The Hodge conjecture for self-products of certain K3 surfaces
aa r X i v : . [ m a t h . AG ] J u l THE HODGE CONJECTURE FOR SELF-PRODUCTS OF CERTAIN K3SURFACES
ULRICH SCHLICKEWEI
Abstract.
We use a result of van Geemen [vG4] to determine the endomorphism algebra ofthe Kuga–Satake variety of a K3 surface with real multiplication. This is applied to prove theHodge conjecture for self-products of double covers of P which are ramified along six lines. Introduction
Let S be a complex K3 surface, i.e. a smooth, projective surface over C satisfying H ( S, O S ) =0 and ω S ≃ O S . Let T ( S ) ⊂ H ( S, Q ) be the rational transcendental lattice of S , defined asthe orthogonal complement of the N´eron–Severi group with respect to the intersection form.The algebra E S := End Hdg ( T ( S )) of endomorphisms of T ( S ) which preserve the Hodge de-composition can be interpreted as a subspace of the space of (2,2)-classes on the self-product S × S . The Hodge conjecture for S × S predicts that E S consists of linear combinations offundamental classes of algebraic surfaces in S × S . Using the Lefschetz theorem on (1,1)-classes,it is easily seen that conversely the Hodge conjecture for S × S holds if E S is generated byalgebraic classes.Mukai [Mu1] used his theory of moduli spaces of sheaves to prove that if the Picard numberof S is at least 11, then any ϕ ∈ E S which preserves the intersection form on T ( S ) can berepresented as a linear combination of fundamental classes of algebraic cycles. Later this resultwas improved by Nikulin [N] on the base of lattice-theoretic arguments to the case that thePicard number of S is at least 5. In [Mu2], Mukai announced that using the theory of modulispaces of twisted sheaves, the hypothesis on the Picard number could be omitted.But how many isometries do exist in the algebra E S ? Results of Zarhin [Z] imply that E S isan algebraic number field, which is either totally real (we say that S has real multiplication ) ora CM field ( S has complex multiplication ). Isometries of T ( S ) correspond to elements of norm 1in E S . If S has complex multiplication, one can use the fact that CM fields are generated as Q -vector spaces by elements of norm 1 to see that E S is generated by isometries. In combinationwith Mukai’s results, this proves the Hodge conjecture for self-products of K3 surfaces withcomplex multiplication and with Picard number at least 5. This was noticed by Ram´on-Mar´ı[RM]. If S has real multiplication, the only Hodge isometries in E S are plus or minus the This work was supported by the SFB/TR 45 ‘Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’of the DFG (German Research Foundation) and by the Bonn International Graduate School in Mathematics(BIGS). identity. Thus, Mukai’s results are no longer sufficient to prove the algebraicity of interestingclasses in E S .In order to approach the case of real multiplication one passes from K3 surfaces to Abelianvarieties by associating to a K3 surface S its Kuga–Satake Abelian variety A . By construction,see [KS], there exists an inclusion of Hodge structures T ( S ) ⊂ H ( A × A, Q ). Van Geemen[vG4] studied the Kuga–Satake variety of a K3 surface with real multiplication. He discoveredthat the corestriction of a certain Clifford algebra over E S plays an important role for theKuga–Satake variety of S . We rephrase and slightly improve his result which then reads asfollows: Theorem 1.
Let S be a K3 surface with real multiplication by a totally real number field E S of degree d over Q . Let A be a Kuga–Satake variety of S .Then there exists an Abelian variety B such that A is isogenous to B d − . The endomorphismalgebra of B is End Q ( B ) = Cores E/ Q C ( Q ) . Here, Q : T × T → E S is a quadratic form on T which already appeared in Zarhin’s paper[Z] and which will be reintroduced in Section 2.4, C ( Q ) is the even Clifford algebra of Q over E S and Cores E/ Q C ( Q ) is the corestriction of this algebra. The corestriction of algebras willbe reviewed in Section 3.2.Theorem 1 leads to a better understanding of the phenomenon of real multiplication for K3surfaces by allowing us to calculate the endomorphism algebra of the corresponding Kuga–Satake varieties. However, since the Kuga–Satake construction is purely Hodge-theoretic, thisstill gives no geometric explanation. Therefore, we focus on one of the few families of K3surfaces for which the Kuga–Satake correspondence has been understood geometrically. Thisis the family of double covers of P ramified along six lines. Paranjape [P] found an explicitcycle on S × A × A which realizes the inclusion of Hodge structures T ( S ) ⊂ H ( A × A, Q ).Building on the decomposition result for Kuga–Satake varieties we derive Theorem 2.
Let S be a K3 surface which is a double cover of P ramified along six lines.Then the Hodge conjecture is true for S × S . As pointed out by van Geemen [vG4], there are one-dimensional sub-families of the familyof such double covers with real multiplication by a totally real quadratic number field. Inconjunction with our Theorem 2, this allows us to produce examples of K3 surfaces S withnon-trivial real multiplication for which End Hdg ( T ( S )) is generated by algebraic classes. Wecould not find examples of this type in the existing literature.The plan of the paper is as follows: In Section 2 we review Zarhin’s results on the endomor-phism algebra and on the special Mumford–Tate group of an irreducible Hodge structure ofK3 type. Also, we recall from [vG4] how a Hodge structure of K3 type with real multiplicationsplits over a finite extension of Q . HE HODGE CONJECTURE FOR SELF-PRODUCTS OF CERTAIN K3 SURFACES 3
Section 3 is devoted to the proof of Theorem 1. After reviewing the definition of the core-striction of algebras, we explain in detail how the Galois group of a normal closure of E S actson the Kuga–Satake Hodge structure. This is the key of the proof.In the final Section 4 we study double covers of P ramified along six lines. We reviewresults of Lombardo [Lo] on the Kuga–Satake variety of such K3 surfaces, of Schoen [S] andvan Geemen [vG2] on the Hodge conjecture for certain Abelian varieties of Weil type and ofcourse Paranjape’s [P] result on the algebraicity of the Kuga–Satake correspondence. Togetherwith Theorem 1, they lead to the proof of Theorem 2. Acknowledgements.
This work is a part of my Ph.D. thesis prepared at the University of Bonn.It is a great pleasure to thank my advisor Daniel Huybrechts for suggesting this interestingtopic and for constantly supporting me.During a four week stay at the University of Milano I had many fruitful discussions with Bertvan Geemen. I am most grateful to him for his insights.2.
Hodge structures of K3 type with real multiplication
Hodge structures of K3 type and their endomorphisms.
Let U(1) be the one-dimensional unitary group which is a real algebraic group. To fix notations we recall that aHodge structure of weight k is a finite-dimensional Q -vector space T together with a morphismof real algebraic groups h : U(1) → GL( T ) R such that for z ∈ U(1)( R ) ⊂ C the C -linearextension of the endomorphism h ( z ) is diagonalizable with eigenvalues z p z q where p + q = k and p, q ≥ z p z q is denoted by T p,q ⊂ T C .A polarization of a weight k Hodge structure (
T, h ) is a bilinear form q : T × T → Q whichis U(1)-invariant and which has the property that ( − k ( k − / q ( ∗ , h ( i ) ∗ ) : T R × T R → T R is asymmetric, positive definite bilinear form. Definition 2.1.1. A Hodge structure of K3 type ( T, h, q ) consists of a Q -Hodge structure ( T, h :U(1) → GL( T ) R ) of weight 2 with dim C T , = 1 together with a polarization q : T × T → Q . Examples.
The second primitive (rational) cohomology and the (rational) transcendental latticeof a projective K3 surface yield examples of Hodge structures of K3 type. More generally,the second primitive cohomology and the Beauville–Bogomolov orthogonal complement of theN´eron–Severi group of an irreducible symplectic variety are Hodge structures of K3 type [GHJ,Part III].Consider the Hodge decomposition T C := T ⊗ Q C = T , ⊕ T , ⊕ T , . Since the quadratic form q is a polarization, this decomposition is q -orthogonal. Moreover, q is positive definite on ( T , ⊕ T , ) ∩ T R and negative definite on T , ∩ T R .Assume that T is an irreducible Hodge structure. Let E := End Hdg ( T ) be the divisionalgebra of endomorphisms of Hodge structures of T . Let ′ : E → E be the involution given by ULRICH SCHLICKEWEI adjunction with respect to q and let E ⊂ E be the subalgebra of E formed by q -self-adjointendomorphisms. Theorem 2.1.2 (Zarhin [Z]) . The map ǫ : E → C , e eigenvalue of e on the eigenspace T , identifies E with a subfield of C . Moreover, E is a totally real number field and the followingtwo cases are possible: • E = E (in this case we say that T has real multiplication ) or • E ⊂ E is a purely imaginary, quadratic extension and ′ is the restriction of complexconjugation to E (we say that T has complex multiplication ). Splitting of Hodge structures of K3 type with real multiplication. (For this andthe next section see [vG4], 2.4 and 2.5.) Let (
T, h, q ) be an irreducible Hodge structure of K3type and assume that E = End Hdg ( T ) is a totally real number field. Note that by Theorem2.1.2, all endomorphisms in E are q -self-adjoint.By the theorem of the primitive element, there exists α ∈ E such that E = Q ( α ). Let d = [ E : Q ]. Let P be the minimal polynomial of α over Q , denote by e E the splitting field of P in R . Let G = Gal( e E/ Q ) and H = Gal( e E/E ). Choose σ = id , σ , . . . , σ d ∈ G such that G = σ H ⊔ . . . ⊔ σ d H. Note that each coset σ i H induces a well-defined embedding E ֒ → e E . In e E [ X ] we get P ( X ) = d Y i =1 ( X − σ i ( α ))and consequently E ⊗ Q e E = Q [ X ] / ( P ) ⊗ Q e E ≃ d M i =1 e E [ X ] / ( X − σ i ( α )) ≃ d M i =1 E σ i . The symbol E σ i stands for the field e E , the index σ i keeps track of the fact that the e E -linearextension of E ⊂ End Q ( E ) acts on E σ i via e ( x ) = σ i ( e ) · x . See Section 3.2 for anotherinterpretation of E σ i .In the same way, since T is a finite-dimensional E -vector space we get a decomposition T e E = T ⊗ Q e E = d M i =1 T σ i . HE HODGE CONJECTURE FOR SELF-PRODUCTS OF CERTAIN K3 SURFACES 5
This is the decomposition of T e E into eigenspaces of the e E -linear extension of the E -action on T , T σ i being the eigenspace of e e E to the eigenvalue σ i ( e ) for e ∈ E . Since each e ∈ E is q -self-adjoint (that is e ′ = e ), the decomposition is orthogonal. Let q e E be the e E -bilinear extension of q to T e E × T e E . Using the notation T i := T σ i and q i = ( q e E ) | T i × T i , we have an orthogonal decomposition(1) ( T e E , q e E ) = d M i =1 ( T i , q i ) . Galois action on T e E . Letting G act in the natural way on e E , we get a (only Q -linear)Galois action on T e E = T ⊗ Q e E . Under this action, for τ ∈ G we have(2) τ T σ i = T τσ i . This is because the Galois action commutes with the e E -linear extension of any endomorphism e ∈ E ⊂ End Q ( T ) the latter being defined over Q and because for t i ∈ T σ i and e ∈ Ee e E ( τ ( t i )) = τ ( e e E ( t i )) = τ ( σ i ( e ) t i ) = τ ( σ i ( e )) τ ( t i ) = ( τ σ i ( e )) τ ( t i ) , which means that τ permutes the eigenspaces of e e E precisely in the way we claimed. Define ahomomorphism(3) γ : G → S d , τ
7→ { i τ ( i ) where ( τ σ i ) H = σ τ ( i ) H } . (This describes the action of G on G/H .) With this notation, (2) reads(4) τ T i = T τ ( i ) . Interpret T as a subspace of T e E via the natural inclusion T ֒ → T e E , t t ⊗
1. Denote by π i the projection to T i . For t ∈ T and τ ∈ G we have t = τ ( t ). Write t i := π i ( t ⊗ t = P i t i . Using (4) we see that(5) t τi = τ ( t i ) . It follows that(6) ι i : T → T i , t π i ( t ⊗ E -vector spaces ( E acting on T i via σ i : E ֒ → e E ). Equation (5) can berephrased as(7) ι τi = τ ◦ ι i . Since q is defined over Q , we have for t ∈ T e E and τ ∈ Gq e E ( τ t ) = τ q e E ( t ) . This implies that for t ∈ T (8) q i ( ι i ( t )) = σ i q ( ι ( t )) . ULRICH SCHLICKEWEI
The special Mumford–Tate group of a Hodge structure of K3 type with realmultiplication.
Zarhin [Z] also computed the special Mumford–Tate group of an irreducibleHodge structure of K3 type. Recall that for a Hodge structure (
W, h : U(1) → GL( W ) R )the special Mumford–Tate group SMT( W ) is the smallest linear algebraic subgroup of GL( W )defined over Q with h (U(1)) ⊂ SMT( W ) R (cf. [G]).Assume that ( T, h, q ) is an irreducible Hodge structure of K3 type with real multiplication by E = End Hdg ( T ). We continue to use the notations of Section 2.2. Denote by Q the restrictionof q to T ⊂ T (use the inclusion ι of (6)). This is an E -valued (since H -invariant), non-degenerate, symmetric bilinear form on the E -vector space T . Denote by SO( Q ) the E -linearalgebraic group of Q -orthogonal, E -linear transformations of T with determinant 1.Recall that for an E -variety Y the Weil restriction
Res E/ Q ( Y ) is the Q -variety whose K -rational points are the E ⊗ Q K -rational points of Y for any extension field Q ⊂ K (cf. [BLR]). Theorem 2.4.1 (Zarhin, see [Z], see also [vG4], 2.8) . The special Mumford–Tate group of theHodge structure ( T, h, q ) with real multiplication by E is SMT( T ) = Res E/ Q (SO( Q )) . Its representation on T is the natural one, where we regard T as a Q -vector space and use thatany E -linear endomorphism of T is in particular Q -linear. After base change to e E SMT( T ) e E = Y i SO(( T i ) , ( q i )) , its representation on T e E = L i ( T i ) is the product of the standard representations. Kuga–Satake varieties and real multiplication
Kuga–Satake varieties.
Let (
T, h, q ) be a Hodge structure of K3 type. Kuga and Sa-take [KS] found a way to associate to this a polarizable Q -Hodge structure of weight one( V, h s : U(1) → GL( V ) R ), in other words an isogeny class of Abelian varieties, together withan inclusion of Hodge structures(9) T ⊂ V ⊗ V. Set V := C ( q ) where C ( q ) is the even Clifford algebra of q . Define a weight one Hodgestructure on V in the following way: Choose f , f ∈ ( T , ⊕ T , ) R such that C ( f + if ) = T , and q ( f i , f j ) = δ i,j (recall that q | ( T , ⊕ T , ) R is positive definite). Define J : V → V, v f f v ,then we see that J = − id. Now we can define a homomorphism of algebraic groups h s : U(1) → GL( V ) R , exp( xi ) exp( xJ ) , and this induces the Kuga–Satake Hodge structure. One can check that h s is independent ofthe choice of f , f (see [vG3, Lemma 5.5]).It can be shown that the Kuga–Satake Hodge structure admits a polarization (cf. [vG3, Prop.5.9]) and that there is an embedding of Hodge structures as in (9) (see [vG3, Prop. 6.3]). HE HODGE CONJECTURE FOR SELF-PRODUCTS OF CERTAIN K3 SURFACES 7
Corestriction of algebras.
Let
E/K be a finite, separable extension of fields of degree d and let A be an E -algebra. We use the notations of Section 2.2, so e E is a normal closure of E over K , σ , . . . , σ d is a set of representatives of G/H where G = Gal( e E/K ) and H = Gal( e E/E ).For σ ∈ G define the twisted e E -algebra as the ring A σ := A ⊗ E e E which carries an e E -algebra structure given by λ · ( a ⊗ e ) = a ⊗ σ − ( λ ) e. Note that A σ ≃ A ⊗ E E σ .Let V be an E -vector space and W an e E -vector space, let σ ∈ G . A homomorphism of K -vector spaces ϕ : V → W is called σ -linear if ϕ ( λv ) = σ ( λ ) ϕ ( v ) for all v ∈ V and λ ∈ E . Ifboth, V and W are e E -vector spaces, there is a similar notion of an σ -linear homomorphism. Lemma 3.2.1.
The map κ σ : A → A σ , a a ⊗ is a σ -linear ring homomorphism and the pair ( A σ , κ σ ) has the following universal property:For all e E -algebras B and for all σ -linear ring homomorphisms ϕ : A → B there exists a unique e E -algebra homomorphism e ϕ : A σ → B making the diagram A κ σ / / ϕ AAAAAAAA A σ e ϕ (cid:15) (cid:15) B commutative.Proof. We only check the universal property. To give a K -linear map α : A ⊗ E e E → B is thesame as to give a K -bilinear map β : A × e E → B satisfying(10) β ( λa, e ) = β ( a, λe )for all a ∈ A, e ∈ e E and λ ∈ E . These maps are related by the condition α ( a ⊗ e ) = β ( a, e ) . Now given ϕ as in the lemma, we define ψ : A × e E → B, ( a, e ) σ ( e ) ϕ ( a ) . This is a K -bilinear map which satisfies (10) and therefore, it induces a K -linear map e ϕ : A ⊗ E e E → B, a ⊗ e σ ( e ) ϕ ( a ) . It is clear that e ϕ is a ring homomorphism and that it respects the e E -algebra structures if weinterpret e ϕ as a map e ϕ : A σ → B . The uniqueness of this map is immediate. ✷ Remark. (i) The lemma shows that up to unique e E -algebra isomorphism, the twisted algebra A σ i depends only on the coset σ i H . Indeed, for σ ∈ σ i H the inclusion A ֒ → A σ i is σ -linear ULRICH SCHLICKEWEI because σ and σ i induce the same embedding of E into e E . By the lemma, there exists an e E -algebra isomorphism α σ,σ i : A σ ∼ → A σ i , a ⊗ e σ ( e ) · ( a ⊗
1) = a ⊗ σ − i σ ( e ).(ii) In Section 2.2 we were in the situation E = Q ( α ). There we discussed the splitting E ⊗ Q e E ≃ L i e E [ X ] / ( X − σ i ( α )) ≃ L i E σ i and we used the symbol E σ i for the field e E with E -action via e ( x ) = σ i ( e ) · x . This is precisely our twisted e E -algebra E σ i on which E acts viathe inclusion κ σ i .For τ ∈ G there is a unique τ -linear ring isomorphism τ : A σ i → A σ τi which extends theidentity on A ⊂ A σ i (in the sense that τ ◦ κ σ i = κ σ τi ). This map is given as the compositionof the following two maps: First apply the identity map A σ i → A τσ i , a ⊗ e a ⊗ e which is a τ -linear ring isomorphism. Then apply the isomorphism α τσ i ,σ τi (by definition of the G -actionon { , . . . , d } we have τ σ i ∈ σ τi H ). On simple tensors the map τ takes the form(11) a ⊗ e a ⊗ σ − τi τ σ i ( e ) . These maps induce a natural action of G on Z G ( A ) := A σ ⊗ e E . . . ⊗ e E A σ d where(12) τ (( a ⊗ e ) ⊗ . . . ⊗ ( a d ⊗ e d ))= (cid:0) a τ − ⊗ σ − τ σ τ − ( e τ − ) (cid:1) ⊗ . . . ⊗ (cid:0) a τ − d ⊗ σ − d τ σ τ − d ( e τ − d ) (cid:1) . Definition 3.2.2 ([D], §
8, Def. 2 or [T], 2.2) . The corestriction of A to K is the K -algebra of G -invariants in Z G ( A ) Cores E/K ( A ) := Z G ( A ) G . Remark. (i) By [D, §
8, Cor. 1] there is a natural isomorphismCores
E/K ( A ) ⊗ K e E ≃ Z G ( A )In particular, with d = [ E : K ] one gets dim K Cores
E/K ( A ) = (dim E ( A )) d .(ii) Let X = Spec( A ) for a commutative L -algebra A . Then for any K -algebra B we get achain of isomorphisms, functorial in B Hom K − Alg (Cores
E/K ( A ) , B ) ≃ (cid:16) Hom e E − Alg ( Z G ( A ) , B ⊗ K e E ) (cid:17) G ≃ Hom E − Alg ( A, B ⊗ K E ) . Here, the last isomorphism is given by composing f ∈ (cid:16) Hom e E − Alg ( Z G ( A ) , B ⊗ K e E ) (cid:17) G withthe inclusion j : A ֒ → Z G ( A ) , a κ σ ( a ) ⊗ ⊗ . . . ⊗
1. (The image of this composition iscontained in the H -invariant part of B ⊗ K e E which is B ⊗ K E .) This map is an isomorphism,since Z G ( A ) is generated as an e E -algebra by elements of the form σ ◦ j ( a ) with a ∈ A and σ ∈ G . HE HODGE CONJECTURE FOR SELF-PRODUCTS OF CERTAIN K3 SURFACES 9
It follows that Res
E/K (Spec( A )) ≃ Spec(Cores
E/K ( A )) , i.e. the Weil restriction of affine E -schemes is the same as the corestriction of commutative E -algebras.3.3. The decomposition theorem.
We will assume from now to the end of the section that(
T, h, q ) is an irreducible Hodge structure of K3 type with E = End Hdg ( T ) a totally real numberfield.Recall that in this case T is an E -vector space which carries a natural E -valued quadraticform Q (see 2.4). Let C ( Q ) be the even Clifford algebra of Q over E . It was van Geemen (see[vG4, Prop. 6.3]) who discovered that the algebra Cores E/ Q ( C ( Q )) appears as a sub-Hodgestructure in the Kuga–Satake Hodge structure of ( T, h, q ). We are going to show that thiscontains all information on the Kuga–Satake Hodge structure.
Theorem 3.3.1.
Denote by ( V, h s ) the Kuga–Satake Hodge structure of ( T, h, q ) . (i) The special Mumford–Tate group of ( V, h s ) is the image of Res E/ Q (Spin( Q )) in Spin( q ) under a morphism m of rational algebraic groups which after base change to e E becomes m e E : Spin( q ) × . . . × Spin( q d ) → Spin( q ) e E , ( v , . . . , v d ) v · . . . · v d . (ii) Let W := Cores E/ Q ( C ( Q )) . Then W can be canonically embedded in V and the imageis SMT( V ) -stable and therefore, it is a sub-Hodge structure. Furthermore, there is a (non-canonical) isomorphism of Hodge structures V ≃ W d − . (iii) We have
End
Hdg ( W ) = Cores E/ Q ( C ( Q )) and consequently End
Hdg ( V ) = Mat d − (cid:0) Cores E/ Q ( C ( Q )) (cid:1) . The proof will be given in Section 3.5. The theorem tells us that the Kuga–Satake variety A of ( T, h, q ) is isogenous to a self-product B d − of an Abelian variety B with End Q ( B ) =Cores E/ Q ( C ( Q )) and therefore, it proves Theorem 1.Note that B is not simple in general. We will see examples below where B decomposesfurther.3.4. Galois action on C ( q ) e E . By Section 2.2 there is a decomposition(
T, q ) e E = d M i =1 ( T i , q i ) . This in turn yields an isomorphism C ( q ) e E ≃ C ( q e E ) ≃ C ( q ) b ⊗ e E . . . b ⊗ e E C ( q d ) . Here, the symbol b ⊗ denotes the graded tensor product of algebras, which on the level of vectorspaces is just the usual tensor product, but which twists the algebra structure by a suitablesign (see [vG3, ??]).Decompose C ( q i ) = C ( q i ) ⊕ C ( q i ) in the even and the odd part. If we forget the algebrastructure and only look at e E -vector spaces, we get C ( q ) e E = M a ∈{ , } d C a ( q ) ⊗ e E . . . ⊗ e E C a d ( q d ) . For a = ( a , . . . , a d ) ∈ { , } d define C a ( q ) = C a ( q ) ⊗ . . . ⊗ C a d ( q d ) . We introduced an action of G = Gal( e E/ Q ) on { , . . . , d } (see (3)). This induces an action(13) G × { , } d → { , } d , ( τ, ( a , . . . , a d )) ( a τ − , . . . , a τ − d ) . The next lemma describes the Galois action on C ( q ) e E . Lemma 3.4.1. (i)
Via the map C ( q i ) ⊂ C ( q ) e E , v i ⊗ . . . ⊗ v i ⊗ . . . ⊗ we interpret C ( q i ) as a subalgebra of C ( q ) e E . Then the restriction of τ ∈ G to C ( q i ) induces anisomorphism of Z / Z -graded Q -algebras τ : ( C ( q i )) ∼ → C ( q τi ) . (ii) For τ ∈ G and a ∈ { , } d we get τ ( C a ( q )) = C τ a ( q ) . Proof.
Tensor the natural inclusion
T ֒ → C ( q ) with e E to get a G -equivariant inclusion T ⊗ Q e E = d M i =1 T i → C ( q ) e E . Using (4), we find for t i ∈ C ( q i ) that τ ( t i ) ∈ C ( q τ ( i ) ). Now, C ( q i ) is spanned as a Q -algebraby products of the form t · . . . · t k = ± (1 ⊗ . . . ⊗ t ⊗ . . . ⊗ · . . . · (1 ⊗ . . . ⊗ t k ⊗ . . . ⊗ t , . . . , t k ∈ T i . Since G acts by Q -algebra homomorphisms on C ( q ) e E , this implies (i).Item (ii) is an immediate consequence of (i): The space C a ( q ) is spanned as Q -vector spaceby products of the form v · . . . · v d = ± v ⊗ . . . ⊗ v d with v i ∈ C a i ( q i ). Then use again, that G acts by Q -algebra homomorphisms. ✷ Lemma 3.4.2.
For i ∈ { , . . . , d } the twisted algebra C ( Q ) σ i is canonically isomorphic as an e E -algebra to C ( q i ) . Thus Z G ( C ( Q )) ≃ C ( q ) ⊗ e E . . . ⊗ e E C ( q d ) . HE HODGE CONJECTURE FOR SELF-PRODUCTS OF CERTAIN K3 SURFACES 11
On both sides there are natural G -actions: On the left hand side G acts via the action introducedin (12), whereas on the right hand side it acts via the restriction of its action on C ( q ) e E (useLemma 3.4.1). Then the above isomorphism is G -equivariant.Proof. Fix i ∈ { , . . . , d } . The composition of the canonical inclusion C ( Q ) ⊂ C ( q ) ≃ C ( Q ) e E with the restriction to C ( Q ) of the map σ i : C ( q ) → C ( q i ) from Lemma 3.4.1induces a σ i -linear ring homomorphism ϕ i : C ( Q ) ֒ → C ( q i ) . By Lemma 3.2.1 we get an e E -algebra homomorphism e ϕ i : C ( Q ) σ i → C ( q i ) . Recall that there are inclusions ι i : T ֒ → T i (see (6)) which satisfy τ ◦ ι i = ι τi (see (7)). Let t , . . . , t m ∈ T such that ι ( t ) , . . . , ι ( t m ) form a q -orthogonal basis of T . Then the vectors ι i ( t ) , . . . , ι i ( t m ) form a q i -orthogonal basis of T i (use (8)). By definition of e ϕ i (14) e ϕ i (cid:0) ι ( t ) i · . . . · ι ( t m ) i d (cid:1) = ι i ( t ) i · . . . · ι i ( t m ) i m . This implies that e ϕ i maps an e E -basis of C ( Q ) σ i onto an e E -basis of C ( q i ), whence e ϕ i is anisomorphism of e E -algebras.As for the G -equivariance, we have to check that for all τ ∈ G the diagram C ( Q ) σ i e ϕ i −−−−→ C ( q i ) τ y y τ C ( Q ) σ τi e ϕ τi −−−−→ C ( q τi )is commutative. It is enough to check this on an e E -basis of C ( Q ) σ i because the vertical mapsare both τ -linear whereas the horizontal ones are e E -linear. Since τ : C ( Q ) σ i → C ( Q ) σ τi wasdefined as the extension of the identity map on C ( Q ) ⊂ C ( Q ) σ i , we have e ϕ τi ◦ τ (cid:0) ι ( t ) i · . . . · ι ( t m ) i m (cid:1) = e ϕ τi (cid:0) ι ( t ) i · . . . · ι ( t m ) i m (cid:1) = ι τi ( t ) i · . . . · ι τi ( t m ) i m = ( τ ◦ ι i )( t ) i · . . . · ( τ ◦ ι i )( t m ) i m = τ (cid:0) ι i ( t ) i · . . . · ι i ( t m ) i m (cid:1) = τ ◦ e ϕ i (cid:0) ι ( t ) i · . . . · ι m ( t m ) i m (cid:1) . This completes the proof of the lemma. ✷ Proof of the decomposition theorem.
Let K be a field and ( U, r ) be a quadratic K -vector space. Recall that the spin group of r comes with two natural representations:First there is the covering representation ρ : Spin( r ) → SO( r ) which over an extension field K ⊂ L maps y ∈ Spin( r )( L ) = { x ∈ ( C ( r ) ⊗ K L ) ∗ | xι ( x ) = 1 and xU x − ⊂ U } to theendomorphism U → U, u xux − . Here, ι : C ( r ) → C ( r ) is the natural involution of theClifford algebra. Secondly, the spin representation realizes Spin( r ) as a subgroup of GL( C ( r )) by sending y ∈ Spin( r )( L ) to the endomorphism of C ( r ) given by x y · x . Proof of (i).
By [vG3, Prop. 6.3], there is a commutative diagram(15) U(1) h s −−−−→ Spin( q ) R −−−−→ GL( C ( q )) R (cid:13)(cid:13)(cid:13) y ρ U(1) −−−−→ h SO( q ) R −−−−→ GL( T ) R . (Van Geemen works with the Mumford–Tate group, therefore he gets a factor t in 6.3.2. Thisfactor is 1 if one restricts the attention to the special Mumford–Tate group, moreover it is thenclear that h s ( C ∗ ) ⊂ CSpin(q) = { v ∈ C ( q ) ∗ | vT v − ⊂ T } implies h s (U(1)) ⊂ Spin( q ).) Claim:
There is a Cartesian diagramSMT( V ) −−−−→ Spin( q ) ρ | SMT( V ) y y ρ SMT( T ) −−−−→ SO( q ) . where the horizontal maps are appropriate factorizations of the inclusions SMT ⊂ GL whoseexistence is guaranteed by (15).
Proof of the claim.
It is clear by looking at (15) and at the definition of the special Mumford–Tate group that SMT( V ) ⊂ SMT( T ) × SO( q ) Spin( q ) . In the same way we see that SMT( T ) ⊂ ρ (SMT( V ))and hence we have a chain of inclusionsSMT( V ) ⊂ SMT( T ) × SO( q ) Spin( q ) ⊂ ρ (SMT( V )) × SO( q ) Spin( q ) . But over any field, the kernel of ρ consists of {± } ⊂ SMT( V ) (because h s ( −
1) = −
1) andthus SMT( V ) = ρ (SMT( V )) × SO( q ) Spin( q ) . This proves the claim. (Claim) ✷ To continue the proof of (i) we have to define the morphism of rational algebraic groups m : Res E/ Q (Spin( Q )) → Spin( q ) . HE HODGE CONJECTURE FOR SELF-PRODUCTS OF CERTAIN K3 SURFACES 13
For that sake, note first that there is a natural isomorphism of e E -algebras(16) C ( Q ) ⊗ Q e E ≃ C ( Q ) ⊗ E ( E ⊗ Q e E ) ≃ M i C ( Q ) ⊗ E E σ i ≃ M i C ( Q ) σ i ≃ C ( q ) ⊕ . . . ⊕ C ( q d )where we use the notations of Section 3.2 and for the last identification Lemma 3.4.2. Considerthe natural G -action on C ( q ) ⊕ . . . ⊕ C ( q d ) given by( τ, ( v , . . . , v d )) ( τ v τ − , . . . , τ v τ − d ) . On C ( Q ) ⊗ Q e E , the Galois group G acts by its natural action on e E . Then the identificationmade in (16) is G -equivariant and we get an isomorphism of Q -vector spaces C ( Q ) ≃ (cid:0) C ( q ) ⊕ . . . ⊕ C ( q d ) (cid:1) G , v ( σ ( v ) , . . . , σ d ( v )) . Now, look at the morphism of e E -affine spaces C ( q ) ⊕ . . . ⊕ C ( q d ) → C ( q ) e E , ( v , . . . , v d ) v · . . . · v d . This morphism is G -equivariant on the e E -points and hence it comes from a morphism of Q -varieties Res E/ Q C ( Q ) → C ( q ) . The restriction of this latter to Res E/ Q (Spin( Q )) is the morphism m we are looking for. It isa morphism of algebraic groups which after base change to e E takes the form m e E : Res E/ Q (Spin( Q )) e E ≃ Spin( q ) × . . . × Spin( q d ) → Spin( q ) e E , ( v , . . . , v d ) v · . . . · v d . It remains to show that the image of m in Spin( q ) is SMT( V ). Using the claim we have toshow that the following diagram exists and that it is Cartesian(17) im( m ) −−−−→ Spin( q ) ρ | im( m ) y y ρ Res E/ Q (SO( Q )) −−−−→ SO( q ) . Here, the lower horizontal map is the one coming from Zarhin’s Theorem 2.4.1.It is enough to study (17) on Q -points. It is easily seen that over e E ⊂ Q the composition ρ ◦ m factorizes over ρ × . . . × ρ d : Spin( q ) × . . . × Spin( q d ) → SO( q ) × . . . × SO( q d ) ≃ Res E/ Q (SO( Q )) e E ⊂ SO( q ) e E . This shows that (17) exists. Moreover we see that ρ | im( m ) surjects onto SMT( T )( Q ) because ρ × . . . × ρ d does so. Since ker( ρ ) = {± } ⊂ im( m ), the diagram (17) is Cartesian. Thiscompletes the proof of (i). (i) ✷ Proof of (ii).
Choose a = (0 , . . . , , . . . , a r ∈ { , } d such that ( a ∈ { , } d | X i a i ≡ ) = G a ⊔ . . . ⊔ G a r , where G acts on { , } d via the action introduced in (13). Let G a j ⊂ G be the stabilizer of a j .Then(18) C ( q ) e E = r M j =0 M [ τ ] ∈ G/G a j C τ a j ( q ) = r M j =0 D a j with D a j = L [ τ ] ∈ G/G a j C τ a j ( q ).By Lemma 3.4.1 this is a decomposition of G -modules. Moreover, recall that Spin( q ) × . . . × Spin( q d ) acts on C ( q ) e E by sending ( v , . . . , v d ) to the endomorphism of C ( q ) e E given by leftmultiplication with m ( v , . . . , v d ) = v · . . . · v d . Under this action each C a ( q ) is (Spin( q ) × . . . × Spin( q d ))-stable. Thus, by (i) the decomposition (18) is also a decomposition of SMT( V )( e E )-modules. Hence, by passing to G -invariants, (18) leads to a decomposition of Hodge structures.Denote by R := D a = C a ( q ) = C ( q ) ⊗ e E . . . ⊗ e E C ( q d ) . By Lemma 3.4.2, using the notations of Section 3.2, we have R = Z G ( C ( Q ))as G -modules and hence R G = Cores E/ Q ( C ( Q )). Thus we have recovered W = Cores E/ Q ( C ( Q )) ⊂ C ( q ) = V as a sub-Hodge structure. We now prove that after passing to G -invariants, the remainingsummands in (18) are isomorphic to sums of copies of W .Denote by d j = ♯ ( G/G a j ) and choose a set of representatives µ , . . . , µ d j of G/G a j in G . Weconsider three group actions on R ⊕ d j : • First there is a natural (Spin( q ) × . . . × Spin( q d ))-action which is just the diagonal actionof the one on R . • Let α : G × R ⊕ d j → R ⊕ d j be the diagonal action of the G -action on R . • Finally define the G -action β by β : G × d j M l =1 R [ µ l ] → d j M l =1 R [ µ l ] ( τ, ( r [ µ ] , . . . , r [ µ d ])) ( τ r [ τ − µ ] , . . . , τ r [ τ − µ dj ] ) . HE HODGE CONJECTURE FOR SELF-PRODUCTS OF CERTAIN K3 SURFACES 15
Now we will proceed in two steps:(a) We show that D a j is isomorphic as G -module and as (Spin( q ) × . . . × Spin( q d ))-moduleto R ⊕ d j where G acts on the latter via β .(b) We show that R ⊕ d j is isomorphic as G -module and as (Spin( q ) × . . . × Spin( q d ))-modulewith G acting via α to R ⊕ d j with G acting via β .Note that neither of these two isomorphisms is canonical. Once (a) and (b) are proved, wehave an isomorphism V e E = C ( q ) e E ≃ R ⊕ d − of G -modules and of SMT( V )( e E )-modules, G acting diagonally on the right hand side. Herewe use that X j d j = ♯ ( a ∈ { , } d | X i a i ≡ ) = 2 d − . The proof of (ii) is then accomplished by passing to G -invariants. Proof of (a).
Denote by F j the field e E G aj . As C a j ( q ) ⊂ D a j is G a j -stable, C a j ( q ) = W j ⊗ F j e E for some F j -vector space W j . Since C a j contains units in C ( q ) e E , so does W j ⊂ C a j . (Veryformally: There is a linear map C a j → End( C ( q ) e E ) , w
7→ { v v · w } which is defined over F j .The image of this map over e E intersects the Zariski-open subset of automorphisms of C ( q ) e E ,hence this must happen already over F j .)Choose a unit w j ∈ W j . Then for τ ∈ G , since w j is G a j -invariant, τ w j ∈ C τ a j ( q ) dependsonly on the coset τ G a j and is again a unit in C ( q ) e E .Define an isomorphism of e E -vector spaces ϕ : D a j = d j M l =1 C µ l a j ( q ) → d j M l =1 R [ µ l ] ( v µ , . . . , v µ dj ) ( v µ · µ ( w j ) , . . . , v µ dj · µ d j ( w j )) . This map is clearly (Spin( q ) × . . . × Spin( q d ))-equivariant since this group acts by multiplicationon the left whereas we multiply on the right.As for the G -equivariance ( G acting via β on the right hand side), we find for( v [ µ ] , . . . , v [ µ dj ] ) ∈ D a j and τ ∈ G : ϕ (cid:0) τ ( v [ µ ] , . . . , v [ µ dj ] ) (cid:1) = ϕ (cid:0) τ v [ τ − µ ] , . . . , τ v [ τ − µ d ] (cid:1) = (cid:0) τ v [ τ − µ ] · µ w j , . . . , τ v [ τ − µ dj ] · µ d j w j (cid:1) = (cid:0) τ ( v [ τ − µ ] · τ − µ w j ) , . . . , τ ( v [ τ − µ dj ] · τ − µ d j w j ) (cid:1) = β (cid:0) τ, ( v µ · µ w j , . . . , v µ d · µ d j w j ) (cid:1) = β (cid:0) τ, ϕ ( v [ µ ] , . . . , v [ µ dj ] ) (cid:1) . Here we used in the penultimate equality that σw j depends only on the coset σG a j . Thisproves (a). (a) ✷ Proof of (b).
Choose a Q -basis f , . . . , f d j of F j . For i = 1 , . . . , d j define an e E -vector spacehomomorphism by ψ i : R ֒ → d j M l =1 R [ µ l ] r (cid:0) µ ( f i ) · r, . . . , µ d j ( f i ) · r (cid:1) . As (Spin( q ) × . . . × Spin( q d ))( e E ) acts by e E -linear automorphisms on R , the ψ i are equivariantfor the Spin-action.Let’s show that ψ i is G -equivariant, G acting on the right hand side via β . For τ ∈ G and r ∈ R we get ψ i ( τ r ) = (cid:0) µ ( f i ) · τ r, . . . , µ d j ( f i ) · τ r (cid:1) = (cid:0) τ ( τ − µ ( f i ) · r ) , . . . , τ ( τ − µ d j ( f i ) · r ) (cid:1) = β (cid:0) τ, ( µ ( f i ) · r, . . . , µ d j ( f i ) · r ) (cid:1) = β ( τ, ψ i ( r )) . Once more, we used the fact that σf i depends only on the coset σG a j .Finally, using Artin’s independence of characters (see [La, Thm. VI.4.1]), we getdet(( µ l ( f i )) l,i ) = 0 . Consequently, the map ⊕ d j i =1 ψ i : R ⊕ d j → R ⊕ d j is an isomorphism which has the equivariance properties we want and (b) is proved. (ii) ✷ Proof of (iii).
Using that endomorphisms of Hodge structures are precisely those endomor-phisms which commute with the special Mumford–Tate group, we have to show thatEnd
SMT( V ) ( W ) = Cores E/ Q ( C ( Q )) . Denote by g the Lie algebra of SMT( V ). ThenEnd SMT( V ) ( W ) = End g ( W )= { f ∈ End Q ( W ) | Xf − f X = 0 for all X ∈ g } . Since for any field extension K/ Q we have Lie(SMT( V ) K ) = g ⊗ Q K this implies that(19) End SMT( V ) K ( W K ) = End SMT( V ) ( W ) ⊗ Q K. Now SMT( V )( e E ) = Spin( q ) × . . . × Spin( q d )( e E ) acts on W e E = C ( q ) ⊗ . . . ⊗ C ( q d ) byfactorwise left multiplication: (cid:0) ( v , . . . , v d ) , w ⊗ . . . ⊗ w d (cid:1) ( v · w ) ⊗ . . . ⊗ ( v d · w d ) . HE HODGE CONJECTURE FOR SELF-PRODUCTS OF CERTAIN K3 SURFACES 17
Therefore, using multiplication on the right, we get an inclusion (cid:0) C ( q ) ⊗ . . . ⊗ C ( q d ) (cid:1) op ֒ → End
SMT( V )( e E ) ( W e E ) , w
7→ { w ′ w ′ · w } . Now, ( C ( q ) ⊗ . . . ⊗ C ( q d )) op ≃ C ( q ) op ⊗ . . . ⊗ C ( q d ) op ≃ C ( q ) ⊗ . . . ⊗ C ( q d ) and hencepassing to G -invariants we have an inclusion(20) Cores E/ Q ( C ( Q )) ֒ → End
SMT( V )( Q ) ( W ) . We will now show that this is an isomorphism over e E . Using (19) and comparing dimensionsthis will prove (iii).To show that (20) is an isomorphism over e E we have to determine the Spin( q ) × . . . × Spin( q d )-invariants in End e E (cid:0) C ( q ) ⊗ . . . ⊗ C ( q d ) (cid:1) = End e E C ( q ) ⊗ . . . ⊗ End e E C ( q d ) . Using the next lemma inductively, this is equal toEnd
Spin( q ) C ( q ) ⊗ . . . ⊗ End
Spin( q d ) C ( q d ) . Now by [vG3, Lemma 6.5], End
Spin( q i ) C ( q i ) = C ( q i ). This proves (iii). ✷ Lemma 3.5.1.
Let G and H be two reductive linear algebraic groups over a field K of charac-teristic . Let M resp. N be finite-dimensional representations over K of G resp. H . Then ( M ⊗ K N ) G × H = M G ⊗ K N H . Proof.
Decompose M = L i M i and N = L j N j in irreducible representations. Then M i ⊗ N j is an irreducible representation of G × H since fixing 0 = m ∈ M i and 0 = n ∈ N i the orbit( G × H ) m ⊗ n generates M i ⊗ N j .To conclude the proof note that the space of invariants is the direct sum of trivial, one-dimensional sub representations. ✷ The Brauer–Hasse–Noether theorem.
Let k be a field of characteristic = 2, let A be a central simple k -algebra (i.e. a finite-dimensional k -algebra with center k which has nonon-trivial two-sided ideals). By Wedderburn’s theorem, there exists a central division algebra D over k and an integer n > A ≃ Mat n ( D ). Let d be the dimension of D over k (this is a square because after base change, D becomes isomorphic to a matrix algebra). Then d is the index of A , denoted by i ( A ). The class of A in the Brauer group of k has finite order.This integer is called the exponent of A , it is denoted by e ( A ). In general, we have e ( A ) | i ( A ).Let K/k be a cyclic extension of degree n , let σ be a generator of the Galois group Gal( K/k ),let a ∈ k ∗ . There is a central simple k -algebra ( σ, a, K/k ) which as a k -algebra is generated by K and an element y ∈ ( σ, a, K/k ) such that y n = a and r · y = y · σ ( r ) for r ∈ K. This algebra is called the cyclic algebra associated with σ, a and
K/k . A cyclic algebra over k of dimension 4 is a quaternion algebra. Theorem 3.6.1 (Brauer, Hasse, Noether [BHN]) . Let k be an algebraic number field. Thenany central division algebra A over k is a cyclic algebra (for an appropriate cyclic extension K/k and σ and a as above). Moreover, the exponent and the index of A coincide. In particular,a central division algebra of exponent 2 is a quaternion algebra. An example.
We continue to assume that (
T, h, q ) is a Hodge structure of K3 type with E = End Hdg ( T ) a totally real number field of degree d over Q . By [vG4, Prop. 3.2] we havedim E T ≥
3. We will consider now the case that dim E T = 3.Then T is an 3-dimensional e E -vector space with quadratic form q of signature (2+ , − ).The 3-dimensional quadratic spaces ( T , q ) , . . . , ( T d , q d ) are negative definite. This implies that C ( q ) R = Mat ( R ) and C ( q i ) R = H for i ≥ E/ Q ( C ( Q )) ⊗ Q e E = Z G ( C ( Q )) = C ( q ) ⊗ e E . . . ⊗ e E C ( q d )we get Cores E/ Q ( C ( Q )) ⊗ Q R = Mat ( R ) ⊗ R H ⊗ R . . . ⊗ R H . Now, since H ⊗ H ≃ Mat ( R ) this becomes(21) Cores E/ Q ( C ( Q )) ⊗ Q R ≃ ( Mat d − ( H ) for even d Mat d ( R ) for odd d. On the other hand, the corestriction induces a homomorphism of Brauer groupscores : Br( E ) → Br( Q )(cf. [D, §
9, Thm. 5]). Therefore, the exponent of Cores E/ Q ( C ( Q )) in the Brauer group of Q is 2. By the Brauer–Hasse–Noether Theorem 3.6.1 there exists a (possibly split) quaternionalgebra D over Q with(22) Cores E/ Q ( C ( Q )) ≃ Mat d − ( D ) . Combining (21) with (22) we see that D is a definite quaternion algebra over Q in case d iseven and an indefinite quaternion algebra in case d is odd. The endomorphism algebra of aKuga–Satake variety of ( T, h, q ) is Mat d − ( D ). Since the dimension of a Kuga–Satake varietyis 2 dim Q ( T ) − = 2 d − , we have proved Corollary 3.7.1.
Let ( T, q, h ) be a Hodge structure of K3 type with E = End Hdg ( T ) a totallyreal number field of degree d over Q . Assume that dim E ( T ) = 3 . Then for any Kuga–Satakevariety A of ( T, h, q ) there exists an isogeny A ∼ B d − where B is a d -dimensional Abelian variety.If d is even, B is a simple Abelian variety of type III, i.e. End Q ( B ) = D for a definitequaternion algebra D over Q . HE HODGE CONJECTURE FOR SELF-PRODUCTS OF CERTAIN K3 SURFACES 19 If d is odd, B has endomorphism algebra End Q ( B ) = D for an indefinite (possibly split)quaternion algebra D over Q .Remark. (i) In the case d = 2 and dim E ( T ) = 3, van Geemen showed in [vG4, Prop. 5.7] thatthe Kuga–Satake variety of T is isogenous to a self-product of an Abelian fourfold with definitequaternion multiplication and Picard number 1. It is this case which will be of interest in thenext section.(ii) The case d = dim E ( T ) = 3 was also treated by van Geemen (see [vG4, 5.8 and 6.4]). Heconsiders the case D ≃ Mat ( Q ) and relates this to work of Mumford and Galluzzi. Note thatin this case the Abelian variety B of the corollary is not simple. Example.
In [vG4, 3.4], van Geemen constructs a one-dimensional family of six-dimensionalK3 type Hodge structures with real multiplication by a quadratic field E = Q ( √ d ) for somesquare-free integer d > d = c + e for rational c, e > P , see Section 4. Pick a member S of this family. Then T ( S ) ⊗ Q E splits in the direct sum of two three-dimensional E -vector spaces T and T . It turns out thatthe quadratic space ( T , q ) = ( T , Q ) is isometric to ( E , √ dX + √ dX − ( d − √ dc ) X ).Consequently C ( Q ) = ( − d, √ d ( d − √ dc )) E ≃ ( − , √ d − c ) E . Here for a, b ∈ E ∗ , the symbol ( a, b ) E denotes the quaternion algebra over E generated byelements 1 , i and j subject to the relations i = a, j = b and ij = − ji (see [vG3, Ex. 7.5]).The projection formula for central simple algebras (see [T, Thm. 3.2]) implies thatCores E/ Q ( C ( Q )) ≃ ( − , N E/ Q ( √ d − c )) Q ≃ ( − , c − d ) Q ≃ ( − , − e ) Q ≃ ( − , − Q which are simply Hamilton’s quaternions over Q . Here, N E/ Q : E → Q is the norm map.Hence, a Kuga–Satake variety for T ( S ) is isogenous to a self-product B where B is a simpleAbelian fourfold with End Q ( B ) = ( − , − Q .4. Double covers of P branched along six lines Let S be a K3 surface which admits a morphism p : S → P such that the branch locus of p is the union of six lines.In this section we use the decomposition theorem to prove Theorem 2 which states that theHodge conjecture holds for S × S .4.1. Abelian varieties of Weil type.
By a result of Lombardo [Lo], the Kuga–Satake varietyof S is of Weil type. We briefly recall what this means.Let K = Q ( √− d ) for some square-free d ∈ N . A polarized Abelian variety ( A, H ) of dimen-sion 2 n is said to be of Weil type for K if there is an inclusion K ⊂ End Q ( A ) mapping √− d to ϕ such that • the restriction of ϕ ∗ : H ( A, C ) → H ( A, C ) to H , ( A ) is diagonalizable with eigenvalues √− d and −√− d , both appearing with multiplicity n , • ϕ ∗ H = dH .There is a natural K -valued Hermitian form on the K -vector space H ( A, Q ) which is definedby e H : H ( A, Q ) × H ( A, Q ) → K, ( v, w ) H ( ϕ ∗ v, w ) + √− dH ( v, w ) . By definition, the discriminant of a polarized Abelian variety of Weil type (
A, H, K ) isdisc(
A, H, K ) = disc( e H ) ∈ Q ∗ /N K/ Q ( K ∗ )where N K/ Q : K → Q is the norm map.Polarized Abelian varieties of Weil type come in n -dimensional families (see [vG2, 5.3]).Weil introduced such varieties as examples of Abelian varieties which carry interesting Hodgeclasses. He constructs a two-dimensional space, called the space of Weil cycles W K ⊂ H n,n ( A, Q ) . For the definition of W K see [vG2, 5.2]. In general, the algebraicity of the classes in W K is notknown. Nonetheless there are some positive results. Here we mention one which we will usebelow. Theorem 4.1.1 (Schoen [S] and van Geemen [vG1], Thm. 3.7) . Let ( A, H ) be a polarizedAbelian fourfold of Weil type for the field Q ( i ) . Assume that the discriminant of ( A, H, Q ( i )) is . Then the space of Weil cycles W Q ( i ) is spanned by classes of algebraic cycles. Van Geemen uses a six-dimensional eigenspace in the complete linear system of the uniquetotally symmetric line bundle L with c ( L ) = H to get a rational (2:1) map of A onto a quadric Q ⊂ P . Then the projection on W Q ( i ) of the classes of the pullbacks of the two rulings of Q generate the space W Q ( i ) .4.2. Abelian varieties with quaternion multiplication.
Let D be a definite quaternionalgebra over Q . Such a D admits an involution x x which after tensoring with R becomesthe natural involution on Hamilton’s quaternions H .A polarized Abelian variety ( A, H ) of dimension 2 n has quaternion multiplication by D ifthere is an inclusion D ⊂ End Q ( A ) such that • H ( A, Q ) becomes a D -vector space and • for x ∈ D we have x ∗ H = xxH .We say that ( A, H, D ) is an Abelian variety of definite quaternion type. Polarized Abelianvarieties of dimension 2 n with quaternion multiplication by the same quaternion algebra comein n ( n − / HE HODGE CONJECTURE FOR SELF-PRODUCTS OF CERTAIN K3 SURFACES 21
Let K ⊂ D be a quadratic extension field of Q . Then K is a CM field and ( A, H, K ) is apolarized Abelian variety of Weil type (see [vGV, Lemma 4.5]). The space of quaternion Weilcycles of (
A, H, D ) W D ⊂ H n,n ( A, Q )is defined as the span of x ∗ W K where x runs over D . It can be shown that this is independentof the choice of K (see [vGV, Prop. 4.7]). For the general member of the family of polarizedAbelian varieties with quaternion multiplication these are essentially all Hodge classes: Theorem 4.2.1 (Abdulali, see [A], Thm. 4.1) . Let ( A, H, D ) be a general Abelian variety ofquaternion type. Then the space of Hodge classes on any self-product of A is generated byproducts of divisor classes and quaternion Weil cycles on A .In particular, if for one quadratic extension field K ⊂ D the space of Weil cycles W K isknown to be algebraic, then the Hodge conjecture holds for any self-product of A . In Abdulali’s theorem, a triple (
A, H, D ) is general if the special Mumford–Tate group of H ( A, Q ) is the maximal one. In the moduli space of triples ( A, H, D ) the locus of generaltriples is everything but a countable union of proper, closed subsets.4.3.
The transcendental lattice of S . We now turn back to our K3 surface S . Let p : S → P be the (2:1) morphism which is ramified over six lines.The N´eron–Severi group of S contains the 15 classes e , . . . , e corresponding to the excep-tional divisors over the intersection points of the six lines. Let h be the class of the pullbackof O P (1).Define e T ( S ) := h e , . . . , e , h i ⊥ ⊂ H ( S, Q ). The (rational) transcendental lattice of S isdefined to be T ( S ) := NS( S ) ⊥ ⊂ H ( S, Q ). Then we have T ( S ) ⊂ e T ( S ) . Both, T ( S ) and e T ( S ) are Hodge structures of K3 type. In addition, T ( S ) is irreducible. Sincethe second Betti number of S is 22, the Q -dimension of e T ( S ) is 6.4.4. The Kuga–Satake variety of e T ( S ) . Denote by A the Kuga–Satake variety associatedwith e T ( S ). Theorem 4.4.1 (Lombardo, see [Lo], Cor. 6.3 and Thm. 6.4) . There is an isogeny A ∼ B where B is an Abelian fourfold with Q ( i ) ⊂ End Q ( B ) . Moreover, B admits a polarization H such that ( B, H, Q ( i )) is a polarized Abelian variety of Weil type with disc( B, H, Q ( i )) = 1 . Paranjape [P] explains in a very nice way how this variety B is geometrically related to S .He shows that there exists a triple ( C, E, f : C → E ) where C is a genus five curve, E an elliptic curve and f a (4 : 1) map such thatPrym( f ) = B. Then S can be obtained as the resolution of a certain quotient of C × C . It is noteworthy thatParanjape does not construct explicitly a triple ( C, E, f ) starting with a K3 surface S in thefamily π . His proof goes the other way round. He associates to any triple a K3 surface andshows then that letting vary the triple he obtains all surfaces in the family π .Paranjape’s construction establishes that the Kuga–Satake inclusion(23) e T ( S ) ֒ → H ( B × B , Q )is given by an algebraic cycle on S × B × B .4.5. Proof of Theorem 2.
As pointed out in the introduction, we have to prove that E S :=End Hdg ( T ( S )) is spanned by algebraic classes. Since the Picard number of S is at least 16, wecan apply Ram´on-Mar´ı’s corollary [RM] of Mukai’s theorem [Mu1] which proves the assertionin the case that S has complex multiplication.Therefore, we may assume that S has real multiplication. Note that T ( S ) is an E S -vectorspace and that dim E S T ( S ) · [ E S : Q ] = dim Q T ( S ) ≤
6. On the other hand, by [vG4, Lemma3.2], we know that dim E S T ( S ) ≥
3. It follows that either E S = Q or E S = Q ( √ d ) for somesquare-free d ∈ Q > . In the first case we use the fact, that the class of the diagonal ∆ ⊂ S × S induces the identity on the cohomology and that the K¨unneth projectors are algebraic onsurfaces so that Q id ⊂ E S is spanned by an algebraic class.It remains to study the case E S = Q ( √ d ). The idea is to consider the Kuga–Satake variety A ( S ) of e T ( S ) = T ( S ). By Paranjape’s theorem the inclusion e T ( S ) ⊂ H ( A ( S ) × A ( S ) , Q )is algebraic. It follows that there is an algebraic projection π : H ( A ( S ) × A ( S ) , Q ) → e T ( S )(see [K, Cor. 3.14]) and therefore it is enough to show that there is an algebraic class α ∈ H ( A ( S ) × A ( S ) , Q ) ⊗ H ( A ( S ) × A ( S ) , Q ) ⊂ H ( A ( S ) , Q )with π ⊗ π ( α ) = √ d .Combining Corollary 3.7.1 with Lombardo’s theorem 4.4.1 we see that A ( S ) ∼ B where B is an Abelian fourfold with End Q ( B ) = D for a definite quaternion algebra and Q ( i ) ⊂ D .Moreover, there is a polarization H of B such that ( B, H, Q ( i )) is a polarized Abelian variety ofWeil type of discriminant 1. Since by [BL, Prop. 5.5.7], the Picard number of B is 1, ( B, H, D )is a polarized Abelian variety of quaternion type.There is a one-dimensional family (
B, H, D ) t of deformations of ( B, H, D ) and this corres-ponds to a one-dimensional family S t of deformations of S which parametrizes K3 surfaces withreal multiplication by the same class. By Abdulali’s Theorem 4.2.1, for t general the space ofHodge classes on ( B t ) ∼ A ( S t ) is generated by products of divisors and quaternion Weilcycles, that is by products of H and classes in W D . Denote the span of these products in H ( A ( S t ) , Q ) by F t . HE HODGE CONJECTURE FOR SELF-PRODUCTS OF CERTAIN K3 SURFACES 23
Since the class corresponding to √ d ∈ e T ( S t ) ⊗ e T ( S t ), the projection π : H ( A ( S t ) , Q ) → e T ( S t ) and the space F t are locally constant, there exists a locally constant class α t ∈ H ( A S t , Q )with the properties: • for all t we have π ⊗ π ( α t ) = √ d , • for all t we have α t ∈ F t .Now by Schoen’s and van Geemen’s Theorem 4.1.1 the space of Weil cycles W Q ( i ) is generatedby algebraic classes on any B t . It follows that W D is generated by algebraic classes andconsequently F t is generated by algebraic classes for any t . In particular, α t ∈ F t is algebraic.This proves the theorem. ✷ References [A] S. Abdulali,
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