Moduli stack of principal bundles

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

${\mathrm{Bun}}_G(X)(R)=$ the category of principal G-bundles over the relative curve $X{\times }_{{\mathbf{F}}_q}\mathrm{Spec}R$.

In particular, the category of ${\mathbf{F}}_q$-points of ${\mathrm{Bun}}_G(X)$, that is, ${\mathrm{Bun}}_G(X)({\mathbf{F}}_q)$, is the category of G-bundles over X. Similarly, ${\mathrm{Bun}}_G(X)$ can also be defined when the curve X is over the field of complex numbers. Roughly, in the complex case, one can define ${\mathrm{Bun}}_G(X)$ 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 ${\mathrm{Bun}}_G(X)$. In the finite field case, it is not common to define the homotopy type of ${\mathrm{Bun}}_G(X)$. But one can still define a (smooth) cohomology and homology of ${\mathrm{Bun}}_G(X)$.

Basic properties

It is known that ${\mathrm{Bun}}_G(X)$ is a smooth stack of dimension $(g(X)-1)\dim G$ where $g(X)$ 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 ${\pi }_0({\mathrm{Bun}}_G(X))$ is in a natural bijection with the fundamental group ${\pi }_1(G)$.

The Atiyah–Bott formula

Main article: Atiyah–Bott formula

Behrend's trace formula

See also: Weil conjecture on Tamagawa numbers and Behrend's formula

This is a (conjectural) version of the Lefschetz trace formula for ${\mathrm{Bun}}_G(X)$ 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

$\mathrm{\#}{\mathrm{Bun}}_G(X)({\mathbf{F}}_q)=q^{\dim {\mathrm{Bun}}_G(X)}\mathrm{tr}({\varphi }^{-1}|H^{*}({\mathrm{Bun}}_G(X);{\mathbb{\mathbb{Z}}}_l))$

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

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

• Heinloth, Jochen (2010), "Lectures on the moduli stack of vector bundles on a curve" (PDF), in Schmitt, Alexander (ed.), Affine flag manifolds and principal bundles, Trends in Mathematics, Basel: Birkhäuser/Springer, pp. 123–153, doi:10.1007/978-3-0346-0288-4_4, ISBN 978-3-0346-0287-7, MR 3013029
• 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/.
• Gaitsgory, Dennis; Lurie, Jacob (2019), Weil's conjecture for function fields, Vol. 1 (PDF), Annals of Mathematics Studies, vol. 199, Princeton, NJ: Princeton University Press, ISBN 978-0-691-18214-8, MR 3887650

Further reading

• C. Sorger, Lectures on moduli of principal G-bundles over algebraic curves

See also

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