Moduli stack of principal bundles

In algebraic geometry, given a smooth projective curve X over a finite field [math]\displaystyle{ \mathbf{F}_q }[/math] and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by [math]\displaystyle{ \operatorname{Bun}_G(X) }[/math], is an algebraic stack given by: for any [math]\displaystyle{ \mathbf{F}_q }[/math]-algebra R,

[math]\displaystyle{ \operatorname{Bun}_G(X)(R) = }[/math] the category of principal G-bundles over the relative curve [math]\displaystyle{ X \times_{\mathbf{F}_q} \operatorname{Spec}R }[/math].

In particular, the category of [math]\displaystyle{ \mathbf{F}_q }[/math]-points of [math]\displaystyle{ \operatorname{Bun}_G(X) }[/math], that is, [math]\displaystyle{ \operatorname{Bun}_G(X)(\mathbf{F}_q) }[/math], is the category of G-bundles over X.

Similarly, [math]\displaystyle{ \operatorname{Bun}_G(X) }[/math] can also be defined when the curve X is over the field of complex numbers. Roughly, in the complex case, one can define [math]\displaystyle{ \operatorname{Bun}_G(X) }[/math] 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 [math]\displaystyle{ \operatorname{Bun}_G(X) }[/math].

In the finite field case, it is not common to define the homotopy type of [math]\displaystyle{ \operatorname{Bun}_G(X) }[/math]. But one can still define a (smooth) cohomology and homology of [math]\displaystyle{ \operatorname{Bun}_G(X) }[/math].

Basic properties

It is known that [math]\displaystyle{ \operatorname{Bun}_G(X) }[/math] is a smooth stack of dimension [math]\displaystyle{ (g(X) - 1) \dim G }[/math] where [math]\displaystyle{ g(X) }[/math] is the genus of X. It is not of finite type but locally of finite type; one thus usually uses a stratification by open substacks of finite type (cf. the Harder–Narasimhan stratification), also for parahoric G over curve X see and for G only a flat group scheme of finite type over X see.

If G is a split reductive group, then the set of connected components [math]\displaystyle{ \pi_0(\operatorname{Bun}_G(X)) }[/math] is in a natural bijection with the fundamental group [math]\displaystyle{ \pi_1(G) }[/math].

Behrend's trace formula

This is a (conjectural) version of the Lefschetz trace formula for [math]\displaystyle{ \operatorname{Bun}_G(X) }[/math] when X is over a finite field, introduced by Behrend in 1993. It states: if G is a smooth affine group scheme with semisimple connected generic fiber, then

[math]\displaystyle{ \# \operatorname{Bun}_G(X)(\mathbf{F}_q) = q^{\dim \operatorname{Bun}_G(X)} \operatorname{tr} (\phi^{-1}|H^*(\operatorname{Bun}_G(X); \mathbb{Z}_l)) }[/math]

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

• l is a prime number that is not p and the ring [math]\displaystyle{ \mathbb{Z}_l }[/math] of l-adic integers is viewed as a subring of [math]\displaystyle{ \mathbb{C} }[/math].
• [math]\displaystyle{ \phi }[/math] is the geometric Frobenius.
• [math]\displaystyle{ \# \operatorname{Bun}_G(X)(\mathbf{F}_q) = \sum_P {1 \over \# \operatorname{Aut}(P)} }[/math], the sum running over all isomorphism classes of G-bundles on X and convergent.
• [math]\displaystyle{ \operatorname{tr}(\phi^{-1}|V_*) = \sum_{i = 0}^\infty (-1)^i \operatorname{tr}(\phi^{-1}|V_i) }[/math] for a graded vector space [math]\displaystyle{ V_* }[/math], 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.

References

• J. Heinloth, Lectures on the moduli stack of vector bundles on a curve, 2009 preliminary version
• J. Heinloth, A.H.W. Schmitt, The Cohomology Ring of Moduli Stacks of Principal Bundles over Curves, 2010 preprint, available at http://www.uni-essen.de/~hm0002/.
• E. Arasteh Rad, U. Hartl, Uniformizing The Moduli Stacks of Global G-Shtukas, 2013 preprint, available at [1].
• Gaitsgory, D; Lurie, J.; Weil's Conjecture for Function Fields. 2014, [2]

Further reading

• Tamagawa number for functional fields
• C. Sorger, Lectures on moduli of principal G-bundles over algebraic curves

See also

• Geometric Langlands conjectures
• Ran space
• Moduli stack of vector bundles