Moduli Stack Of Vector Bundles

In algebraic geometry, the moduli stack of rank-n vector bundles Vect_n 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 [math]\displaystyle{ -n^2 }[/math].^[1] Moreover, viewing a rank-n vector bundle as a principal [math]\displaystyle{ GL_n }[/math]-bundle, Vect_n is isomorphic to the classifying stack [math]\displaystyle{ BGL_n = [\text{pt}/GL_n]. }[/math]

Definition

For the base category, let C be the category of schemes of finite type over a fixed field k. Then [math]\displaystyle{ \operatorname{Vect}_n }[/math] is the category where

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

Let [math]\displaystyle{ p: \operatorname{Vect}_n \to C }[/math] be the forgetful functor. Via p, [math]\displaystyle{ \operatorname{Vect}_n }[/math] 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 [math]\displaystyle{ \operatorname{Vect}_n(U) = p^{-1}(U) }[/math] 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

• classifying stack
• moduli stack of principal bundles

References

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

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