Moduli stack of principal bundles

In algebraic geometry, given a smooth projective curve X over a finite field ${\displaystyle \mathbf {F} _{q}}$ and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by ${\displaystyle \operatorname {Bun} _{G}(X)}$, is an algebraic stack given by:^[1] for any ${\displaystyle \mathbf {F} _{q}}$-algebra R,

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

In particular, the category of ${\displaystyle \mathbf {F} _{q}}$-points of ${\displaystyle \operatorname {Bun} _{G}(X)}$, that is, ${\displaystyle \operatorname {Bun} _{G}(X)(\mathbf {F} _{q})}$, is the category of G-bundles over X.

Similarly, ${\displaystyle \operatorname {Bun} _{G}(X)}$ can also be defined when the curve X is over the field of complex numbers. Roughly, in the complex case, one can define ${\displaystyle \operatorname {Bun} _{G}(X)}$ 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 ${\displaystyle \operatorname {Bun} _{G}(X)}$.

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

It is known that ${\displaystyle \operatorname {Bun} _{G}(X)}$ is a smooth stack of dimension ${\displaystyle (g(X)-1)\dim G}$ where ${\displaystyle g(X)}$ 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 ^[2] and for G only a flat group scheme of finite type over X see.^[3]

If G is a split reductive group, then the set of connected components ${\displaystyle \pi _{0}(\operatorname {Bun} _{G}(X))}$ is in a natural bijection with the fundamental group ${\displaystyle \pi _{1}(G)}$.^[4]

This is a (conjectural) version of the Lefschetz trace formula for ${\displaystyle \operatorname {Bun} _{G}(X)}$ when X is over a finite field, introduced by Behrend in 1993.^[5] It states:^[6] if G is a smooth affine group scheme with semisimple connected generic fiber, then

${\displaystyle \#\operatorname {Bun} _{G}(X)(\mathbf {F} _{q})=q^{\dim \operatorname {Bun} _{G}(X)}\operatorname {tr} (\phi ^{-1}|H^{*}(\operatorname {Bun} _{G}(X);\mathbb {Z} _{l}))}$

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

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

• 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

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

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