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 $\operatorname{Bun}_G(X)$ , is an algebraic stack given by:{{citation |url=http://www.math.harvard.edu/~lurie/283notes/Lecture2-FunctionFields.pdf|title=Tamagawa Numbers in the Function Field Case (Lecture 2)|first=Jacob|last=Lurie|date=April 3, 2013 |access-date=2014-01-30 |archive-url=https://web.archive.org/web/20130411033546/http://www.math.harvard.edu/~lurie/283notes/Lecture2-FunctionFields.pdf |archive-date=2013-04-11 |url-status=dead }} for any $\mathbf{F}_q$ -algebra R,

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

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

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

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

Basic properties

It is known that $\operatorname{Bun}_G(X)$ 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 {{harvnb|Heinloth|2010|loc=Proposition 2.1.2}} and for G only a flat group scheme of finite type over X see.{{citation

| last1 = Arasteh Rad | first1 = E.

| last2 = Hartl | first2 = Urs

| arxiv = 1302.6351

| doi = 10.1093/imrn/rnz223

| issue = 21

| journal = International Mathematics Research Notices

| mr = 4338216

| pages = 16121–16192

| title = Uniformizing the moduli stacks of global G-shtukas

| year = 2021}}; see Theorem 2.5

If G is a split reductive group, then the set of connected components $\pi_0(\operatorname{Bun}_G(X))$ is in a natural bijection with the fundamental group $\pi_1(G)$ .{{harvnb|Heinloth|2010|loc=Proposition 2.1.2}}

The Atiyah–Bott formula

Behrend's trace formula

{{see also|Weil conjecture on Tamagawa numbers|Behrend's formula}}

This is a (conjectural) version of the Lefschetz trace formula for $\operatorname{Bun}_G(X)$ when X is over a finite field, introduced by Behrend in 1993.{{citation|url=http://www.math.ubc.ca/~behrend/thesis.pdf|title=The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles|first=Kai A.|last=Behrend|type=PhD thesis|publisher=University of California, Berkeley|year=1991}} It states:{{harvnb|Gaitsgory|Lurie|2019|loc=Chapter 5: The Trace Formula for Bun_G(X), p. 260}} if G is a smooth affine group scheme with semisimple connected generic fiber, then

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

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

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

Notes

References

{{citation

| last = Heinloth | first = Jochen

| editor-last = Schmitt | editor-first = Alexander

| contribution = Lectures on the moduli stack of vector bundles on a curve

| contribution-url = https://www.uni-due.de/~hm0002/Artikel/StacksCourse_v2.pdf

| doi = 10.1007/978-3-0346-0288-4_4

| isbn = 978-3-0346-0287-7

| mr = 3013029

| pages = 123–153

| publisher = Birkhäuser/Springer | location = Basel

| series = Trends in Mathematics

| title = Affine flag manifolds and principal bundles

| year = 2010}}

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/.
{{citation

| last1 = Gaitsgory | first1 = Dennis

| last2 = Lurie | first2 = Jacob

| isbn = 978-0-691-18214-8

| mr = 3887650

| publisher = Princeton University Press | location = Princeton, NJ

| series = Annals of Mathematics Studies

| title = Weil's conjecture for function fields, Vol. 1

| url = https://people.math.harvard.edu/~lurie/papers/tamagawa-abridged.pdf

| volume = 199

| year = 2019}}