Simpson's construction of varieties with many local systems
aa r X i v : . [ m a t h . AG ] O c t SIMPSON’S CONSTRUCTION OF VARIETIES WITH MANYLOCAL SYSTEMS
DONU ARAPURA
To Steve Zucker
One of the goals of this note is to say something about the fundamental group ofa smooth complex projective variety in terms of the quantity of local systems onit. Given a finitely generated group Γ, let d N (Γ) be the dimension of the spaceof irreducible rank N representations. The number d (Γ) coincides with the firstBetti number, so one may think of d N (Γ) as a nonabelian generalization. The basicproblem is to see how these numbers behave when Γ is the fundamental group of asmooth projective variety X . In this case, these numbers are always even [A]. If X is a curve of genus at least two, or even if it maps onto such a curve, then d N (Γ) > N . If X is an abelian variety, then d (Γ) > d N (Γ) = 0 for all N > d = 0 butsome higher d N >
0. Some cheap examples are given in the first section. However,they are not very interesting in the sense that they are very close to the exampleswe already know. In the second section I will turn to a beautiful construction dueto Carlos Simpson [S], which also produces smooth projective varieties such that d N ( π ( X )) > N >
1. In fact, the real purpose of this article is to makeSimpson’s construction a bit more accessible and explicit, with the hope that theseexamples will be studied more thoroughly in the future. Some specific problemsare suggested in the last section.1.
Representation varieties
For Γ a group with generators g , . . . , g n , an element of Hom (Γ , GL N ( C )) isgiven by n matrices subject to the relations of the group. In this way, the setbecomes an affine scheme of finite type, called the representation “variety”. (Forthe present purposes, a scheme will be identified with the set of its closed points.)The algebraic group GL N ( C ) acts on the representation variety by conjugation,and the GIT quotient M (Γ , N ) = Hom (Γ , GL N ( C )) //GL N ( C ):= Spec O ( Hom (Γ , GL N ( C ))) GL N ( C ) can be identified with the set of isomorphism classes of semisimple representationsof rank N [LM]. This is often called the character variety. Let M (Γ , N ) irred ⊂ M (Γ , N )denote the possibly empty open subset of irreducible representations. We havequasifinite (i.e. set theoretically finite to one) morphisms M (Γ , N ) × M (Γ , N ) → M (Γ , N + N ) Partially supported by the NSF . given by direct sum. We can decompose(1) M (Γ , N ) = [ N + ... + N r = N im M (Γ , N ) irred × . . . × M (Γ , N r ) irred Let d N (Γ) = dim M (Γ , N ) irred where we take it to be zero if it is empty. From (1), we obtain: Lemma 1.1.
We have dim M (Γ , N ) = max N + ... + N r = N d N (Γ) + . . . + d N r (Γ) Therefore dim M (Γ , N ) > if and only if d M (Γ) > for some M ≤ N . We have M (Γ ,
1) = dim
Hom (Γ , C ∗ ), therefore d (Γ) = rank Γ / [Γ , Γ]. For higher N , these numbers are usually very difficult to calculate, although there are someeasy cases. We have d N (Γ) = 0 when N > d N (Γ) > N . Thisremark applies to the fundamental group of a smooth projective curve of genus atleast two.When Γ = π ( X ) is the fundamental group of a smooth projective variety X ,Hodge theory tells us that d (Γ) = dim H ( X ) is even. More generally, nonabelianHodge theory implies that M (Γ , N ) irred carries a quaternionic or hyperk¨ahler struc-ture, therefore every d N (Γ) is even [A, thm 3.1]. Here is the example promised inthe introduction. Theorem 1.2.
There exists a smooth projective variety X with d ( π ( X )) = 0 and d N ( π ( X )) ≥ d for any given N > and d > .Proof. Let C → P be a cyclic cover of the form y N = f ( x ), where f has distinctroots. Let x denote one of the roots. By choosing deg f sufficiently large, we canassume that the genus g of C is greater than or equal to d . The group G = Z /N Z will act on C with C/G ∼ = P . If follows that H ( C, Q ) G = 0. Consequently,if γ ∈ G denotes a generator, it will act nontrivially on H ( C, Z ). By Serre [Se,prop 15], there exists a simply connected variety Y on which G acts freely. Let X = ( C × Y ) /G , where G acts diagonally. The projection X → Y /G is a fibrationwith fibre C and section given by y ( x , y ). Therefore we have split exactsequence 1 → π ( C ) → π ( X ) ← → G → H ( G, H ( π ( C ) , Q )) → H ( π ( X ) , Q ) → H ( G, H ( π ( C ) , Q ))As noted above, the group on the right vanishes. Since G is finite, the group on theleft also vanishes. Therefore H ( π ( X ) , Q ) = 0, which means that d ( π ( X )) = 0.Let ρ ∈ Hom ( π ( C ) , C ∗ ) = ( C ∗ ) g be a one dimensional character. For a genericchoice of ρ , the characters ρ, ρ ◦ γ, . . . ρ ◦ γ N − are all distinct. Let C ρ denote the C [ π ( C )]-module associated to ρ . The induced representation V ρ = Ind C ρ gives arank N C [ π ( X )]-module. As an C [ π ( C )]-module(2) V ρ = C ρ ⊕ C ρ ◦ γ ⊕ . . . IMPSON’S CONSTRUCTION OF VARIETIES WITH MANY LOCAL SYSTEMS 3 and γ acts by cyclically permuting the factors. It follows easily that V ρ is anirreducible π ( X )-module for generic ρ . Also by computing characters, using (2),we see that V ρ ∼ = V ρ ′ only if ρ ′ ∈ { ρ, γρ, . . . } . Therefore the map ρ V ρ is aquasifinite morphism from an open subset of ( C ∗ ) g to M ( π ( X ) , N ) irred . Thus d N ( π ( X )) ≥ g ≥ d . (cid:3) The drawback of this method is that it does not produce any really new examplesof fundamental groups of smooth projective varieties. I will describe a more subtleconstruct in the next section, but first I want to record the following useful factwhich was used implicitly above.
Lemma 1.3.
Suppose that Γ ⊂ Γ is a subgroup of index r < ∞ . (a) If W ρ is a nontrivial (i.e. nonconstant) family of representations in M (Γ , N ) ,then the restrictions Res W ρ give a nontrivial family in M (Γ , N ) . (b) Conversely if
Res W ρ is a nontrivial family, then so is W ρ . (c) If V ρ is a nontrivial family of representations in M (Γ , N ) , then Ind V ρ isa nontrivial family in M (Γ , rN ) Proof.
The first two items are the content of lemma 1.5 of [S]. For (c), we havethat Res(Ind V ρ ) = V ρ ⊕ . . . is nontrivial, so Ind V ρ is nontrivial by (b). (cid:3) Simpson’s construction
Let Z be a smooth projective variety with dimension 2 n +1 ≥ Z ⊂ P K such that O Z (1) is sufficiently amplein the sense that it is a high enough power of a given ample bundle. Sufficientampleness is needed for the proofs of proposition 2.1 and theorem 2.4. Let P ⊂ ˇ P K be a general linear subspace of the dual space of dimension d ≥
2. Then we canform the incidence variety Y = { ( z, H ) ∈ Z × P | z ∈ H } with projections and inclusions labelled as follows Y f / / π (cid:15) (cid:15) ι ❋❋❋❋❋❋❋❋ ZP Z × P F O O Π o o Denote the fibre of π over t by X t . Let D = { t ∈ P | X t is singular } be the(reduced) discriminant. The following standard fact is stated in [DL] and variousother places. A proof, assuming sufficient ampleness, can be found in [S, prop 6.1]. Proposition 2.1.
The discriminant D is an irreducible hypersurface and for ageneric dimensional plane Q ⊂ P , the singularities of D ∩ Q are nodes and cusps. The next step is to form a double cover branched over D . If g ( x , . . . , x d ) = 0 isan affine equation of D , then the cover y = g may acquire additional ramificationat infinity. It is better to control this in advance by defining D = ( D if deg D is even D + D otherwise, where D is a hyperplane in general position DONU ARAPURA
Let U = P − D . Let p ′ : X ′ → P be the double cover branched along D . As ascheme X ′ = Spec (cid:18) O P ⊕ O P (cid:18) deg D (cid:19)(cid:19) where the sheaf in parantheses is made into an algebra in the standard way (cf [EV,p 22]). This will usually be singular but the singularities are normal local completeintersections. The singular set X ′ sing ⊆ Σ = p ′− D sing . Let q : X → X ′ be adesingularization which is an isomorphism on the complement of Σ. This varietyis what we are after. It is very similar to, although not identical to, Simpson’sconstruction in [S, lemma 6.3]. The difference is that Simpson’s variety is a branchedcover of P of indeterminate degree, on which, by design, the local systems V ρ constructed below extend. This makes it simpler for the purpose of constructinglocal systems. However, the lack of explicitness makes it harder to do precisecomputations. Theorem 2.2.
The first Betti number of X is zero. For some M > , d M ( π ( X )) > . The rest of this section will be devoted to the proof of this theorem.
Proposition 2.3.
The first Betti number of X is zero.Proof. By Hodge theory, the proposition is equivalent to H ( X, O X ) = 0. We provethe last equation by induction on d starting with d = 2. In this case, Σ consistsof a finite set of singular points. The local analytic germ of X ′ at p ∈ Σ is eitherof the form y = x x or y = x − x . These are the well known singularitiesof type A n for n = 1 , H ( X, O X ) = H ( X ′ , O X ′ ). The last group H ( X ′ , O X ′ ) ∼ = H ( P , O P ) ⊕ H ( P , O P (deg D/ d >
2, choose a general hyperplane H ⊂ P . By the Bertini, G = p − H is smooth. By induction, we can assume that H ( G, O G ) = 0. We have an exactsequence H ( X, O X ( − G )) → H ( X, O X ) → H ( X, O G ) = 0The first group H ( X, O X ( − G )) = H d − ( X, ω X ( G )) is zero by the Kawamata-Viehweg vanishing theorem [EV, p 49]. (cid:3) We turn to the second part of theorem. By assumption Z carries a positivedimensional family of rank one local systems. Fix a generic such system C ρ , andconsider the sheaf V ρ = coker( R n Π ∗ ( F ∗ C ρ ) ι ∗ → R n π ∗ ( f ∗ C ρ )) | U This is a local system of some rank
N >
1. The stalk of V ρ over t is the primitive n th cohomology of X t with coefficients in C ρ . The rank N is just the dimension ofthis space. Let R ρ : π ( U ) → GL N C denote the representation corresponding to V ρ . Theorem 2.4 (Simpson [S, thm 5.1]) . As ρ varies, V ρ gives a nontrivial familyin M ( π ( U ) , N ) . IMPSON’S CONSTRUCTION OF VARIETIES WITH MANY LOCAL SYSTEMS 5
The proof is rather involved, so we will be content to make a few brief commentsabout it. The key ingredient is nonabelian Hodge theory, which sets up a correspon-dence between semisimple local systems and certain Higgs bundles, which for ourpurposes can be viewed as sheaves on the cotangent bundle. Simpson then checksthat as the ρ vary, the supports of the Higgs bundles corresponding to V ρ , calledspectral varieties, also vary nontrivially. When Z is an abelian variety, there is amore elementary argument which avoids Higgs bundles [S, p 358], and this alreadysuffices for constructing nontrivial examples.Let γ be a loop going once around a smooth point D . This involves a choice,but any two choices are conjugate because D is irreducible. We have(3) R ρ ( γ ) = I by the Picard-Lefschetz formula or see [S, lemma 6.5]. Let γ be a loop around D when it exists. Then(4) R ρ ( γ ) = I because V ρ extends to a local system on P − D . Let p = p ′ ◦ q and ˜ U = p − U .We can identify ˜ U = p ′− U ⊂ X ′ . This is an ´etale double cover of U correspondingto an index two subgroup π ( ˜ U ) ⊂ π ( U ). This subgroup contains γ i . We canidentify π ( X ′ − Σ) with the quotient of π ( ˜ U ) by the normal subgroup generatedby the γ i . Combining this with (3) and (4) yields Lemma 2.5.
The pullback of the local system p ′∗ V ρ extends to X ′ − Σ . Let X ′ Q = X ′ ∩ p ′∗ Q where Q ⊂ P is a general 2-plane. Lemma 2.6. π ( X ′ ) ∼ = π ( X ′ Q ) Proof.
Since X has local complete intersection singularities, we can apply the Lef-schetz theorem of [FL, p 28] to deduce the above isomorphism. (cid:3) To simplify notation, replace Σ by its restriction to X ′ Q . Then Σ consists of afinite set of points. For each p ∈ Σ, let L p denote the link which is the boundary ofa small contractible neigbourhood of p . The group π ( X ′ ) = π ( X ′ Q ) is the quotientof π = π ( X ′ Q − Σ) by the normal subgroup N generated by S p π ( L p ). For anygroup Γ, let K (Γ) = ker[Γ → b Γ]where b Γ is the profinite completion. This can also be characterized as the inter-section of all finite index subgroups, or as the smallest normal subgroup for whichΓ /K (Γ) is residually finite. Lemma 2.7.
There exists a normal subgroup of finite index Γ ⊆ π such that π ( L p ) ∩ Γ ⊆ K ( π ) for each p ∈ Σ .Proof. As noted above, Σ consists of a finite set of singular points of type A or A . These singularities can also be described as quotients of ( C ,
0) by an action of Z / Z or Z / Z [D]. Therefore π ( L p ) must either be Z / Z or Z / Z and in particularfinite. Since π/K ( π ) is residually finite, we can find a finite index subgroup ¯Γ of itavoiding the nonzero elements im( π ( L p )). Let Γ be the preimage. (cid:3) Let Ψ ⊆ π ( X ′ ) denote the image of Γ. Lemma 2.8.
DONU ARAPURA (a) If ¯Γ and ¯ N denote the images of Γ and N in π/K ( π ) , then ¯Γ ∩ ¯ N = 1 . (b) Γ /K ( π ) ∼ = Ψ /K ( π ( X ′ )) .Proof. Item (a) follows immediately from lemma 2.7. The canonical map Γ /K ( π ) → Ψ /K ( π ( X ′ )) is clearly surjective. The kernel is ¯Γ ∩ ¯ N . So (b) follows from (a). (cid:3) Lemma 2.9.
The restriction of R ρ to Γ is the pull back of a representation of Ψ .Proof. By a theorem of Malˇcev [M, p 309], any finitely generated linear group isresidually finite. Therefore the restriction Res V ρ = R ρ | Γ factors through Γ /K ( π ) ∼ =Ψ /K ( π ( X ′ )). (cid:3) To finish the proof of theorem 2.2, observe that by the above results, the restric-tion Res V ρ comes from a Ψ-module W ρ . We can form the induced π ( X ′ )-moduleInd W ρ . This corresponds to a nontrivial family of semisimple local systems on X ′ by lemma 1.3, which pulls back to a nontrivial family on X . Therefore by lemma 1.1 d M ( π ( X )) > M . By proposition 2.3, M >
1, and this concludes theproof. 3.
Problems
I will end by discussing a few follow up problems.
Problem 3.1.
Determine (a presentation for) the fundamental group of X , con-structed in section two, for some explicit choice of Z ⊂ P K , such as when it is anabelian variety. My hope is that this will give a genuinely new and interesting example of agroup in the class of fundamental groups of smooth projective varieties. It is clearthat it would differ from most of the standard known examples which either havepositive first Betti number or are rigid in the sense that all d N = 0. Furthermore π ( X ) would be different from the examples constructed in section one. Simpson’sarguments [S] show that in his terminology that X , with the local system I = q ∗ Ind W ρ above, has the nonfactorization property N F . This means that I isnot the pull back of a local system on a curve even if we allow X to be replacedby another variety mapping surjectively to it. This will imply that π ( X ) cannotcontain the fundamental group of a curve as a subgroup of finite index. Problem 3.2.
Find an example of a smooth projective variety with an infinitefamily of irreducible unitary representations which do not come from curves, i.e.that satisfy
N F This is equivalent to asking for a variety with an infinite family of stable vectorbundles, with vanishing Chern classes, which do not come from curves. This canbe rephrased as asking for an infinite family of stable Higgs bundles of the abovetype with zero Higgs fields. Simpson’s construction described above yields Higgsbundles with nonzero Higgs fields. This is clear from his proof of theorem 2.4.For applications to the fundamental group, it suffices to stick with dimension d = 2. One reason for allowing d > Problem 3.3.
Study the birational geometry of these varieties.
IMPSON’S CONSTRUCTION OF VARIETIES WITH MANY LOCAL SYSTEMS 7
For instance, although they have zero first Betti number, I suspect that theybehave like varieties with large Albanese. One way to try to make this precise isby using the notion of Shafarevich maps in the sense of Campana and Koll´ar [K].In most cases, I suspect that this map should be birational. This would be ananalogue of the Albanese map being generically finite.
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