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 ${\displaystyle \operatorname {Bun} _{G}(X)}$.

Basic properties

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]

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 ${\displaystyle \operatorname {Bun} _{G}(X)}$ 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.

Notes

1. Lurie, Jacob (April 3, 2013), Tamagawa Numbers in the Function Field Case (Lecture 2) (PDF), archived from the original (PDF) on 2013-04-11, retrieved 2014-01-30
2. Heinloth 2010, Proposition 2.1.2
3. Arasteh Rad, E.; Hartl, Urs (2021), "Uniformizing the moduli stacks of global G-shtukas", International Mathematics Research Notices (21): 16121–16192, arXiv:1302.6351, doi:10.1093/imrn/rnz223, MR 4338216; see Theorem 2.5
4. Heinloth 2010, Proposition 2.1.2
5. Behrend, Kai A. (1991), The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles (PDF) (PhD thesis), University of California, Berkeley
6. Gaitsgory & Lurie 2019, Chapter 5: The Trace Formula for Bun_G(X), p. 260

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