Moduli stack of principal bundles

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

In algebraic geometry, given a smooth projective curve X over a finite field and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by , is an algebraic stack given by:[1] for any -algebra R,

the category of principal G-bundles over the relative curve .

In particular, the category of -points of , that is, , is the category of G-bundles over X.

Similarly, can also be defined when the curve X is over the field of complex numbers. Roughly, in the complex case, one can define as the quotient stack of the space of holomorphic connections on X by the gauge group. Replacing the quotient stack (which is not a topological space) by a homotopy quotient (which is a topological space) gives the homotopy type of .

In the finite field case, it is not common to define the homotopy type of . But one can still define a (smooth) cohomology and homology of .

Basic properties[edit]

It is known that is a smooth stack of dimension where is the genus of X. It is not of finite type but of locally of finite type; one thus usually uses a stratification by open substacks of finite type (cf. the Harder–Narasimhan stratification.) If G is a split reductive group, then the set of connected components is in a natural bijection with the fundamental group .[2]

The Atiyah–Bott formula[edit]

Main article: Atiyah–Bott formula

Behrend's trace formula[edit]

This is a (conjectural) version of the Lefschetz trace formula for when X is over a finite field, introduced by Behrend in 1993.[3] It states:[4] if G is a smooth affine group scheme with semisimple connected generic fiber, then

where (see also Behrend's trace formula for the details)

  • l is a prime number that is not p and the ring of l-adic integers is viewed as a subring of .
  • is the geometric Frobenius.
  • , the sum running over all isomorphism classes of G-bundles on X and convergent.
  • for a graded vector space , provided the series on the right absolutely converges.

A priori, neither left nor right side in the formula converges. Thus, the formula states that the two sides converge to finite numbers and that those numbers coincide.

Notes[edit]

References[edit]

Further reading[edit]

See also[edit]