Construction of MB , MDol , MDR
Hendrik Orem
Talbot Workshop, Spring 2011
Contents
1 Some Moduli Space Theory
1.1 Moduli of Sheaves: Semistability and Boundedness
1.2 Geometric Invariant Theory . . . . . . . . . . . . .
1.3 Moduli of Semistable Sheaves . . . . . . . . . . . .
1.4 Λ-Modules . . . . . . . . . . . . . . . . . . . . . . .
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with Flat Connection
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3 Betti Moduli Space
3.1 The Betti Moduli Space over Spec C . . . . . . . . . . . . . . . . . . . . . .
3.2 Interlude: Local Systems of Schemes . . . . . . . . . . . . . . . . . . . . . .
3.3 The Relative Betti Moduli Space . . . . . . . . . . . . . . . . . . . . . . . .
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2 The
2.1
2.2
2.3
Moduli of Higgs Bundles & Vector Bundles
Moduli of Λ-Modules . . . . . . . . . . . . . . . .
Moduli of Higgs Bundles . . . . . . . . . . . . . .
Moduli of Vector Bundles with Flat Connection .
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11
Some Moduli Space Theory
I will begin this section, following [4], by sketching the general theory needed to construct
moduli spaces of sheaves. This will be followed by some comments on Λ-modules, of which
vector bundles with flat connection and Higgs bundles are examples. This will set the stage
for constructing some of the moduli spaces of interest to us in these notes.
1.1
Moduli of Sheaves: Semistability and Boundedness
Throughout, X will be a projective scheme over Spec C, and we fix a very ample invertible
sheaf OX (1). The Hilbert polynomial of a coherent sheaf E is the unique polynomial p(E, x) ∈
Q[x] such that p(E, n) = dim H 0 (X, E(n)) for all n sufficiently large.
We say that a coherent sheaf E on X is of pure dimension d if, for all coherent subsheaves
F ⊂ E, we have that the dimension of the support of E and the dimension of the support
1
of F are both equal to d. Support of E here means the set of points where the stalk of E is
nonzero. The degree of the Hilbert polynomial of E is the dimension of the support of E.
In [4], Simpson defines two different notions of semistability for coherent sheaves. First,
if we write
nd−1
nd
+ ···
p(E, n) = r + a
d!
(d − 1)!
then we call r = r(E) the rank of E and a(E) the degree of E. If E is locally free (i.e.,
a vector bundle), then this coincides with the usual notion of degree. We now say that E
is µ-semistable if, for all coherent F ⊂ E, we have a(F)/r(F) ≤ a(E)/r(E). We call E
p-semistable if instead for sufficiently large n we have
p(E, n)
p(F, n)
≤
r(F)
r(E)
for all coherent F ⊂ E.
A family of coherent sheaves on X is bounded if there exists some finite type C-scheme T
and a coherent sheaf F on T ×X such that the family of sheaves is contained in {F|Spec k(t)×X :
t ∈ T a closed point}. The idea is that constructing moduli spaces for a family of sheaves
requires that the set isn’t “too large”; requiring that the family arise as restrictions of a fixed
coherent sheaf imposes this [2].
If X is projective over S, where S is a finite type C-scheme, we say that a p-semistable
sheaf E on X/S with Hilbert polynomial P is a coherent sheaf E on X which is flat over S and
which, on each fiber Xs over a closed point s ∈ S, is p-semistable with Hilbert polynomial
P and of pure dimension d. The following basic result will be needed below.
Theorem 1.1 ([4], Corollary 1.6). The set of p-semistable sheaves on X over S is bounded.
1.2
Geometric Invariant Theory
This subsection summarizes the basic ideas of Mumford’s geometric invariant theory, as
described in [4]. Let Y \ be a functor from schemes to sets, Y a scheme, and ϕ : Y \ → Y a
natural transformation (viewing Y as its functor of points); we say that Y \ is corepresented
by Y if, given any scheme W and natural transformation ψ : Y \ → W , there is a unique
map of schemes f : Y → W such that ψ = f ◦ ϕ.
A more general notion, and the one we will use to characterize the moduli spaces below,
is the notion of Y universally corepresenting the functor Y \ : if V → Y is a map of schemes,
we define the fiber product of functors
[V ×Y Y \ ](S 0 ) = V (S 0 ) ×Y (S 0 ) Y \ (S 0 ).
Now say that Y universally corepresents Y \ if V corepresents V ×Y Y \ for all schemes V → Y .
We say that a map of functors is a local isomorphism if it induces an isomorphism of
sheafifications in the étale topology. The importance of this notion for us is that, if there
exists a local isomorphism between two functors, then a scheme universally corepresents one
functor if and only if it universally corepresents the other; in constructing the moduli space
of semistable sheaves, we will first show that the desired moduli functor is locally isomorphic
to a functor with a moduli space whose construction is easier.
2
Let G be a reductive algebraic group (i.e., trivial unipotent radical) which acts on a
scheme Z (for now, take the base scheme to be Spec C). Now define the quotient functor
Y \ to be Y \ (S 0 ) = Z(S 0 )/G(S 0 ). Given a G-invariant map of schemes ϕ : Z → Y , Y is a
categorical quotient if it corepresents Y \ , and a universal categorical quotient if it universally
corepresents Y \ .
With G and Z as above, suppose that we additionally have an equivariant line bundle
L on G. Given z ∈ Z, we call z semistable if, for some n, there exists a G-invariant section
f ∈ H 0 (Z, L⊗n )G such that f (z) 6= 0 and {x : f (x) 6= 0} is affine. The following important
result will be used in the construction in § 1.3.
Theorem 1.2 ([4], Proposition 1.11). With the above notation, there exists a universal
categorical quotient ϕ : Z ss → Y of the semistable points of Z under the action of G.
1.3
Moduli of Semistable Sheaves
In these notes, we’ll be interested in a moduli space which universally corepresents the
following functor: take X to be projective over a finite type C-scheme S. Then M\ (OX , P )
is the functor which associates to an S-scheme S 0 the set of semistable sheaves on X 0 → S 0 of
pure dimension d with Hilbert polynomial P , where X 0 is obtained from X by base change
to S 0 .
Sketching the construction of the moduli space associated to this functor requires some
preliminaries. First, the Hilbert scheme Hilb(W, P ) is a scheme which represents the following functor. Fix a coherent sheaf W on X and polynomial P . Then for any σ : S 0 → S,
the S 0 -valued points of Hilb(W, P ) are the isomorphism classes of quotients on X 0 ,
σ∗W → F → 0
where F is flat over S 0 with Hilbert polynomial P . Moreover, the fiber of Hilb(W, P ) over
any closed point s ∈ S is Hilb(Ws , P ). It is a theorem of Grothendieck that this scheme is
projective over S.
Let V be a finite dimensional vector space. Then we can see that the group Sl(V ) acts
naturally on Hilb(V ⊗W, P ). This group action preserves the morphism Hilb(V ⊗W, P ) →
S.
Now we return to the functor of interest to us, M\ (OX , P ). Let X 0 be the scheme obtained
from X → S via base-change to S 0 . Define W = OX (−N ) for a fixed N , and let V = CP (N ) .
The S 0 -valued points of Hilb(V ⊗ W, P ) are the set of pairs (E, α) where E is a coherent
sheaf on X 0 which is flat over S 0 with Hilbert polynomial P and α is a morphism
α : V ⊗ OS 0 → H 0 (X 0 /S 0 , E(N ))
such that the image generates E(N ).
Define Q1 ⊂ Hilb(V ⊗W, P ) to be the subset with the property that E has pure dimension
d and is p-semistable; it is a nontrivial result which I will not prove that Q1 is open. Because
the family of p-semistable sheaves on the fibers with Hilbert polynomial P is bounded by
Theorem 1.1, we may choose the above N sufficiently large to guarantee the following:
1. Every p-semistable sheaf with Hilbert polynomial P appears as a point of Q1 ;
3
2. Every p-semistable sheaf E with Hilbert Polynomial P has H 0 (X 0 /S 0 , E(N )) locally free
over S 0 with rank P (N ), where the H 0 (X 0 /S 0 , F) denotes the direct image of the sheaf
F on the base;
3. Formation of the H 0 commutes with base change.
Given the above choice of N , we now set Q2 to be the open subset of Q1 where the map α is
an isomorphism. Summarizing, we now have that Q2 represents the functor which associates
to an S-scheme S 0 the set of all (E, α) with E p-semistable on X 0 with Hilbert polynomial P
and α : V ⊗ OS 0 ∼
= H 0 (X 0 /S 0 , E(N )).
Notice that there is an action of the corresponding functors, i.e., Sl(V )\ acts on Q\2 . Since
Q2 ⊂ Q1 , there is a natural map of functors Q\2 → M\ (OX , P ) (given an (E, α) in Q2 (S 0 ),
simply forget α); this map is invariant under the action of Sl\ (V ). Thus the map descends
to a map of quotient functors
ϕ
Q\2 /Sl(V )\ → M\ (OX , P )
where the left hand side takes S 0 to the set-theoretic quotient of Q2 (S 0 )\ by Sl\ (V ).
Theorem 1.3 ([4], Theorem 1.21 (1)). The above map ϕ is a local isomorphism.
Proof. (Sketch) By definition, we argue étale-locally. Because we may choose local frames
for H 0 (X 0 /S 0 , E(N )), the natural transformation Q\2 /Gl(V )\ → M\ (OX , P ) is a local isomorphism (i.e., the group acts by locally identifying the possible choices of frame, which is
certainly étale local). However, in this action the center Gm acts trivially on Q2 , so we in
fact have an action of PGl(V ) on Q2 .
Now note that the maps Gl(V ) → PGL(V ) and Sl(V ) → PGl(V ) are étale-locally surjective, hence the corresponding map on the quotient Q\2 /Sl(V ) → Q\2 /PGl(V ) is also a local
ismorphism. Thus we obtain the following diagram
M\ (OO X , P ) o
Q\2 /Sl(V )\
/
Q\2 /Gl(V )\
Q\2 /PGl(V )\
where we have that all but the top arrow are local isomorphisms, hence the top arrow is
also.
From our generalities on geometric invariant theory, we know that in order to construct
a scheme M(OX , P ) universally corepresenting M\ (OX , P ) it thus suffices to construct a
scheme which universally corepresents the quotient Q\2 /Sl(V )\ . In order to apply Mumford’s
results, which will give us a scheme universally corepresenting the quotient functor, we will
exhibit Q2 as the set of semistable of a scheme on which Sl(V ) acts.
In particular, let Hilb(V ⊗ W, P, d) be the closure of the points of Hilb(V ⊗ W, P ) with
the property that the sheaf E is of pure dimension d. By definition, we have Q2 as a subset
of this scheme. The desired fact now follows from a technical and lengthy calculation [4]:
4
Theorem 1.4. The scheme Q2 is the set of semistable points of Hilb(V ⊗ W, P, d) under
the action of Sl(V ).
The above results now give us the existence of M(OX , P ) as a scheme via Theorem 1.2.
I will close this section by stating some further results, whose proof requires considerable
additional work and will not be discussed (see [4], Theorem 1.21).
Theorem 1.5. The scheme M(OX , P ) is projective and its points represent equivalence
classes of semistable sheaves, where E1 ∼ E2 when the associated graded sheaves are equal,
gr(E1 ) = gr(E2 ) (i.e., M(OX , P ) parametrizes Jordan equivalence classes of semistable
sheaves).
1.4
Λ-Modules
In this section, I will define Λ-modules, which are sheaves of modules over a sheaf of rings of
differential operators. Higgs bundles are examples of Λ-modules, and the following section
will discuss the construction of the moduli space of Λ-modules in some generality. This
follows §2 of [4].
A sheaf of rings of differential operators on X over some base scheme f : X → S (taken
to be Noetherian over C) is a sheaf of OX -algebras Λ over X together with a filtration
Λ0 ⊂ Λ1 ⊂ · · · such that:
1. Λ = ∪i Λi ;
2. Λi Λj ⊂ Λi+j ;
3. The image of the given map OX → Λ is precisely Λ0 ;
4. f −1 (OS ) ⊂ OX is contained in Z(Λ);
5. The left and right OX -module structures on the quotients Gri (Λ) coincide;
6. All of the sheaves Gri (Λ) are coherent;
7. The sheaf Gr(Λ) = ⊕∞
i=0 Gri (Λ) is generated by Gr1 (Λ) in the sense that
Gr1 (Λ) ⊗OX · · · ⊗OX Gr1 (Λ) Gri (Λ).
Now we can define a Λ-module as a sheaf E of left Λ-modules on X which is coherent
as a sheaf of OX -modules. There are two examples of central importance to these notes,
described below.
We may view vector bundles with flat connection as Λ-modules under appropriate circumstances: assume that f : X → S is smooth, and take Λ = DX/S , the usual sheaf of
relative differentials. The order filtration on differentials is known to satisfy the properties
required of a sheaf of rings of differential operators (indeed, the axioms are modeled on D).
It is a basic result of the theory of D-modules that any sheaf of OX -coherent modules E over
Λ will be locally free of finite rank, i.e., the sheaf of sections of a vector bundle. The action
5
of Λ defines a map ∇ from differential operators to endomorphisms of E such that ∇2 = 0,
and thus a flat connection [1].
A related choice of Λ yields Higgs bundles as an example of Λ-modules: let Λ = ΛHiggs
denote Sym· (T (X/S)). This is related to the previous choice in that, for smooth varieties
over C, ΛHiggs is the associated graded algebra of DX/S above. Now we define Λ-modules to be
Hitchin pairs (E, ϕ) where E is an OX -coherent sheaf and ϕ : E → ΩX/S ⊗E has the additional
property that, when the matrices
P defining ϕ are written out in a local trivialization, they
pairwise commute (i.e., if ϕ =
ϕi dzi in local coordinates zi , then the ϕi commute [3]).
This latter condition is typically written ϕ ∧ ϕ = 0. ϕ is sometimes called a Higgs field. If
we impose the additional condition that E is locally free, the above Hitchen pair is called a
Higgs bundle. The realization of these as Λ-modules will be described when we address their
moduli.
2
The Moduli of Higgs Bundles & Vector Bundles with
Flat Connection
I will begin by discussing the moduli space of Λ-modules, as in [4]. As vector bundles with
flat connection and Higgs bundles are both examples of Λ-modules, this will give us existence
of both of the moduli spaces.
2.1
Moduli of Λ-Modules
Consider X projective over S, where S is finite type over C, and denote by Xs = X ×S s the
fiber over a geometric point s ∈ S. Let Λ be some sheaf of differential operators on X and
Λs the corresponding sheaf of differential operators on Xs obtained via base change.
We call a Λ-module E p-semistable (or respectively µ-semistable) if
1. E is flat over S;
2. the restrictions to geometric fibers Es are of pure dimension d and p-semistable (resp.
µ-semistable) Λs -modules;
3. each restriction Es has the same Hilbert polynomial.
A crucial fact in the construction of the moduli spaces of interest, as in the construction
of the moduli of general semistable sheaves, is the following boundedness result.
Theorem 2.1 ([4], Proposition 3.5). The set of µ-semistable Λs -modules on geometric fibers
Xs with a fixed Hilbert polynomial P is bounded.
The construction here is very similar to that given in § 1.3. Again we begin with a Hilbert
scheme with an action of Sl(V ), and construct the desired scheme as the set of semi-stable
points of a subscheme.
Theorem 2.2. Fix a polynomial P and let N be sufficiently large. There exists a scheme
Q, quasi-projective over S, which represents the functor that associates to an S-scheme S 0
6
the set of isomorphism classes of pairs (E, α) where E is p-semistable Λ-module with Hilbert
polynomial P on X 0 and α is a map
∼
α : (OS 0 )P (N ) → H 0 (X 0 /S 0 , E(N )).
The idea of the proof is to begin with the Hilbert scheme of quotients
Λk ⊗OX OX (−N )P (N ) → E → 0.
The procedure of finding appropriate subschemes of this, eventually obtaining the desired
Q, is similar to (but much more involved than) what happened in § 1.3, so I will omit it.
Let M\ (Λ, P ) denote the functor which associates to an S-scheme S 0 the set of isomorphism classes of p-semistable Λ0 -modules on X 0 over S 0 with Hilbert polynomial P . Just as
in § 1.3, one can show that the points of Q are semistable for the action of Sl(V ), so we
obtain the following.
Theorem 2.3 ([4], Theorem 4.7). The universal categorical quotient M(Λ, P ) = Q/Sl(V )
exists as a scheme and universally corepresents M\ (Λ, P ). The scheme is a quasi-projective
variety and the geometric points represent the Jordan equivalence classes of p-semistable
Λ-modules with Hilbert polynomial P on fibers Xs .
2.2
Moduli of Higgs Bundles
Let Λ = ΛHiggs . Before proceeding with the following description of Higgs bundles.
Theorem 2.4 ([5], Lemma 6.5). A Hitchin pair on X over S is the same thing as an OX coherent Λ-module on X. In particular, a Higgs bundle on X over S is the same thing as
an OX -coherent, locally free Λ-module on X.
Then Theorem 2.3 immediately gives us that the functor M\Higgs (X/S, P ), associating
to an S-scheme S 0 the set of isomorphism classes of p-semistable Hitchin pairs on X 0 over
S 0 with Hilbert polynomial P is universally corepresented by a scheme MHiggs (X/S, P ) =
M(ΛHiggs , P ). Its points parametrize Jordan equivalence classes of p-semistable Hitchin pairs
with Hilbert polynomial P on fibers Xs .
Now let’s consider the Dolbeault moduli space: let M\Dol (X/S, n) be the functor which
associates to an S-scheme S 0 the set of isomorphism classes of p-semistable Higgs bundles on
X 0 over S 0 with vanishing Chern classes and Hilbert polynomial nP0 , where P0 is the Hilbert
polynomial of OX . One can prove that the assumption that the Higgs bundles were locally
free was unnecessary (i.e., if we had defined M\Dol (X/S, n) to simply give us Hitchin pairs
rather than Higgs bundles, we would have the same functor). From this description we can
see that there exists a scheme MDol (X/S, n) ⊂ MHiggs (X/S, nP0 ) universally corepresenting
M\Dol (X/S, n). Its points correspond to direct sums of µ-stable Higgs bundles with vanishing
rational Chern classes on the fibers (µ-stable means that we take the inequality defining µsemistability to be strict); see [5], Proposition 6.6 and Corollary 6.7
7
2.3
Moduli of Vector Bundles with Flat Connection
The description of vector bundles with flat connection as Λ-modules given in § 1.4 allows us
to immediately apply our results about moduli of Λ-modules for Λ = DX/S . We thus have
the desired theorem about the functor M\DR (X/S, n), which assigns to an S-scheme S 0 the
set of isomorphism classes of vector bundles with flat connection on X 0 /S 0 of rank n:
Theorem 2.5 ([5], Theorem 6.13). If X is smooth and projective over S, then there is a
scheme MDR (X/S, n), quasi-projective over S, which universally corepresents M\DR (X/S, n).
3
Betti Moduli Space
In this section, I will sketch the construction of the Betti moduli space associated to representations of π1 (X an , x). I will follow the treatment of [5].
3.1
The Betti Moduli Space over Spec C
Given a finitely generated group Γ, we will consider the set of complex representations of
degree n:
R(Γ, n) = {Hom (Γ, Gl(n, C))}.
I claim that this has the structure of a scheme, as follows. If we write Γ as generated by
γ1 , . . . , γk modulo the relations W , then clearly R(Γ, n) is the subset of the k-fold product
of Gl(n, C) consisting of those (m1 , . . . , mk ) such that r(m1 , . . . , mk ) = 1 for each r ∈ W .
Thus we have a Zariski-closed subset of the affine scheme Gl(n, C) × · · · × Gl(n, C), hence is
also an affine scheme.
Recall that the Jordan-Hölder Theorem states that the quotients in a composition series
of a representation do not depend on the choice of composition series. Thus it makes sense to
define two representations to be Jordan equivalent if their associated graded representations
are isomorphic, i.e., if they have the same sets of composition factors.
Theorem 3.1. There exists a universal categorical quotient R(Γ, n) → M(Γ, n) under the
action of Gl(n, C). M(Γ, n) is an affine scheme of finite type over C, and its closed points
correspond to the Jordan equivalence classes of representations.
The construction, though I will not prove this statement, is given by letting M(Γ, n) =
Spec B, where B is the ring of invariants inside the coordinate ring of R(Γ, n).
Now if we consider X a connected, smooth, projective variety over C and choose x ∈ X,
then we can form the Betti representation space
RB (X, x, n) = R(Γ, n)
where Γ = π1 (X an , x). The Betti moduli space, MB (X, n) is defined analogously. It does
not depend on the choice of basepoint, as change of basepoint corresponds to conjugation,
compatible with the action of Gl(n, C).
8
3.2
Interlude: Local Systems of Schemes
A local system of schemes Z over a topological space T is a functor from the category of
schemes over C to the category of sheaves of sets over T satisfying the following condition:
there is a covering ∪α Uα = T so that on each Uα , the action of Z on open subsets V of Uα
is represented by a scheme Z(V ) (that is, for fixed V ⊂ Uα , Z(V ) is a functor from schemes
to sets, and we require this to be representable), and if W ⊂ V ⊂ Uα , the restriction map
Z(V ) → Z(W ) is an isomorphism. We can thus be assured that the stalks
Zt = lim Z(V )
−→
t∈V
will be schemes.
With basic point-set and covering space topology results, one may prove the following. I
will sketch the direction which will be used later.
Theorem 3.2 ([5], Lemma 6.2). π1 (T, t) acts on Zt by C-scheme automorphisms. If T is
connected and locally simply connected, then Z 7→ Zt defines an equivalence of categories
between the category of local systems of schemes over T and the category of schemes with an
action of π1 (T, t).
Proof. (Sketch) If we have π1 (T, t) acting on a scheme Zt and T has a universal cover T̃ ,
then form the constant local system Z̃ on T̃ given by Z̃(U )(S) = Zt (S) for U open and
connected. For U open now in the base T , we set Z(U )(S) to be the invariants under the
action of π1 (T, t) inside Z̃(Ũ )(S); here Ũ denotes the inverse image of U under the covering
map. All restriction maps are defined to be the identity, so we obtain a local system of
schemes Z on T , as desired.
We may define the total analytic space Z (an) of a local system of schemes as follows.
Given a local system of schemes Z on T , choose an open covering of T = ∪Ti such that Z|Ti
is a constant local system of stalk Zi . For each overlap Ti ∩ Tj , we have an isomorphism
Zi ∼
= Zj (obtained via comparison with the stalk of Z at a point in the intersection); these
isomorphisms obey the cocycle compatibility condition with respect to triple overlaps, so
we may form Z (an) by gluing together Zian × Ti along these relations (here Zian denotes the
complex analytic space corresponding to Zi ).
3.3
The Relative Betti Moduli Space
We now generalize the construction in §3.1 to the situation where X is smooth and projective
over S, a finite type C-scheme. Let S and the fibers Xs be connected, and choose basepoints
t ∈ S and x ∈ Xt . Denote by Γ the group π1 (Xtan , x). I will describe an action of π1 (S an , t)
on the scheme MB (Γ, n) which, by Theorem 3.2, gives us a local system of schemes over S an ;
this local system will give us the desired relative moduli space.
Let Aut(Γ) denote the group of automorphisms, Inn(Γ) its inner automorphisms, and
Out(Γ) = Aut(Γ)/Inn(Γ). Define a map π1 (S an , t) → Out(Γ) as follows. First note that the
map f an corresponding to f : X → S is a fibration. Given a loop σ : [0, 1] → S an based at
9
t ∈ S, we obtain another fibration σ ∗ (X an ) over [0, 1]:
[0, 1] × X an = σ ∗ (X an )
[0, 1]
σ
/
X an
Xt
f an
/ S an
t
It is a fibration over [0, 1], hence trivial. As σ is a loop based at t, the fibers over 0
and 1 give two copies of Xt × {point}, and hence a self-homeomorphism of Xt . We thus
get an action of π1 (S an , t) on π1 (Xt , x) which is well-defined up to inner automorphism (the
self-homeomorphism of Xt might move the basepoint x, but change of basepoint corresponds
to an inner automorphism). This gives the desired map π1 (S an , t) → Out(Γ).
Notice that Aut(Γ) acts on RB (Xt , x, n) for each n, and that this descends to an action of
Inn(Γ) on MB (Xt , n), as we obtain MB from RB via modding out by conjugation. Combining
this with the above observation, we have an action of π1 (S an , t) on MB (Γ, n), which, by
Theorem 3.2, gives a local system of schemes MB (X/S, n) over S an . We call this local system
of schemes the relative version of the Betti moduli space. It has the property (justifying its
status as a moduli space) that
MB (X/S, n)s = MB (Xs , n),
i.e., the stalk of this local system of schemes over s ∈ S is the (usual, non-relative) Betti
moduli space of the fiber Xs .
10
References
[1] R. Hotta, K. Takeuchi, and T. Tanisaki. D-modules, perverse sheaves, and representation
theory. Springer, 2008.
[2] D. Huybrechts and M. Lehn. The geometry of moduli spaces of sheaves. Cambridge Univ
Pr, 2010.
[3] C.T. Simpson. Higgs bundles and local systems. Publications mathématiques, 75:5–95,
1992.
[4] C.T. Simpson. Moduli of representations of the fundamental group of a smooth projective
variety I. Publications mathématiques de l’IHES, 79(1):47–129, 1994.
[5] C.T. Simpson. Moduli of representations of the fundamental group of a smooth projective
variety. II. Publications Mathématiques de l’IHÉS, 80(1):5–79, 1994.
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