is a stack over C is precisely the statement "vector bundles have the descent property". Note that each fiber Vect n ⁡ ( U ) = p − 1 ( U ) {\displaystyle \operatorname {Vect} _{n}(U)=p^{-1}(U)} over U is the category of rank-n vector bundles over U where every morphism is an isomorphism (i.e., each fiber of p is a groupoid).

See also

  • Behrend, Kai (2002). "Localization and Gromov-Witten Invariants". In de Bartolomeis; Dubrovin; Reina (eds.). Quantum Cohomology. Lecture Notes in Mathematics. Vol. 1776. Berlin: Springer. pp. 3–38. doi:10.1007/978-3-540-45617-9_2. ISBN 978-3-540-43121-3.

References

  1. Behrend 2002, Example 20.2. - Behrend, Kai (2002). "Localization and Gromov-Witten Invariants". In de Bartolomeis; Dubrovin; Reina (eds.). Quantum Cohomology. Lecture Notes in Mathematics. Vol. 1776. Berlin: Springer. pp. 3–38. doi:10.1007/978-3-540-45617-9_2. ISBN 978-3-540-43121-3. https://doi.org/10.1007%2F978-3-540-45617-9_2 ↩