Gerbe


In mathematics, a gerbe is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of fibre bundles where the fibre is the classifying stack of a group. 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 word that literally means wheat sheaf.

Definitions

Gerbe

A gerbe on a topological space X is a stack G of groupoids over X which is locally non-empty and transitive.
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. As principal bundles glue together, 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

Algebraic geometry

Let be a variety over an algebraically closed field, an algebraic group, for example. Recall that a G-torsor over is an algebraic space with an action of and a map, such that locally on is a direct product. 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 is a direct product. Here denotes the classifying stack of, i.e. a quotient of a point by a trivial -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.

Moduli stack of stable bundles on a curve

Consider a smooth projective curve over of genus. Let be the moduli stack of stable vector bundles on of rank and degree. It has a coarse moduli space, 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 the automorphism group 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.. It is a trivial gerbe if and only if and are coprime.

Root stacks

Another class of gerbes can be found using the construction of root stacks. Informally, the -th root stack of a line bundle over a scheme is a space representing the -th root of and is denoted
pg 52.
The -th root stack of has the property
as gerbes. It is constructed as the stack
sending an -scheme to the category whose objects line bundles of the form
and morphisms are commutative diagrams compatible with the isomorphisms. This gerbe is banded by the algebraic group of roots of unity, where on a cover it acts on a point by cyclically permuting the factors of in. Geometrically, these stacks are formed as the fiber product of stacks
where the vertical map of comes from the Kummer sequence
This is because is the moduli space of line bundles, so the line bundle corresponds to an object of the category .
Root stacks with sections
There is another related construction of root stacks with sections. Given the data above, let be a section. Then the -th root stack of the pair is defined as the lax 2-functor
sending an -scheme to the category whose objects line bundles of the form
and morphisms are given similarly. These stacks can be constructed very explicitly, and are well understood for affine schemes. In fact, these form the affine models for root stacks with sectionspg 4. Given an affine scheme, all line bundles are trivial, hence and any section is equivalent to taking an element. Then, the stack is given by the stack quotient
pg 9
with
If then this gives an infinitesimal extension of.

Examples throughout algebraic geometry

These and more general kinds of gerbes arise in several contexts as both geometric spaces and as formal bookeeping tools:
Gerbes first appeared in the context of algebraic geometry. They were subsequently developed in a more traditional geometric framework by Brylinski. 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.