HENDRIK AND PIETRO (NOTES TAKEN BY HIRO, EDITED BY
HENDRIK)
2011 TALBOT WORKSHOP
1. Hendrik
I’ll be talking for the first half of this hour.
I’ll be constructing MB , MDol , MDR , MHod . All of them.
I don’t want to make any pretense of being an expert in this. The ratio of minutes to
moduli spaces is not very favorable.
Okay, so. A preliminary remark I wanted to make: I’ll be making these as coarse
moduli schemes, instead of a fine moduli space to get a algebraic stack. Why? The
reason is that we’ll be talking about complex analytic equivalences between these moduli
spaces, so you want to be talking about something like a scheme rather than a stack.
I need a couple definitions.
1.1. Notions of (semi)stability. If I have X a scheme over C and projective, and E
a coherent sheaf oN X, I’m going to give a generalization of semistability.
We will say that E is µ-semistable if the slope of E, which is defined to be the degree
of E divided by the rank of E
slopepE q degpE q
rkpE q.
These are given as two coefficients of the polynomial of the Hilbert polynomial. If that
doesn’t help you, just think of a vector bundle.
If I have F a coherent subsheaf of E, then I’d like the slope of F to be ¤ the slope of
E. When this holds, I will call E µ-semistable. If it’s strictly less, we’ll have stable.
This is just a generalization of what you saw earlier today.
For p-semistable. Same thing, except now we require
ppF, nq
rkpF q
¤ prkpE,pEnqq
for n ¡¡ 0, where p are the respective Hilbert polynomials.
This is now not just two coefficients of the Hilbert polynomial, but sees more structure.
Now we say E is of pure dimension d if dimpSupppF qq d dimpSupppE qq where
supp is the set where the stalk is non-zero. So it means if I take the dimension of the
support of a subsheaf, the dimension doesn’t shrink.
Now, if I have a functor from schemes to sets, called Y 7 , then a scheme Y universally
corepresents Y 7 if there’s a map Y 7 Ñ Y , a natural transformation (thinking of Y as a
functor) then if V ever maps to V , then the fiber product
V
Y Y 7
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2011 TALBOT WORKSHOP
V co-represents the fiber product. We’re talking about things as contravariant functors.
Whenever a scheme maps into one, then the functor I get is represented by that
scheme. It’s the coarse moduli space I’m interested in.
If I have G a reductive group acting on X, I can form X 7 {G7 , set theoretic quotient
inducing ‘quotients on functors.’ We say that this map of schemes
X
ÑY
is a universal categorical quotient if Y universally corepresents X 7 {G7 . This is the sense
in which I’ll be taking quotient; not stack. If a group acts on a scheme, and I want to
put an equivalent relation and mod out by orbits, I’ll look for the universal categorical
quotient rather than the actual stack quotient you might want. This is your coarse
moduli space.
(Carlos: Representing and corepresenting are different: Maps to, or from, scheme to
the functor.
Corepresenting is that any map form a scheme to a stack factors through the coarse
moduli space.)
I’m going to define a functor, the moduli functor for semistable schemes.
Let X {S be a projective scheme over S. Let M 7 pOX , P q be the functor
(If S 1 Ñ S, I’ll call X 1 the base change i.e., the pullback.)
which takes S 1 Ñ S to the set of semistable sheaves on X 1 with Hilbert polynomial p
and pure dimension d.
This is the functor; I’d like to talk about how to construct the coarse moduli space.
It takes in S 1 Ñ S and it spits out the semistable sheaves on X 1 with a given Hilbert
polynomial and degree.
I need a starting point on how to universally represent this guy. Use Grothendieck.
It’s a scheme. It’s a theorem: There exists a scheme, Hilbpw, pq where W is a coherent
sheaf on X and p is a polynomial. Such that if I have σ : S 1 Ñ S, then the S 1 -valued
points of the Hilbert scheme are quotients of sheaves like this
σw
ÑF Ñ0
on X 1 , with F flat over S 1 and Hilbert polynomial p.
So the fact that there exists a scheme, I’ll take for granted. This will get us close to
the moduli functor that we want.
Take N P Z and W OX pN q and v CP pN q . I have HilbpV b W, P q. It turns
out that we can choose a subset Q2 € HilbpV b W, P q of this Hilbert scheme which
corepresents the functor which eats pS 1 Ñ S q and you spit out (you need N sufficiently
large, for various reasons, but we can’t talk about it here) pairs pE, αq where E is coherent
on X 1 of pure dimension d and α is an isomorphism
α : OX
1
b V H 0pX 1{S 1, E pN qq.
Here it represents, it’s sill a fine moduli space. This is the direct image.
But now if I want E p-semistable, I don’t need these α. How do I just get semi-stable
guys?
Well, SLpV q acts on Q2 by actin on the α by acting on this V .
HENDRIK AND PIETRO (NOTES TAKEN BY HIRO, EDITED BY HENDRIK)
3
So I want to say two things—(1) there’s a natural map from Q2 , as a functor, the the
moduli space M 7 pOX , P q ’m interested in, and (2) This descends to the quotient
Q72 {SLpV q7
Ñ M 7 pO X , P q.
because SLpV q just acts on the α.
Claim: φ is a local isomorphism. i.e., they induce the same sheafification on the etale
topology. So it means that these two things are universally corepresented by the same
scheme.
Now use some GIT black magic; Carlos shows that Q2 , inside the Hilbert scheme
(more or less), is the set of semistable points for the action, sot he GIT black magic that
I don’t fully understand says that there is a coarse moduli space for this quotient.
THe point is that GIT gives me the existence of this moduli space I want, after I do
Hilbert’s theorem about existence of the Hilbert scheme.
Now I get all three of these guys
MDol , MDR , MHod
in a similar way. And then I’ll move onto the Betti moduli space, which is more concrete.
Same/similar ideas work for the moduli, instead of the functor M which was semistable
sheaves with a fixed Hilbert polynomial, they give you moduli of semistable Λ-modules.
This Λ is a sheaf of rings of differential operators. A Λ-module is a sheaf of modules
over this sheaf of rings.
(1) For instance, if Λ is just DX {S , relative differential operators in the usual sense,
then modules over this are just vector bundles with flat connection. That fact that Λ
acts means you get ∇2 0.
This is a D-modules things if you like.
(2)If I instead taking Λ SympT pX {S qq, the symmetric powers of the tangent bundle,
I get Higgs bundles. The fact that Λ acts will get you φ2 0.
I can’t talk about this too much, but I’d love to.
(3) THe last example is λ-connections. for this, we again take Λ to be the relative
differentials, except instead of X {S we have X A1 over S A1 . The coordinate on A1
scales part of your connection.
I talked about how to get coars emoduli space for semistable sheaves; you can do the
same for semistable Λ-modules. Depending on what Λ is, you get out moduli of vector
bundles with flat connections, moduli of Higgs bundles, and moduli of λ-connections.
I know I’m leaving out details, but that’s the nature of talking about all these moduli
spaces in so many minutes.
Question: Are we talking about sheaves here or vector bundles? Well we’ll be taking
O-coherent sheaves here.
Carlos: For number (3), λ-connections, we need to modify things a little bit to scale
the differential part to 0. This is to take the Rees
I should say that the fiber over 0, if I think of this as a scheme over A1 , the fiber over
0 is the moduli of Higgs bundles, and over 1 is vector bundles with flat connections.
So the ring in (3) is exactly a deformation from (1) to (2).
(2) is Gr of the filtration on differential operators.
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2011 TALBOT WORKSHOP
2.
Let me talk about Betti now, just a little bit.
Let Γ be a finitely generated group. I have generators γ1 , . . . , γk and some relations
w, then the moduli space of its representations
HompΓ, GLpn, Cqq
this set RpΓ, N q of representations not up to isomorphisms. But GLpn, Cq acts on this
space by simultaneous conjugation of the image in GLn pCq. The orbits are isomorphism
classes of representations.
SO if I want to get a space, a moduli space for isomorphism classes of these representations, I need to take this quotient RpΓ, nq{G.
First, RpΓ, nq is clearly a subscheme of GLpn, Cqk . It just consists of elements
m1 , . . . , mk sch that rpm1 , . . . , mk q 1 for all words in w. These are all polynomial
conditions. So yes, RpΓ, nq is a subvariety of GLpn, Cqk .
Now I have a fine moduli space for R which is a crazy thing that has distinct points
for the same isomorphism class of representations.
The main theorem is: There exists a map
RpΓ, nq Ñ M pΓ, nq
which universally corepresents the quotient by GLn pCq.
So I get a coarse moduli space for the isomorphisms classes of representations.
One thing: If A is the coordinate ring for RpΓ, nq
A k rRpΓ, nqs
, let B to be the GLnC invariants of the coordinate ring.
B
AGL
n
pCq
Then you get M pΓ, nq SpecpB q. I just take Spec of the invariants of this guy’s
coordinate ring.
Now if X is smooth and projective over C, fix x P X and let Γ π1 pX an , xq. Then
MB pX, nq M pΓ, nq
because modding out, we’re modding out by the same thing. So this doesn’t depend on
the base point.
This is the Betti moduli space of X over C, it’s a bit harder to construct the moduli
space over an arbitrary S.
3. pietro
I should talk about riemann-hilbert corresondence, an isomrophism between the betti
moduli space for the fundmental gropu of a variety, together with moduli of flat bundles.
I should also talk about some symplectic structure you can put on the Betti moduli space.
I’m going to talk about the simple case when X is a Riemann surface.
HENDRIK AND PIETRO (NOTES TAKEN BY HIRO, EDITED BY HENDRIK)
5
4. Symplectic structure on MB for X a curve
TO do this, we shoudl compute the tangent space to MB . We can do it in a general
setting for Γ a gropu and a reductive aglebraig gropu G.
Fix B a bliniear form on the lie algebra of G such that it is in variant under the
adjoint action of G.
How should we compute teh tangent space? Well before going to the moduli space
(the quotietn), we have this affine scheme. And if we choose a represntation
φ P RpΓ, Gq
and let φt be some path at φ so φ0
The equivariance means
φ.
φt pxq expptupxq
So
upxy q upxq
...qφpxq.
Adφpxqupy q.
This is amp u : Γ Ñ g and we have φ : Γ Ñ G Ñ g (last map via adjoint action) and we
denote by gAdφ the Γ representation coming from this φ.
Then we have gropu ohomolgoy C 1 pΓ, gAd φ q to just be maps from Γ Ñ g with boundary map
δupx, y q upxy q upxq upy q.
This is the defiition o the cochain gorpu for gropu cohomology. This definition simply
says that u is a closed element.
u P Z 1 pΓ, gAdφ q.
Now, φt contained in a G-orbit,
implies that u P B 1 pΓ, gAdu q
and this means that Tψ pMb pΓ, N qq H 1 pΓ, gadψ q. So the tangnt space is given by
gropu cohomology.
Now if Γ is the fundamental gropu of a Riemann surface?
In cohomology we have a cup product H i pΓ, gq H 2i pΓ, g q Ñ H 2 pΓ, Rq. Using the
pairing betwen g and g dual.
Since we work on a surface, this last thing is equal to R. Using the abelian form that
we fixed in the beginning, we can dientify g with g to make a map
H i pΓ, gq H 2i pΓ, gq
Ñ H 2pΓ, Rq.
And so now we have a map on the tangent space
H 1 pΓ, gq H 1 pΓ, gq
Ñ R.
The propositoini s that this gives a symplectic structure on the tangent space.
In general you shoudl be able to define something like this just using the Kahler form,
so for X not a curve.
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2011 TALBOT WORKSHOP
5.
Now let X̃ G{Γ be the Pφ principal G budnel over X.
Then ad Pφ , give it a flat conneciton dφ .
Then
H 1 pΓ, gAdφ q H 1 pX, ad Pφ q.
Now we just have to show that this symplectic structure is closed.
Now the sapce of all connections on P is an affine space over A1 pX, ad P q. Since it’s
affine, we can identify all tangent spaces with the vector space A1 pX, adP q. So we take
two elements there, and we take their wedge product
η, θ
ÞÑ η ^ θ P A1pX, ad P b ad P q
but then we have the killing form
B :ggÑR
and we can apply to make a map
b ad P Ñ R.
This gives us an alternating two-form out of η ^ θ.
ad P
Integrating this over the Riemann sruface X gives us a number.
If η, θ are in Z 1 pX, ad P q, then this number is exactly ω B pη, θq.
This ω B is what I defined earlier as a bracket on H 1 pΓ, gq.
This is closed because it does not depend on A.
In fac thtis defines a symplecitc structure ont eh sapce of all connections. The restriction to flat connections, then modding out, is the process fo symplectic reduction.
Sam: Is this symplectic form algebraic for the algebraic structure on the Betti moduli
space? Carlos says yes.
AND if you think about it as holmoprhic symplectic form, it’s also algebraic on the
DeRham moduli space.
You expect that if something’s algebraic, it should be algebraic on all poitns of view,
except Dolbeaut, because the isomrophism to Deolbeaut is not algebraic. The metric is
the same for everything, but the symplectic structure is different essentially.
6.
I’m two minutes overtime so let me just say what Riemann-Hilbert correspondence is,
without proving it.
MB pX {S, nq MDR pX {S, nq
This is a complex analytic equivalence.
On the left we have reps of X over S, over GLpnq, which is purely topological. i.e.,
local systems.
On the right hand side we have pE, ∇q, flat bundles.
This correspondence is given as follows: Taking a local system L, you tensor over
constant sheaf
L ÞÑ L bC OX
and the flat connection is just given by 1 b A.
HENDRIK AND PIETRO (NOTES TAKEN BY HIRO, EDITED BY HENDRIK)
7
To go the other way, we begin with a flat bundle pE, ∇q and the local system is given
by ∇-invariant sections of E. So
E∇
ts P E |∇s 0u.