Construction of M B, M Dol, M DR

Construction of M B, M Dol, M DR Hendrik Orem Talbot Workshop, Spring 2011 Contents 1 Some Moduli Space Theory 1 1.1 Moduli of Sheaves: Semistability and Boundedness.............. 1 1.2 Geometric Invariant Theory........................... 2 1.3 Moduli of Semistable Sheaves.......................... 3 1.4 Λ-Modules..................................... 5 2 The Moduli of Higgs Bundles & Vector Bundles with Flat Connection 6 2.1 Moduli of Λ-Modules............................... 6 2.2 Moduli of Higgs Bundles............................. 7 2.3 Moduli of Vector Bundles with Flat Connection................ 8 3 Betti Moduli Space 8 3.1 The Betti Moduli Space over Spec C...................... 8 3.2 Interlude: Local Systems of Schemes...................... 9 3.3 The Relative Betti Moduli Space........................ 9 Bibliography 11 1 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 O X (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 p(e, n) = r nd d! + a nd 1 (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(f, n) r(f) p(e, n) 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 X s 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 ) = V (S ) Y (S ) Y (S ). 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 ) = Z(S )/G(S ). 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) 0 and {x : f(x) 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 (O X, P ) is the functor which associates to an S-scheme S the set of semistable sheaves on X S of pure dimension d with Hilbert polynomial P, where X is obtained from X by base change to S. 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 S, the S -valued points of Hilb(W, P ) are the isomorphism classes of quotients on X, σ W F 0 where F is flat over S with Hilbert polynomial P. Moreover, the fiber of Hilb(W, P ) over any closed point s S is Hilb(W s, 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 (O X, P ). Let X be the scheme obtained from X S via base-change to S. Define W = O X ( N) for a fixed N, and let V = C P (N). The S -valued points of Hilb(V W, P ) are the set of pairs (E, α) where E is a coherent sheaf on X which is flat over S with Hilbert polynomial P and α is a morphism α : V O S H 0 (X /S, E(N)) such that the image generates E(N). Define Q 1 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 Q 1 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 Q 1 ; 3

2. Every p-semistable sheaf E with Hilbert Polynomial P has H 0 (X /S, E(N)) locally free over S with rank P (N), where the H 0 (X /S, 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 Q 2 to be the open subset of Q 1 where the map α is an isomorphism. Summarizing, we now have that Q 2 represents the functor which associates to an S-scheme S the set of all (E, α) with E p-semistable on X with Hilbert polynomial P and α : V O S = H 0 (X /S, E(N)). Notice that there is an action of the corresponding functors, i.e., Sl(V ) acts on Q 2. Since Q 2 Q 1, there is a natural map of functors Q 2 M (O X, P ) (given an (E, α) in Q 2 (S ), simply forget α); this map is invariant under the action of Sl (V ). Thus the map descends to a map of quotient functors Q 2/Sl(V ) ϕ M (O X, P ) where the left hand side takes S to the set-theoretic quotient of Q 2 (S ) 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 /S, E(N)), the natural transformation Q 2/Gl(V ) M (O X, 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 G m acts trivially on Q 2, so we in fact have an action of PGl(V ) on Q 2. 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 (O X, P ) Q 2/Gl(V ) Q 2/Sl(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(O X, P ) universally corepresenting M (O X, 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 Q 2 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 Q 2 as a subset of this scheme. The desired fact now follows from a technical and lengthy calculation [4]: 4

Theorem 1.4. The scheme Q 2 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(O X, 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(O X, P ) is projective and its points represent equivalence classes of semistable sheaves, where E 1 E 2 when the associated graded sheaves are equal, gr(e 1 ) = gr(e 2 ) (i.e., M(O X, 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 O X -algebras Λ over X together with a filtration Λ 0 Λ 1 such that: 1. Λ = i Λ i ; 2. Λ i Λ j Λ i+j ; 3. The image of the given map O X Λ is precisely Λ 0 ; 4. f 1 (O S ) O X is contained in Z(Λ); 5. The left and right O X -module structures on the quotients Gr i (Λ) coincide; 6. All of the sheaves Gr i (Λ) are coherent; 7. The sheaf Gr(Λ) = i=0gr i (Λ) is generated by Gr 1 (Λ) in the sense that Gr 1 (Λ) OX OX Gr 1 (Λ) Gr i (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 O X -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 = D X/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 O X -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 = Λ Higgs denote Sym (T (X/S)). This is related to the previous choice in that, for smooth varieties over C, Λ Higgs is the associated graded algebra of D X/S above. Now we define Λ-modules to be Hitchin pairs (E, ϕ) where E is an O X -coherent sheaf and ϕ : E Ω X/S E has the additional property that, when the matrices defining ϕ are written out in a local trivialization, they pairwise commute (i.e., if ϕ = ϕ i dz i in local coordinates z i, then the ϕ i 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 X s = X S s the fiber over a geometric point s S. Let Λ be some sheaf of differential operators on X and Λ s the corresponding sheaf of differential operators on X s 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 E s are of pure dimension d and p-semistable (resp. µ-semistable) Λ s -modules; 3. each restriction E s 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 Λ s -modules on geometric fibers X s 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 6

the set of isomorphism classes of pairs (E, α) where E is p-semistable Λ-module with Hilbert polynomial P on X and α is a map α : (O S ) P (N) H 0 (X /S, E(N)). The idea of the proof is to begin with the Hilbert scheme of quotients Λ k OX O X ( 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 the set of isomorphism classes of p-semistable Λ -modules on X over S 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 X s. 2.2 Moduli of Higgs Bundles Let Λ = Λ Higgs. 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 O X - coherent Λ-module on X. In particular, a Higgs bundle on X over S is the same thing as an O X -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 the set of isomorphism classes of p-semistable Hitchin pairs on X over S with Hilbert polynomial P is universally corepresented by a scheme M Higgs (X/S, P ) = M(Λ Higgs, P ). Its points parametrize Jordan equivalence classes of p-semistable Hitchin pairs with Hilbert polynomial P on fibers X s. Now let s consider the Dolbeault moduli space: let M Dol (X/S, n) be the functor which associates to an S-scheme S the set of isomorphism classes of p-semistable Higgs bundles on X over S with vanishing Chern classes and Hilbert polynomial np 0, where P 0 is the Hilbert polynomial of O X. 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 M Dol (X/S, n) M Higgs (X/S, np 0 ) 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 µ- semistability 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 = D X/S. We thus have the desired theorem about the functor M DR (X/S, n), which assigns to an S-scheme S the set of isomorphism classes of vector bundles with flat connection on X /S of rank n: Theorem 2.5 ([5], Theorem 6.13). If X is smooth and projective over S, then there is a scheme M DR (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 π 1 (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 γ 1,..., γ k modulo the relations W, then clearly R(Γ, n) is the subset of the k-fold product of Gl(n, C) consisting of those (m 1,..., m k ) such that r(m 1,..., m k ) = 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 R B (X, x, n) = R(Γ, n) where Γ = π 1 (X an, x). The Betti moduli space, M B (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 U α = T so that on each U α, the action of Z on open subsets V of U α is represented by a scheme Z(V ) (that is, for fixed V U α, Z(V ) is a functor from schemes to sets, and we require this to be representable), and if W V U α, the restriction map Z(V ) Z(W ) is an isomorphism. We can thus be assured that the stalks Z t = lim t V Z(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). π 1 (T, t) acts on Z t by C-scheme automorphisms. If T is connected and locally simply connected, then Z Z t defines an equivalence of categories between the category of local systems of schemes over T and the category of schemes with an action of π 1 (T, t). Proof. (Sketch) If we have π 1 (T, t) acting on a scheme Z t and T has a universal cover T, then form the constant local system Z on T given by Z(U)(S) = Z t (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 π 1 (T, t) inside Z(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 = T i such that Z Ti is a constant local system of stalk Z i. For each overlap T i T j, we have an isomorphism Z i = 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 Zi an complex analytic space corresponding to Z i ). 3.3 The Relative Betti Moduli Space T i along these relations (here Z an i denotes the 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 X s be connected, and choose basepoints t S and x X t. Denote by Γ the group π 1 (Xt an, x). I will describe an action of π 1 (S an, t) on the scheme M B (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 π 1 (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 ) X an X t [0, 1] f an σ 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 X t {point}, and hence a self-homeomorphism of X t. We thus get an action of π 1 (S an, t) on π 1 (X t, x) which is well-defined up to inner automorphism (the self-homeomorphism of X t might move the basepoint x, but change of basepoint corresponds to an inner automorphism). This gives the desired map π 1 (S an, t) Out(Γ). Notice that Aut(Γ) acts on R B (X t, x, n) for each n, and that this descends to an action of Inn(Γ) on M B (X t, n), as we obtain M B from R B via modding out by conjugation. Combining this with the above observation, we have an action of π 1 (S an, t) on M B (Γ, n), which, by Theorem 3.2, gives a local system of schemes M B (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 M B (X/S, n) s = M B (X s, 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 X s. 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. 11