Characteristic classes of bundles of K3 manifolds and the Nielsen realization problem
aa r X i v : . [ m a t h . A T ] D ec CHARACTERISTIC CLASSES OF BUNDLES OF K3 MANIFOLDS ANDTHE NIELSEN REALIZATION PROBLEM
JEFFREY GIANSIRACUSA, ALEXANDER KUPERS, AND BENA TSHISHIKU
Abstract.
Let K be the K3 manifold. In this note, we discuss two methods to prove that certaingeneralized Miller–Morita–Mumford classes for smooth bundles with fiber K are non-zero. Asa consequence, we fill a gap in a paper of the first author, and prove that the homomorphismDiff( K ) → π Diff( K ) does not split. One of the two methods of proof uses a result of Frankeon the stable cohomology of arithmetic groups that strengthens work of Borel, and may be ofindependent interest. Introduction
In this paper K denotes the K manifold , which is the underlying oriented manifold of a complex K C P cut out by the homogeneous equation z + z + z + z = 0. For each element c ∈ H i ( B SO(4); Q ), there is a characteristic class κ c of smooth oriented manifold bundles withfiber K , called a generalized Miller–Morita–Mumford class : given such a bundle E → B we takethe vertical tangent bundle T v E and integrate the class c ( T v E ) ∈ H i ( E ; Q ) over the fibers to get κ c ( E ) ∈ H i − ( B ; Q ).Let Diff( K ) denote the group of orientation-preserving C -diffeomorphisms, in the C -topology.Its classifying space B Diff( K ) carries a universal smooth manifold bundle with fiber K , and hencethere are classes κ c ∈ H ∗ ( B Diff( K ); Q ) which may or may not be zero. Letting L = (7 p − p )denote the second Hirzebruch L -polynomial, we prove the following: Theorem A.
The generalized Miller–Morita–Mumford-class κ L ∈ H ( B Diff( K ); Q ) is nonzero. The Hirzebruch L-polynomials are related to signatures of manifolds and as a corollary of TheoremA, there exists a smooth bundle of K K K ) to be the group π Diff( K ) of path components ofDiff( K ). Theorem B.
The surjection p : Diff( K ) → Mod( K ) does not split, i.e. there is no homomor-phism s : Mod( K ) → Diff( K ) so that p ◦ s = Id . This is an instance of the Nielsen realization problem; see e.g. [MT18]. Theorem B first appearedin [Gia09], but the proof was flawed (see [Gia19]). However, it can be repaired with smallmodifications and many of the ideas in this paper derive from [Gia09].
Acknowledgements.
The authors thank H. Grobner and M. Krannich, as well as the anonymousreferees, for helpful comments. The third author thanks B. Farb for introducing him to the paper[Gia09].
Date : December 16, 2019.2000
Mathematics Subject Classification.
Key words and phrases.
Characteristic classes, K3 surfaces, arithmetic groups, cohomology.
Table 1.
The class κ L i +1 ( X d ) in terms of the class λ for i ≤ i κ L i +1 ( X d ) 24 8 λ λ λ
45 8 λ
315 16 λ λ λ λ Quasi-polarized K surfaces Suppose that π : E → B is an oriented manifold bundle with closed fibers of dimension d . Thishas a vertical tangent bundle T v E with corresponding characteristic classes c ( T v E ) ∈ H i ( E ; Q )for each c ∈ H i ( B SO( d ); Q ). The generalized Miller–Morita–Mumford classes are obtained byintegration of these classes along the fibers: κ c ( E ) := Z π c ( T v E ) ∈ H i − d ( B ; Q ) . Applying this construction to the universal bundle of K B Diff( K ) results inclasses κ c ∈ H i − ( B Diff( K ); Q ) for each c ∈ H i ( B SO(4); Q ) = Q [ e, p ].These classes are natural in the bundle: for any continuous map f : B ′ → B , κ c ( f ∗ E ) = f ∗ κ c ( E ).To prove κ c = 0 ∈ H ∗ ( B Diff( K ); Q ), it therefore suffices to find a single bundle E → B suchthat κ c ( E ) = 0.We shall use the moduli space M d of quasi-polarized K d (the value of d plays no role in our arguments). This is actually a stack with finite automorphism groupsof bounded order, but since we are interested in its rational cohomology we may ignore thesetechnical details. We shall not go into the details of its construction, but recall some facts from[vdGK05, Pet16]. There is a universal family π : X d → M d of K t i := c i ( T v X d ) ∈ H i ( X d ; Q ).The class t is the pullback of a class λ ∈ H ( M d ; Q ). The main result of [vdGK05] is that λ = 0 but λ = 0, in the Chow ring of M d . Petersen gives the corresponding result in rationalcohomology [Pet16], and attributes it to van der Geer and Katsura. We shall use this to provethe following improvement of Theorem A: Proposition 1.
The generalized Miller–Morita–Mumford-class κ L i +1 ∈ H i ( B Diff( K ); Q ) isnon-zero for i ≤ .Proof. It suffices to prove that κ L i +1 ( X d ) = 0. Since the K p , p are the only non-zero Pontryagin classes of the vertical tangent bundle. These can be expressedin terms of the Chern classes using [MS74, Corollary 15.5]: p ( T v X d ) = t − t and p ( T v X d ) = t . We substitute these into the first nine Hirzebruch L -polynomials, as computed by McTague[McT14]. Since integration along fibers is linear, it suffices to compute R π t i t j . As t = π ∗ λ ,the push-pull formula gives λ i R π t j , and [vdGK05, Section 3] used Grothendieck–Riemann–Rochto determine that R π t j = a j − λ j − for particular integers a j − . Using this, we compute that κ L i +1 ( X d ) is a non-zero multiple of λ i for 1 ≤ i ≤ (cid:3) Example . Let us do the computation for i = 3 as an example: L = − p + 22 p p − p p i with i ≥ L ( T v X d ) = − t + 24 t t − t t + 8 t t + 21 t κ L i +1 ( X d ) = Z π L ( T v X d ) = 24 λ · − λ · λ + 8 λ · λ + 21 · λ λ . HARACTERISTIC CLASSES OF BUNDLES OF K3 MANIFOLDS 3
Remark . The classes κ L i +1 remain non-zero when pulled back to H i ( B Diff( K rel ∗ ); Q ), be-cause H ∗ ( B Diff( K ) , Q ) → H ∗ ( B Diff( K rel ∗ ) , Q ) is injective: its composition with the Becker–Gottlieb transfer is given by multiplication with χ ( K ) = 24. We do not know whether κ L i +1 remains non-zero when pulled back to H i ( B Diff( K rel D ); Q ).3. Miller–Morita–Mumford classes and the action on homology
One can also approach the group of diffeomorphisms of K through its action on H ( K ; Z ). Inparticular, we shall explain a relationship between the generalized Miller–Morita–Mumford classesand the arithmetic part of the mapping class group.The middle-dimensional homology group H ( K ; Z ) ∼ = Z has intersection form given by M = H ⊕ H ⊕ H ⊕ − E ⊕ − E , with H the hyperbolic form and − E the negative of the E -form.This is equivalent over R to the symmetric (22 × B = (cid:18) I − I (cid:19) , where I n is the ( n × n ) identity matrix. In particular, we can consider Aut( M ) as a subgroup ofthe Lie group O(3 , K ) on H ( K ; Z ) preserves the intersection form and hence induces a homo-morphism α : Mod( K ) → Aut( M ), whose image Γ K is the index 2 subgroup of Aut( M ) of thoseelements such that the product of the determinant and the spinor norm equals 1, cf. [Gia09, § L -polynomials L i ∈ H i ( B SO; Q ), whose pullback to H i ( B SO(4); Q ) we shall denote in the same manner, can beobtained from the arithmetic group Γ K . We will now justify this claim.There are homomorphismsΓ K −→ Aut( M ) −→ O(3 , ≃ ←− O(3) × O(19) . Thus we get, up to homotopy, a map w : B Γ K −→ B O(3) × B O(19) which classifies a bundle η with fibers M ⊗ R , which decomposes as a direct sum η + ⊕ η − of a 3- and a 19-dimensionalsubbundle. We define a class x i := w ∗ (ph i ⊗ − ⊗ ph i ) ∈ H i ( B Γ K ; Q ) , where ph i denotes the degree 4 i component of the Pontryagin character.By definition x i is pulled back from B O(3) × B O(19), but it is in fact pulled back from B O(3)[Gia09, Proposition 2.2]. By Chern–Weil theory the Pontryagin classes of the flat bundle η vanish[MS74, Corollary C.2]. This implies ph( η + ) + ph( η − ) = 0, and thus x i = ph i ( η + ) − ph i ( η − ) =2 ph i ( η + ), which is evidently pulled back along B Γ K −→ B O(3) × B O(19) π −→ B O(3) . Lemma 4.
The pullback of x i ∈ H i ( B Γ K ; Q ) along the map B Diff( K ) → B Γ K is equal to / i +1 κ L i +1 ∈ H i ( B Diff( K ); Q ) .Proof. Atiyah proved that x i ∈ H i ( B Γ K ; Q ) pulls back to κ ˜ L i +1 ∈ H i ( B Diff( K ); Q ) alongthe map B Diff( K ) → B Γ K [Ati69, § L i +1 is the Atiyah–Singer modification of theHirzebruch L -polynomials: while the latter has generating series √ z/ tanh( √ z ), this modificationhas generating series √ z/ tanh( √ z/ i +1 ˜ L i +1 = L i +1 . (cid:3) Let Γ
Ein < Γ K be the index 2 subgroup of those elements such that both the determinant andthe spinor norm are 1; it has index 4 in Aut( M ) and is the maximal subgroup contained inthe identity component of O(3 , (3 ,
19) in O(3 , B Γ Ein −→ B SO (3 , ≃ ←− B SO(3) × B SO(19) π −→ B SO(3) . JEFFREY GIANSIRACUSA, ALEXANDER KUPERS, AND BENA TSHISHIKU
To understand the induced map H ∗ ( B SO(3); Q ) → H ∗ ( B Γ Ein ; Q ), we introduce the space X u = SO(22)SO(3) × SO(19) . In Section 4 we shall discuss the Matsushima homomorphism µ : H ∗ ( X u ; C ) −→ H ∗ ( B Γ Ein ; C ) . The principal SO(3)-bundle SO(22) / SO(19) → X u is classified by a 39-connected map X u → B SO(3) that factors over the map X u → B SO(3) × B SO(19). By [Gia09, Lemma 3.4] (a specialcase of [Bor77, Proposition 7.2]), the Matsushima homomorphism fits in a commutative diagram(1) H ∗ ( B SO(3) × B SO(19); C ) H ∗ ( X u ; C ) H ∗ ( B SO(3); C ) H ∗ ( B Γ Ein ; C ) . µ Changing coefficients to the complex numbers and pulling back x i from B Γ K to B Γ Ein , we get x i ∈ H i ( B Γ Ein ; C ). From this we will conclude: Lemma 5.
The class x i ∈ H i ( B Γ Ein ; C ) is in the image of the Matsushima homomorphism.Proof. The argument preceding Lemma 4 tells us that in the commutative diagram H ∗ ( B SO(3); C ) H ∗ ( B Γ Ein ; C ) H ∗ ( B O(3); C ) H ∗ ( B Γ K ; C ) , the element x i ∈ H ∗ ( B Γ K ; C ) is pulled back from B O(3), and hence x i ∈ H ∗ ( B Γ Ein ; C ) ispulled back from B SO(3). The results then follows from the commutative diagram (1). (cid:3) Results of Franke and Grobner
In this section we explain a result about the Matsushima homomorphism, which implies:
Proposition 6.
The homomorphism H ∗ ( B SO(3); C ) → H ∗ ( X u ; C ) → H ∗ ( B Γ Ein ; C ) is injectivein degrees ∗ ≤ . Let G be a connected semi-simple linear algebraic group over Q . The real points G ( R ) form asemi-simple Lie group. Fix maximal compact subgroups K < G ( R ) and U < G ( C ) with K ⊂ U ,let Y ∞ := G ( R ) /K be the symmetric space of G , and X u := U/K be the compact dual symmetricspace of G . Fixing an arithmetic lattice Γ < G ( Q ), by work of Matsushima and Borel [Mat62,Bor74] there is a homomorphism H ∗ ( X u ; C ) → H ∗ (Γ \ Y ∞ ; C ) constructed using differential forms.Since Γ acts on the contractible space Y ∞ with finite stabilizers, H ∗ (Γ \ Y ∞ ; C ) ∼ = H ∗ ( B Γ; C ). Weshall call the composition(2) µ : H ∗ ( X u ; C ) −→ H ∗ (Γ \ Y ∞ ; C ) ∼ = H ∗ ( B Γ; C )the Matsushima homomorphism . It may be helpful to point out that µ in general is not inducedby a map of spaces, since it does not preserve the rational cohomology as a subset of the complexcohomology [Bor77, Oku01]. Example . The Matsushima homomorphism discussed in the previous section is a particularinstance of this. In this case G = SO(3 , X u as in the previous section. In thisparticular instance µ does preserve the rational cohomology in the range ∗ ≤
39, as a consequenceof the commutative diagram 1.Borel [Bor74] proved that the Matsushima homomorphism is an isomorphism in a range of degrees,and by work of Franke [Fra08] it is injective in a larger range.
HARACTERISTIC CLASSES OF BUNDLES OF K3 MANIFOLDS 5
Theorem 8 (Franke) . The homomorphism (2) is injective in degrees (3) ∗ ≤ min R dim N R , where R ranges over maximal parabolic subgroups of G over Q , and N R ⊂ R is the unipotentradical. This is not stated explicitly in [Fra08], but a similar statement is given in [Gro13], as we nowexplain. We require the following additional setup (see [FS98], [LS04], [SV05] or [Har19, § adelic symmetric space Y A and the adelic locally symmetric space X A by Y A := Y ∞ × G ( A f ) and X A := G ( Q ) \ Y A , where A f is the ring of finite adeles of Q . The (sheaf) cohomology H ∗ ( X A ; C ) can be identifiedwith the colimit colim H ∗ ( X A /K f ; C ), where K f ⊂ G ( A f ) ranges over open compact subgroups.Each X A /K f is a finite disjoint union F i Γ i \ Y ∞ with Γ i < G ( Q ) an arithmetic lattice. Definition 9.
The automorphic cohomology of G is given by(4) H ∗ ( G ; C ) := colim H ∗ ( X A /K f ; C ) . In this framework, there is a map [Gro13, pg. 1062](5) Ψ : H ∗ ( g , K ; C ) −→ H ∗ ( G ; C )where H ∗ ( g , K ; C ) is relative Lie algebra cohomology with trivial coefficients. The constructionof the Matsushima homomorphism (2) passes through the isomorphism H ∗ ( X u ; C ) ∼ = H ∗ ( g , K ; C )[Oku01, § § H ∗ ( g , K ; C ) H ∗ ( G ; C ) H ∗ ( X u ; C ) H ∗ ( B Γ; C ) , Ψ ∼ = µ with right vertical induced by the map B Γ → Γ \ Y ∞ ֒ → F i Γ i \ Y ∞ = X A /K f for suitable K f .We will see that Theorem 8 follows the following result regarding the homomorphism (5). Proposition 10.
The homomorphism (5) is injective in degrees ∗ ≤ min R dim N R . The proposition follows from [Fra08, Gro13]. There is a small amount of work needed to translatethe results of these papers to our setting.
Proof.
First we explain a weaker statement: (5) is injective in degrees ∗ < min R dim N R . This is proved directly in [Gro13], building on [Fra98, FS98, Fra08]. We explain only what isneeded for our argument, and refer to [Gro13] for more details. The cohomology H ∗ ( G ; C ) canbe identified with relative Lie algebra cohomology H ∗ ( G ; C ) = H ∗ ( g , K ; A ( G )), where A ( G ) isa space of automorphic forms [Gro13, Introduction]. (Comparing with Grobner’s notation, weremark that since G is semisimple in our case, the quotient m G in [Gro13] is just the Lie algebra g ; furthermore, since we are only interested in the trivial representation E = C , we will write A ( G ) instead of A J ( G ).)By [Fra98] and [FS98], there is a decomposition A ( G ) = M { P } M φ P A { P } ,φ P ( G ) , and hence also H ∗ ( G ; C ) = M { P } M φ P H ∗ (cid:0) g , K ; A { P } ,φ p ( G ) (cid:1) , where { P } ranges over (associate classes) of Q -parabolic subgroups and φ P ranges over (associateclasses) of cuspidal automorphic representations of the Levi subgroups of elements of { P } ; see Although including this argument is not strictly necessary, this statement is already sufficient for Theorem Band the argument illustrates the connection between the Matsushima homomorphism and automorphic forms.
JEFFREY GIANSIRACUSA, ALEXANDER KUPERS, AND BENA TSHISHIKU [Gro13, § A ( G ) corresponding to P = G is denoted A Eis ( G ), and thecorresponding subspace H ∗ Eis ( G ; C ) ⊂ H ∗ ( G ; C ) is called the Eisenstein cohomology . The constantfunctions span a trivial sub-representation 1 G ( A ) ⊂ A Eis ( G ). This defines a map H ∗ ( g , K ; C ) → H ∗ ( G ; C ), which is precisely the map (5). Necessarily 1 G ( A ) is contained in a unique summand A { P } ,φ P ( G ). Then by [Gro13, Cor. 17], the induced map H ∗ ( g , K ; C ) → H ∗ (cid:0) g , K ; A { P } ,φ P ( G ) (cid:1) is injective in a range 0 ≤ ∗ < q res , where the constant q res = q res ( P, φ P ) is defined in [Gro13, § § q res is equal tothe constant q max = min R dim N R defined in [Gro13, § G is definedover Q , which as only one place, so the sum in Grobner’s definition of q max has only one term).Next we explain how to deduce from [Fra08] that (5) is injective for ∗ ≤ min R dim N R .We define H ∗ c ( G ; C ) to be the colimit of the compactly supported cohomology groups H ∗ c ( X A /K f ; C ).Using Poincar´e duality for each of the symmetric spaces Γ i /Y ∞ , the map Ψ : H ∗ ( X u ; C ) ∼ = H ∗ ( g , K ; C ) → H ∗ ( G ; C ) has a dual mapΨ ′ : H ∗ c ( G ; C ) −→ H ∗ ( X u ; C )on compactly supported cohomology. Then ker(Ψ) = Im(Ψ ′ ) ⊥ , where the orthogonal complementis with respect to the cup product ⌣ on H ∗ ( X u ; C ). Franke [Fra08] gives a precise descriptionof Im(Ψ ′ ). To describe it, fix a minimal parabolic P < G , and consider a parabolic subgroup R ⊃ P . Write R = M AN for the Langlands decomposition, where M is semi-simple, A isabelian, and N is unipotent. When we vary R , we write M R , N R for emphasis. The compactdual symmetric space of M , denoted X M , embeds in X u . Franke proves that Im(Ψ ′ ) = ker(Φ),where Φ : H ∗ ( X u ; C ) −→ Y H ∗ ( X M ; C )is the map induced by the inclusions X M ֒ → X u , ranging over R = M AN maximal parabolicsubgroups (maximal is equivalent to dim A = 1). See [Fra08, (7.2) pg. 59] and also [SV05, § ⊥ . To show that Ψ is injective in low degrees, we use the following observation: if V ⊥ ⊂ ker(Φ) forsome subspace V ⊂ H ∗ ( X u ; C ), then ker(Ψ) = ker(Φ) ⊥ ⊂ V . This implies that Ψ is injective indegrees ∗ < min = v ∈ V deg( v ).Fix R , and consider the inclusion i : X M → X u . For k ≥
1, observe that a ∈ H k ( X u ; C ) belongsto ker( i ∗ ) if and only if a ⌣ PD( i ∗ ( z )) = 0 for every z ∈ H k ( X M ; C ). Here PD( · ) denotesPoincar´e duality. Then V ⊥ ⊂ ker(Φ), where V ⊂ H ∗ ( X u ; C ) is defined as the image of M R M k ≥ H k ( X M R ; C ) i ∗ −→ H ∗ ( X u ; C ) PD −−→ H ∗ ( X u ; C ) , where L R ranges over maximal parabolic subgroups containing P as before. Observe thatclasses in H ∗ ( X M ; C ) of low dimension map to classes in H ∗ ( X u ; C ) of low codimension . Thusif v ∈ V , then deg( v ) ≥ dim X u − dim X M for each M . Therefore, Ψ is injective in degrees ∗ < min R (cid:0) dim X u − dim X M R (cid:1) .Finally, we show the minimum codimension of X M ⊂ X u is equal to 1 + min R dim R . Thisfollows quickly from the Iwasawa decomposition for a semi-simple Lie group and Langlandsdecompositions for a parabolic subgroup. By the Iwasawa decomposition, we can write G = KAN ,where K is maximal compact. For our maximal parabolic R , we have R = M A R N R , andfurthermore, since M is semisimple, it has an Iwasawa decomposition M = K M A M N M . Observethat dim X u = dim AN , dim X M = dim A M N M , and dim AN = dim A M N M + dim A R N R . Thendim X u − dim X M = dim A R N R = 1 + dim N R . This completes the proof. (cid:3)
Proof of Theorem 8.
For any x ∈ H ∗ ( g , K ; C ) in the given range, by the injectivity of Ψ andthe description (4) of H ∗ ( G ; C ) as a colimit, there is an arithmetic lattice Γ ′ < G ( Q ) so thatΨ( x ) is in the image of H ∗ (Γ ′ ; C ) → H ∗ ( G ; C ), as in (6). By transfer, the same is true for any HARACTERISTIC CLASSES OF BUNDLES OF K3 MANIFOLDS 7 further finite-index subgroup of Γ ′ . Then since H ∗ ( g , K ; C ) is degree-wise finite-dimensional, inthe desired range (5) provides an injective map H ∗ ( g , K ; C ) → H ∗ (Γ ′ ; C ) for some arithmeticlattice Γ ′ ≤ G ( Q ). Any arithmetic lattice Γ ≤ G ( Q ) is commensurable to Γ ′ , and hence Γ and Γ ′ have a common finite index subgroup Γ ′′ . Consider the commutative diagram H ∗ ( B Γ ′ ; C ) H ∗ ( g , K ; C ) H ∗ ( B Γ ′′ ; C ) .H ∗ ( B Γ; C )By a transfer argument the top composition is injective in the desired range, and hence so is H ∗ ( g , K ; C ) → H ∗ ( B Γ; C ), proving that (5) and hence (2) is injective in the desired range. (cid:3) In the remainder of this section we compute Franke’s constant min R dim N R for G = SO( p, q ).We also compute Franke’s constant for G = Sp g and G = SL n , since these are examples ofcommon interest.4.1. Special orthogonal groups.
Fix 1 ≤ p ≤ q , set d = q − p , and consider the algebraicgroup SO( B ) := { g ∈ SL p + q | g t Bg = B } , where B is the ( p + q ) × ( p + q )-matrix given by B = (cid:18) I p − I q (cid:19) . The associated compact dual symmetric space is X u = SO( p + q ) / (SO( p ) × SO( q )), whose coho-mology H ∗ ( X u ; C ) can be computed using [McC01, Theorem 8.2]. Proposition 11.
Fix a finite-index subgroup Γ ≤ SO( B ; Z ) . Then the Matsushima homomor-phism H ∗ ( X u ; C ) → H ∗ ( B Γ; C ) is injective in degrees ∗ ≤ p + q − .Proof. By the preceding discussion, it suffices to provemin R dim N R = p + q − , where R ranges over a maximal parabolic subgroups over Q , and N R is the unipotent radical.Parabolic subgroups of SO( B ; R ) are stabilizers of isotropic flags in ( R p + q , B ). A maximal para-bolic subgroup is specified by a single non-trivial isotropic subspace. Let e , . . . , e p , f , . . . , f q bethe standard basis for R p + q (whose Gram matrix is B ). Denoting u i = e i + f i , let R k < SO( B ; R )be the stabilizer of W = R { u , . . . , u k } for 1 ≤ k ≤ p . Every maximal parabolic subgroup isconjugate to some R k .Fix 1 ≤ k ≤ p . An element of R k preserves the flag 0 ⊂ W ⊂ W ⊥ ⊂ R p + q . The unipotent radical N k ⊂ R k is the subgroup that acts trivially on each of the quotients W/ W ⊥ /W , R p + q /W ⊥ .To determine dim N k , denote v i = e i − f i for 1 ≤ i ≤ p , and work in the ordered basis u , . . . , u k , u k +1 , . . . , u p , f p +1 , . . . , f q , v k +1 , . . . , v p , v , . . . , v k . Then g ∈ N k can be expressed as a block matrix g = I k y z I p + q − k x I k , where y = − x t Q and z + z t = x t Qx and Q is the ( p + q − k ) × ( p + q − k ) matrix Q = I p − k I q − p I p − k . JEFFREY GIANSIRACUSA, ALEXANDER KUPERS, AND BENA TSHISHIKU
The homomorphism N k ∋ g x ∈ R k ( p + q − k ) has kernel the space of skew-symmetric matrices z t = − z , so dim N k = k ( p + q − k ) + k ( k − . For 1 ≤ k ≤ p , this number is smallest when k = 1,which gives the constant claimed in the theorem. (cid:3) Proof of Proposition 6.
Since M , the intersection form of the K B over R with p = 3 and q = 19. Thus when we apply Theorem 8, the same estimates as inProposition 11 holds. Thus the map H ∗ ( X u ; C ) → H ∗ ( B Γ Ein ; C ) → H ∗ ( B Γ; C ) is injective for ∗ ≤
20 and hence so is H ∗ ( X u ; C ) → H ∗ ( B Γ Ein ; C ). (cid:3) Symplectic groups.
We next specialize Theorem 8 to finite index subgroups of symplecticgroups. Take G = Sp n to be the algebraic group defined bySp n := { g ∈ SL n | g t J n g = J n } , where J n is the 2 n × n matrix given by J n := (cid:18) I n − I n (cid:19) . The associated compact dual symmetric space is X u = Sp( n ) / U( n ), whose cohomology in therange below is the polynomial algebra on generators c , c , c , . . . with | c i | = 2 i . Proposition 12.
For any finite-index subgroup Γ ≤ Sp n ( Z ) the Matsushima homomorphism H ∗ ( X u ; C ) → H ∗ ( B Γ; C ) is injective in degrees ∗ ≤ n − .Proof of Proposition 12. The proof follows from Theorem 8 similar to Proposition 11. Let e , . . . , e n , f , . . . , f n be the standard symplectic basis for R n . Let R k be the maximal parabolic subgroupof Sp n defined as the stabilizer of W = R { e , . . . , e k } for 1 ≤ k ≤ n . Working in the ba-sis e , . . . , e k , e k +1 , . . . , e n , f k +1 , . . . , f n , f , . . . , f k , an element of the unipotent radical N k can beexpressed as a block matrix g = I k y z I n − k x I k , where y = x t J ′ and z − z t = y t J ′ y and J ′ = J n − k . It follows that dim N k = 2 k ( n − k ) + k + k ( k − .For 1 ≤ k ≤ n , this number is smallest when k = 1. (cid:3) The tautological ring of A g . Let A g denote the moduli space of principally polarized abelianvarieties. The tautological ring R ∗ CH ( A g ) ⊂ CH ∗ ( A g ; Q ) in the Chow ring is the subalgebragenerated by the λ -classes λ i ∈ CH i ( A g ; Q ), the Chern classes of the Hodge bundle (the 2 g -dimensional vector bundle given at an abelian variety X ∈ A g by the tangent space to its identityelement). Van der Geer proved it has a Q -basis given by the monomials λ a λ a · · · λ a g − g − with a i ∈ { , } [vdG99, Theorem (1.5)] [vdG13, § R ∗ H ( A g ) ⊂ H ∗ ( A g ; Q ) in rational cohomology defined as the subalgebra generated by the λ -classes. In the literature it is claimed van der Geer’s computation also holds in cohomology,but no reference for this is known to the authors. We provide a proof below: Theorem 13.
The tautological ring R ∗ H ( A g ) ⊂ H ∗ ( A g ; Q ) has a Q -basis given by the monomials λ a λ a · · · λ a g − g − with a i ∈ { , } .Proof. R ∗ CH ( A g ) surjects onto R ∗ H ( A g ), so it suffices to prove they have the same dimension.The space A g is the quotient of the contractible Siegel upper half space H g by Sp g ( Z ). Thisaction has finite stabilizers, so there is an isomorphism H ∗ ( A g ; Q ) ∼ = H ∗ (Sp g ( Z ); Q ). Under theisomorphism H ∗ ( A g ; C ) ∼ = H ∗ (Sp g ( Z ); C ), R ∗ H ( A g ) ⊗ C is exactly the image of the Matsushimahomomorphism [vdG13, § u g ) in H ∗ ( X u ; Q ) ∼ = Q [ u , · · · , u g ]((1 + u + u + · · · + u g )(1 − u + u − · · · + ( − g u g ) − . HARACTERISTIC CLASSES OF BUNDLES OF K3 MANIFOLDS 9
This is [SV05, Lemma 8], combined with the description of H ∗ ( X u ; Q ) in [vdG99, § H ∗ ( X u ; Q ) ∼ = R ∗ CH ( A g +1 ) identifying u i with λ i . Inparticular, from the basis given above we see that the kernel of the Matsushima homomorphism isspanned by the monomials u ǫ u ǫ · · · u ǫ g − g − u g with ǫ i ∈ { , } . Thus the image of the Matsushimahomomorphism has the same dimension as R ∗ CH ( A g ), and the result follows. (cid:3) Observe this result in particular describes the image of the Matsushima homomorphism in H ∗ ( B Γ; C ) for finite-index subgroups Γ ⊂ Sp g ( Z ).4.3. Special linear groups.
Finally, we specialize Theorem 8 to finite-index subgroups of speciallinear groups. Now we have G = SL n and X u = SU( n ) / SO( n ), whose cohomology in the rangebelow is the exterior algebra on generators ¯ c , ¯ c , ¯ c , . . . with | ¯ c i | = 2 i − Proposition 14.
For any finite-index subgroup Γ ≤ SL n ( Z ) the Matsushima homomorphism H ∗ ( X u ; C ) → H ∗ ( B Γ; C ) is injective in degrees ∗ ≤ n − . The proof is similar to the proof of Propositions 11 and 12, but simpler; one identifies the maximalparabolic subgroups over Q as the stabilizers of a non-trivial subspace W and observes that thestabilizers of 1-dimensional subspaces have the smallest unipotent radical, of dimension n − A result announced by Lee.
In [Lee78, Theorem 1], Lee announced a result which in par-ticular implies that the range in Proposition 14 can be improved to ∗ ≤ n −
3. His result canbe deduced from page 61 of [Fra08], where Franke describes the kernel of the Matsushima homo-morphism for finite index subgroups of SL n ( O K ), with O K the ring of integers in a number field K : Theorem 15.
For any finite-index subgroup Γ ≤ SL n ( Z ) , the image of the Matsushima homomor-phism H ∗ ( X u ; C ) → H ∗ ( B Γ; C ) is an exterior algebra on the classes ¯ c , · · · , ¯ c n − with | ¯ c i | = 2 i − when n is odd, and an exterior algebra on the classes ¯ c , · · · , ¯ c n − when n is even.Proof. The cohomology of compact dual X u for SL n ( Z ) is given by the following exterior algebras: H ∗ ( X u ; Q ) = ( Λ(¯ c , · · · , ¯ c n ) if n is oddΛ(¯ c , · · · , ¯ c n − , e ) if n is even,with | ¯ c i | = 2 i − | e | = n . According to page 61 of [Fra08], when n is odd the kernel ofMatsushima homomorphism is the ideal generated by ¯ c n , and when n is even it is the idealgenerated by ¯ c n − and e . (cid:3) Remark . Theorem 15 resolves a question in [EVGS13, Remark 7.5]; the Borel class ¯ c is non-zero in H ( B SL n ( Z ); Q ) for n ≥
5, and the Borel class ¯ c is non-zero in H ( B SL n ( Z ); Q ) for n ≥
7. Similarly ¯ c ¯ c is non-zero in H ( B SL n ( Z ); Q ) for n ≥
7. Curiously, the non-zero classthey find in H ( B SL ( Z ); Q ) is not stable.5. Moduli of Einstein metrics
To apply our knowledge of the cohomology of arithmetic groups, we use the global Torelli theoremto study the moduli space M Ein of Einstein metrics on the K § M Ein := T Ein (cid:12) Γ Ein of a moduli space T Ein of marked Einstein metrics by the subgroup Γ
Ein ≤ Γ K . The space T Ein admits a description as a hyperplane complement, but we only use a pair of consequences of this.Fix a finite-index subgroup Γ ′ ≤ Γ K , and assume Γ ′ is contained in Γ Ein . Equivalently, onemay assume it is contained in the identity component of O(3 , Ein := α − (Γ Ein ) and Mod ′ := α − (Γ ′ ). Proposition 17.
The homomorphism H ∗ ( B Γ ′ ; C ) → H ∗ ( B Mod ′ ; C ) is injective for any Γ ′ ≤ Γ K .Proof. We will first prove that the surjection Mod( K ) → Γ K splits over Γ Ein by Giansiracusa’swork: there is a map(7) e : M Ein −→ B Diff( K ) −→ B Mod( K ) −→ B Γ K . The induced homomorphism π ( M Ein ) → Γ K is injective with image Γ Ein by the global Torellitheorem [Gia09, § K ) → Γ K splits over Γ Ein . This proves the case Γ ′ = Γ Ein ; forΓ ′ ⊂ Γ Ein one restricts the splitting to Γ ′ .If Γ ′ Γ ′ ∩ Γ Ein , then Γ ′ ∩ Γ Ein has index 2 in Γ ′ and similarly Mod ′ ∩ Mod
Ein has index 2 inMod ′ . Thus the injective homomorphism H ∗ ( B (Γ ′ ∩ Γ Ein ); C ) → H ∗ ( B (Mod ′ ∩ Mod
Ein ); C ) isone of representations of Z / ∼ = Γ ′ / (Γ ′ ∩ Γ Ein ) = Mod ′ / (Mod ′ ∩ Mod
Ein ), and we can identify H ∗ ( B Γ ′ ; C ) → H ∗ ( B Mod ′ ; C ) with the induced map on Z / Z / (cid:3) To prove Theorem A we must prove that p ∗ x = 0 ∈ H ( B Diff( K ); Q ). To do so, it suffices toprove that is non-zero when pulled back to M Ein : Proposition 18.
For the map e defined in (7) , e ∗ x = 0 ∈ H ( M Ein ; Q ) .Proof. We will prove that e ∗ : H ( B Γ K ; Q ) → H ( M Ein , Q ) is injective. In [Gia09, § T Ein −→ M
Ein = T Ein (cid:12) Γ Ein −→ B Γ Ein . Its E -page is given by E p,q = ( q is odd, Q σ ∈ ∆ q/ / Γ Ein H p ( B Stab( σ ); Q ) if q is even.The description of ∆ q/ / Γ Ein is not important here, as we shall only use the rows 0 ≤ q ≤
3. Ofthese, the following are non-zero: for q = 0 we get H p ( B Γ K ; Q ), and for q = 2 we get a productof the cohomology groups of groups Γ commensurable with O(2 , Z ) or O(3 , Z ). For suchgroups H (Γ; Q ) vanishes [Mar91, Corollary 7.6.17], and thus there can not be any non-zerodifferential into the entry E , . (cid:3) Nielsen realization
We now deduce Theorem B from either Proposition 1 or 6. The argument in fact shows thatDiff( K ) → Mod( K ) does not split over any finite index subgroup of Mod( K ). Proof of Theorem B.
We will show that Diff( K ) → Mod( K ) does not split by contradiction, sowe assume there is a splitting s : Mod( K ) → Diff( K ), which necessarily factors over the discretegroup Diff( K ) δ as Mod( K ) s δ −→ Diff( K ) δ p δ −→ Diff( K ) . Observe that x ∈ H ( B Γ K ; Q ) is non-zero; either one pulls back to B Diff( K ) and uses Propo-sition 1 and Lemma 4, or one pulls back to B Γ Ein and uses Proposition 6. By Proposition 17 itspullback to H ( B Mod( K ); Q ), which we denote by c , is also non-zero. Its pullback under B Mod( K ) s δ −→ B Diff( K ) δ p δ −→ B Diff( K ) p −→ B Mod( K )is c and hence non-zero. By Section 3 we get p ∗ c = κ L and we claim that ( p δ ) ∗ κ L ∈ H ( B Diff( K ) δ ) vanishes. This would contradict c = 0 and finish the proof. To prove theclaim, we use that B Diff( K ) δ classifies flat K -bundles, i.e. bundles with a foliation transverse tothe fibers and of codimension 4. The normal bundle to this foliation is isomorphic to the verticaltangent bundle, and by the Bott vanishing theorem its Pontryagin ring vanishes in degrees > L of degree 12 vanishes. (cid:3) HARACTERISTIC CLASSES OF BUNDLES OF K3 MANIFOLDS 11
Remark . The idea of using Bott vanishing to obstruct Nielsen realization originates in Morita’swork [Mor87, Thm. 8.1].
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