Moduli stack of principal bundles explained
In algebraic geometry, given a smooth projective curve X over a finite field
and a smooth
affine group scheme G over it, the
moduli stack of principal bundles over
X, denoted by
, is an
algebraic stack given by: for any
-algebra
R,
\operatorname{Bun}G(X)(R)=
the category of
principal G-bundles over the relative curve
.In particular, the category of
-points of
, that is,
\operatorname{Bun}G(X)(Fq)
, is the category of
G-bundles over
X.
Similarly,
can also be defined when the curve
X is over the field of complex numbers. Roughly, in the complex case, one can define
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
.
In the finite field case, it is not common to define the homotopy type of
. But one can still define a (
smooth)
cohomology and
homology of
.
Basic properties
It is known that
is a smooth stack of dimension
where
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.
[1] If G is a split reductive group, then the set of connected components
\pi0(\operatorname{Bun}G(X))
is in a natural bijection with the fundamental group
.
The Atiyah–Bott formula
See 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
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
\#\operatorname{Bun}G(X)(Fq)=
| \dim\operatorname{Bun |
| q | |
| G(X)} |
\operatorname{tr}(\phi-1
| *(\operatorname{Bun} |
| |H | |
| G(X); |
Zl))
where (see also
Behrend's trace formula for the details)
- l is a prime number that is not p and the ring
of
l-adic integers is viewed as a subring of
.
is the
geometric Frobenius.
\#\operatorname{Bun}G(X)(Fq)=\sumP{1\over\#\operatorname{Aut}(P)}
, the sum running over all isomorphism classes of
G-bundles on
X and convergent.
\operatorname{tr}(\phi-1|V*)=
(-1)i\operatorname{tr}(\phi-1|Vi)
for a
graded vector space
, 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
- 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/.
Further reading
See also
Notes and References
- see Theorem 2.5
