In algebraic geometry, given a smooth projective curve X over a finite field F q {\displaystyle \mathbf {F} _{q}} and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by Bun G ( X ) {\displaystyle \operatorname {Bun} _{G}(X)} , is an algebraic stack given by: for any F q {\displaystyle \mathbf {F} _{q}} -algebra R,
Bun G ( X ) ( R ) = {\displaystyle \operatorname {Bun} _{G}(X)(R)=} the category of principal G-bundles over the relative curve X × F q Spec R {\displaystyle X\times _{\mathbf {F} _{q}}\operatorname {Spec} R} .In particular, the category of F q {\displaystyle \mathbf {F} _{q}} -points of Bun G ( X ) {\displaystyle \operatorname {Bun} _{G}(X)} , that is, Bun G ( X ) ( F q ) {\displaystyle \operatorname {Bun} _{G}(X)(\mathbf {F} _{q})} , is the category of G-bundles over X.
Similarly, Bun G ( X ) {\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 Bun G ( X ) {\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 Bun G ( X ) {\displaystyle \operatorname {Bun} _{G}(X)} .
In the finite field case, it is not common to define the homotopy type of Bun G ( X ) {\displaystyle \operatorname {Bun} _{G}(X)} . But one can still define a (smooth) cohomology and homology of Bun G ( X ) {\displaystyle \operatorname {Bun} _{G}(X)} .
Basic properties
It is known that Bun G ( X ) {\displaystyle \operatorname {Bun} _{G}(X)} is a smooth stack of dimension ( g ( X ) − 1 ) dim G {\displaystyle (g(X)-1)\dim G} where g ( X ) {\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 π 0 ( Bun G ( X ) ) {\displaystyle \pi _{0}(\operatorname {Bun} _{G}(X))} is in a natural bijection with the fundamental group π 1 ( G ) {\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 Bun G ( X ) {\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
# Bun G ( X ) ( F q ) = q dim Bun G ( X ) tr ( ϕ − 1 | H ∗ ( Bun G ( X ) ; Z l ) ) {\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 Z l {\displaystyle \mathbb {Z} _{l}} of l-adic integers is viewed as a subring of C {\displaystyle \mathbb {C} } .
- ϕ {\displaystyle \phi } is the geometric Frobenius.
- # Bun G ( X ) ( F q ) = ∑ P 1 # Aut ( P ) {\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.
- tr ( ϕ − 1 | V ∗ ) = ∑ i = 0 ∞ ( − 1 ) i tr ( ϕ − 1 | V i ) {\displaystyle \operatorname {tr} (\phi ^{-1}|V_{*})=\sum _{i=0}^{\infty }(-1)^{i}\operatorname {tr} (\phi ^{-1}|V_{i})} for a graded vector space V ∗ {\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
- 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
See also
References
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 https://web.archive.org/web/20130411033546/http://www.math.harvard.edu/~lurie/283notes/Lecture2-FunctionFields.pdf ↩
Heinloth 2010, Proposition 2.1.2 - 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 https://www.uni-due.de/~hm0002/Artikel/StacksCourse_v2.pdf ↩
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 /wiki/ArXiv_(identifier) ↩
Heinloth 2010, Proposition 2.1.2 - 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 https://www.uni-due.de/~hm0002/Artikel/StacksCourse_v2.pdf ↩
Behrend, Kai A. (1991), The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles (PDF) (PhD thesis), University of California, Berkeley http://www.math.ubc.ca/~behrend/thesis.pdf ↩
Gaitsgory & Lurie 2019, Chapter 5: The Trace Formula for BunG(X), p. 260 - 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 https://people.math.harvard.edu/~lurie/papers/tamagawa-abridged.pdf ↩