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 $-n^2$ .{{harvnb|Behrend|2002|loc=Example 20.2.}} Moreover, viewing a rank-n vector bundle as a principal $GL_n$ -bundle, Vect_n is isomorphic to the classifying stack $BGL_n = [\text{pt}/GL_n].$

Definition

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

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

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

References

{{cite book |last=Behrend |first=Kai |year=2002 |chapter=Localization and Gromov-Witten Invariants |title=Quantum Cohomology. Lecture Notes in Mathematics |editor1-last=de Bartolomeis |editor2-last=Dubrovin |editor3-last=Reina |volume=1776 |publisher=Springer |location=Berlin |doi=10.1007/978-3-540-45617-9_2 |pages=3–38|isbn=978-3-540-43121-3 }}

Category:Algebraic geometry

moduli stack of vector bundles

Definition

See also

References