Gerbe

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

In mathematics, a gerbe (/ɜːrb/; French: [ʒɛʁb]) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud (Giraud 1971) following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as a generalization of principal bundles to the setting of 2-categories. Gerbes provide a convenient, if highly abstract, language for dealing with many types of deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them.

"Gerbe" is a French (and archaic English) word that literally means wheat sheaf.

Definitions[edit]

Gerbe[edit]

A gerbe on a topological space X is a stack G of groupoids over X which is locally non-empty (each point in X has an open neighbourhood U over which the section category G(U) of the gerbe is not empty) and transitive (for any two objects a and b of G(U) for any open set U, there is an open covering {Vi}i of U such that the restrictions of a and b to each Vi are connected by at least one morphism).

A canonical example is the gerbe of principal bundles with a fixed structure group H: the section category over an open set U is the category of principal H-bundles on U with isomorphism as morphisms (thus the category is a groupoid). As principal bundles glue together (satisfy the descent condition), these groupoids form a stack. The trivial bundle X x H over X shows that the local non-emptiness condition is satisfied, and finally as principal bundles are locally trivial, they become isomorphic when restricted to sufficiently small open sets; thus the transitivity condition is satisfied as well.

Examples[edit]

Algebraic geometry[edit]

Let M be a variety over an algebraically closed field k, G an algebraic group, for example . Recall that a G-torsor over M is an algebraic space P with an action of G and a map π: PM, such that locally on M (in étale topology or fppf topology) π is a direct product π|U: G × UU. A G-gerbe over M may be defined in a similar way. It is an Artin stack with a map , such that locally on M (in étale or fppf topology) π is a direct product .[1] Here BG denotes the classifying stack of G, i.e. a quotient of a point by a trivial G-action. There is no need to impose the compatibility with the group structure in that case since it is covered by the definition of a stack. The underlying topological spaces of and are the same, but in each point is equipped with a stabilizer group isomorphic to G.

Consider a smooth projective curve C over k of genus g > 1. Let be the moduli stack of stable vector bundles on C of rank r and degree d. It has a coarse moduli space Ms
r, d
, which is a quasiprojective variety. These two moduli problems parametrize the same objects, but the stacky version remembers automorphisms of vector bundles. For any stable vector bundle E the automorphism group Aut(E) consists only of scalar multiplications, so each point in a moduli stack has a stabilizer isomorphic to . It turns out that the map is indeed a -gerbe in the sense above.[2]. It is a trivial gerbe if and only if r and d are coprime.


These and more general kinds of gerbes arise in several contexts:

Differential geometry[edit]

  • and -gerbes: Jean-Luc Brylinski's approach

History[edit]

Gerbes first appeared in the context of algebraic geometry. They were subsequently developed in a more traditional geometric framework by Brylinski (Brylinski 1993). One can think of gerbes as being a natural step in a hierarchy of mathematical objects providing geometric realizations of integral cohomology classes.

A more specialised notion of gerbe was introduced by Murray and called bundle gerbes. Essentially they are a smooth version of abelian gerbes belonging more to the hierarchy starting with principal bundles than sheaves. Bundle gerbes have been used in gauge theory and also string theory. Current work by others is developing a theory of non-abelian bundle gerbes.

See also[edit]

References[edit]

  1. ^ Edidin, Dan; Hassett, Brendan; Kresch, Andrew; Vistoli, Angelo (2001). "Brauer groups and quotient stacks". American Journal of Mathematics: 761–777. arXiv:math/9905049. doi:10.1353/ajm.2001.0024.
  2. ^ Hoffman, Norbert (2010). "Moduli stacks of vector bundles on curves and the King–Schofield rationality proof". Cohomological and geometric approaches to rationality problems: 133–148.

External links[edit]