Algebraic cycles on certain hyperkaehler fourfolds with an order 3 non-symplectic automorphism
aa r X i v : . [ m a t h . AG ] D ec ALGEBRAIC CYCLES ON CERTAIN HYPERK ¨AHLER FOURFOLDS WITH ANORDER NON–SYMPLECTIC AUTOMORPHISM
ROBERT LATERVEERA
BSTRACT . Let X be a hyperk¨ahler variety, and assume X has a non–symplectic automorphism σ of order > dim X . Bloch’s conjecture predicts that the quotient X/ < σ > should havetrivial Chow group of –cycles. We verify this for Fano varieties of lines on certain special cubicfourfolds having an order non–symplectic automorphism.
1. I
NTRODUCTION
Let X be a smooth projective variety over C , and let A i ( X ) := CH i ( X ) Q denote the Chowgroups of X (i.e. the groups of codimension i algebraic cycles on X with Q –coefficients, modulorational equivalence). Let A ihom ( X ) denote the subgroup of homologically trivial cycles. It doesnot seem an exaggeration to say that the field of algebraic cycles is filled with open questions [6],[15], [16], [21], [32]. Among these open questions, a prominent position is occupied by Bloch’sconjecture, proudly and sturdily overtowering the field like an unscalable mountain top. Conjecture 1.1 (Bloch [6]) . Let X be a smooth projective variety of dimension n . Let Γ ∈ A n ( X × X ) be such that Γ ∗ = 0 : H i ( X, O X ) → H i ( X, O X ) ∀ i > . Then Γ ∗ = 0 : A nhom ( X ) → A n ( X ) . A particular case of conjecture 1.1 is the following:
Conjecture 1.2 (Bloch [6]) . Let X be a smooth projective variety of dimension n . Assume that H i ( X, O X ) = 0 ∀ i > . Then A n ( X ) ∼ = Q . The “absolute version” (conjecture 1.2) is obtained from the “relative version” (conjecture 1.1)by taking Γ to be the diagonal. Conjecture 1.2 is famously open for surfaces of general type (cf.[22], [31], [12] for some recent progress). Mathematics Subject Classification.
Primary 14C15, 14C25, 14C30.
Key words and phrases.
Algebraic cycles, Chow groups, motives, Bloch’s conjecture, Bloch–Beilinson filtra-tion, hyperk¨ahler varieties, Fano varieties of lines on cubic fourfolds, multiplicative Chow–K¨unneth decomposition,splitting property, finite–dimensional motive.
Let us now suppose that X is a hyperk¨ahler variety (i.e., a projective irreducible holomorphicsymplectic manifold [2], [3]), say of dimension m . Suppose there exists a non–symplecticautomorphism σ ∈ Aut( X ) of order k > m . This implies that (cid:0) σ + σ + . . . + σ k (cid:1) ∗ = 0 : H i ( X, O X ) → H i ( X, O X ) ∀ i > . Conjecture 1.1 (applied to the correspondence
Γ = P kj =1 Γ σ j ∈ A m ( X × X ) , where Γ f denotes the graph of an automorphism f ∈ Aut( X ) ) then predicts the following: Conjecture 1.3.
Let X be a hyperk¨ahler variety of dimension m . Let σ ∈ Aut( X ) be an order k non–symplectic automorphism, and assume k > m . Then (cid:0) σ + σ + . . . + σ k (cid:1) ∗ = 0 : A mhom ( X ) → A m ( X ) . The main result of this note is that conjecture 1.3 is true for a certain family of hyperk¨ahlerfourfolds:
Theorem (=theorem 3.1) . Let Y ⊂ P ( C ) be a smooth cubic fourfold defined by an equation f ( X , X , X , X ) + g ( X , X ) = 0 , where f and g are homogeneous polynomials of degree . Let X = F ( Y ) be the Fano variety oflines in Y . Let σ ∈ Aut( X ) be the order automorphism induced by P ( C ) → P ( C ) , [ X : . . . : X ] [ X : X : X : X : νX : νX ] (where ν is a primitive rd root of unity).Then (id + σ + σ ) ∗ A hom ( X ) = 0 . As an immediate consequence of theorem 3.1, we find that Bloch’s conjecture 1.2 is verifiedfor the quotient:
Corollary (=corollary 4.1) . Let X and σ be as in theorem 3.1, and let Z := X/ < σ > be thequotient. Then A ( Z ) ∼ = Q . Another consequence (corollary 4.2) is that a certain instance of the generalized Hodge con-jecture is verified.The proof of theorem 3.1 relies on the theory of finite–dimensional motives [18], combinedwith the Fourier decomposition of the Chow ring of X constructed by Shen–Vial [24]. Conventions.
In this article, the word variety will refer to a reduced irreducible scheme of finitetype over C . A subvariety is a (possibly reducible) reduced subscheme which is equidimensional. All Chow groups will be with rational coefficients : we will denote by A j ( X ) the Chowgroup of j –dimensional cycles on X with Q –coefficients; for X smooth of dimension n thenotations A j ( X ) and A n − j ( X ) are used interchangeably.The notations A jhom ( X ) , A jAJ ( X ) will be used to indicate the subgroups of homologicallytrivial, resp. Abel–Jacobi trivial cycles. For a morphism f : X → Y , we will write Γ f ∈ LGEBRAIC CYCLES ON CERTAIN HK FOURFOLDS 3 A ∗ ( X × Y ) for the graph of f . The contravariant category of Chow motives (i.e., pure motiveswith respect to rational equivalence as in [23] , [21] ) will be denoted M rat .We will write H j ( X ) to indicate singular cohomology H j ( X, Q ) .
2. P
RELIMINARIES
Quotient varieties.Definition 2.1. A projective quotient variety is a variety Z = X/G , where X is a smooth projective variety and G ⊂ Aut ( X ) is a finite group. Proposition 2.2 (Fulton [11]) . Let Z be a projective quotient variety of dimension n . Let A ∗ ( Z ) denote the operational Chow cohomology ring. The natural map A i ( Z ) → A n − i ( Z ) is an isomorphism for all i .Proof. This is [11, Example 17.4.10]. (cid:3)
Remark 2.3.
It follows from proposition 2.2 that the formalism of correspondences goes throughunchanged for projective quotient varieties (this is also noted in [11, Example 16.1.13] ). Wemay thus consider motives ( Z, p, ∈ M rat , where Z is a projective quotient variety and p ∈ A n ( Z × Z ) is a projector. For a projective quotient variety Z = X/G , one readily proves (usingManin’s identity principle) that there is an isomorphism of motives h ( Z ) ∼ = h ( X ) G := ( X, ∆ G , in M rat , where ∆ G denotes the idempotent | G | P g ∈ G Γ g . Finite–dimensional motives.
We refer to [18, Definition 3.7] for the definition of finite–dimensional motive (cf. also [1], [13], [16], [21, Chapters 4 and 5] for further context andapplications). The following two results provide a lot of examples:
Theorem 2.4 (Kimura [18]) . Let X be a smooth projective variety, and assume X is dominatedby a product of curves. Then X has finite–dimensional motive.Proof. A smooth projective curve has finite–dimensional motive [18, Corollary 4.4]. Sincefinite–dimensionality is stable under taking products of varieties [18, Corollary 5.11], a prod-uct of curves has finite–dimensional motive. Applying [18, Proposition 6.9], this implies that X has finite–dimensional motive. (cid:3) Theorem 2.5.
Let X be a smooth projective variety, and let e X be the blow–up of X with smoothcenter Y ⊂ X . If X and Y have finite–dimensional motive, then also e X has finite–dimensionalmotive.Proof. This is well–known, and follows from the blow–up formula for Chow motives [23, The-orem 2.8]. (cid:3)
ROBERT LATERVEER
An essential property of varieties with finite–dimensional motive is embodied by the nilpo-tence theorem:
Theorem 2.6 (Kimura [18]) . Let X be a smooth projective variety of dimension n with finite–dimensional motive. Let Γ ∈ A n ( X × X ) be a correspondence which is numerically trivial.Then there is N ∈ N such that Γ ◦ N = 0 ∈ A n ( X × X ) . Proof.
This is [18, Proposition 7.5]. (cid:3)
Actually, the nilpotence property (for all powers of X ) could serve as an alternative definitionof finite–dimensional motive, as shown by Jannsen [16, Corollary 3.9].Conjecturally, any variety has finite–dimensional motive [18, Conjecture 7.1]; we are still farfrom knowing this.2.3. MCK decomposition.Definition 2.7 (Murre [20]) . Let X be a projective quotient variety of dimension n . We say that X has a CK decomposition if there exists a decomposition of the diagonal ∆ X = π + π + · · · + π n in A n ( X × X ) , such that the π i are mutually orthogonal idempotents and ( π i ) ∗ H ∗ ( X ) = H i ( X ) .(NB: “CK decomposition” is shorthand for “Chow–K¨unneth decomposition”.) Remark 2.8.
The existence of a CK decomposition for any smooth projective variety is part ofMurre’s conjectures [20] , [15] . Definition 2.9 (Shen–Vial [24]) . Let X be a projective quotient variety of dimension n . Let ∆ smX ∈ A n ( X × X × X ) be the class of the small diagonal ∆ smX := (cid:8) ( x, x, x ) | x ∈ X (cid:9) ⊂ X × X × X . An MCK decomposition is a CK decomposition { π Xi } of X that is multiplicative , i.e. it satisfies π Xk ◦ ∆ smX ◦ ( π Xi × π Xj ) = 0 in A n ( X × X × X ) for all i + j = k . (NB: “MCK decomposition” is shorthand for “multiplicative Chow–K¨unneth decomposition”.)A weak MCK decomposition is a CK decomposition { π Xi } of X that satisfies (cid:16) π Xk ◦ ∆ smX ◦ ( π Xi × π Xj ) (cid:17) ∗ ( a × b ) = 0 for all a, b ∈ A ∗ ( X ) . Remark 2.10.
The small diagonal (seen as a correspondence from X × X to X ) induces the multiplication morphism ∆ smX : h ( X ) ⊗ h ( X ) → h ( X ) in M rat . Suppose X has a CK decomposition h ( X ) = n M i =0 h i ( X ) in M rat . LGEBRAIC CYCLES ON CERTAIN HK FOURFOLDS 5
By definition, this decomposition is multiplicative if for any i, j the composition h i ( X ) ⊗ h j ( X ) → h ( X ) ⊗ h ( X ) ∆ smX −−→ h ( X ) in M rat factors through h i + j ( X ) .If X has a weak MCK decomposition, then setting A i ( j ) ( X ) := ( π X i − j ) ∗ A i ( X ) , one obtains a bigraded ring structure on the Chow ring: that is, the intersection product sends A i ( j ) ( X ) ⊗ A i ′ ( j ′ ) ( X ) to A i + i ′ ( j + j ′ ) ( X ) .It is expected (but not proven !) that for any X with a weak MCK decomposition, one has A i ( j ) ( X ) ?? = 0 for j < , A i (0) ( X ) ∩ A ihom ( X ) ?? = 0 ; this is related to Murre’s conjectures B and D, that have been formulated for any CK decompo-sition [20] .The property of having an MCK decomposition is severely restrictive, and is closely relatedto Beauville’s “(weak) splitting property” [5] . For more ample discussion, and examples ofvarieties with an MCK decomposition, we refer to [24, Section 8] , as well as [29] , [25] , [10] . The Fourier decomposition.
In what follows, we will make use of the following:
Theorem 2.11 (Shen–Vial [24]) . Let Y ⊂ P ( C ) be a smooth cubic fourfold, and let X := F ( Y ) be the Fano variety of lines in Y . There exists a self–dual CK decomposition { Π Xi } for X , and (Π X i − j ) ∗ A i ( X ) = A i ( j ) ( X ) , where the right–hand side denotes the splitting of the Chow groups defined in terms of the Fouriertransform as in [24, Theorem 2] . Moreover, we have A i ( j ) ( X ) = 0 if j < or j > i or j is odd . In case Y is very general, the Fourier decomposition A ∗ ( ∗ ) ( X ) forms a bigraded ring, andhence { π Xi } is a weak MCK decomposition.Proof. (A matter of notation: what we denote A i ( j ) ( X ) is denoted CH i ( X ) j in [24].)The existence of a self–dual CK decomposition { Π Xi } is [24, Theorem 3.3]. (More in detail:[24, Theorem 3.3] applies to any hyperk¨ahler fourfold F of K [2] type with a cycle class L ∈ A ( F × F ) that represents the Beauville–Bogomolov pairing and satisfies [24, equalities (6), (7),(8), (9)]. For the Fano variety of lines of a cubic fourfold, the cycle L of [24, definition (107)]has these properties, as shown in [24, Section 3].)According to [24, Theorem 3.3], the given CK decomposition agrees with the Fourier decom-position of the Chow groups. The “moreover” part is because the { Π Xi } are shown to satisfyMurre’s conjecture B [24, Theorem 3.3].The statement for very general cubics is [24, Theorem 3]. (cid:3) ROBERT LATERVEER
Remark 2.12.
Unfortunately, it is not yet known that the Fourier decomposition of [24] inducesa bigraded ring structure on the Chow ring for all
Fano varieties X of smooth cubic fourfolds.For one thing, it has not yet been proven that A ( X ) · A ( X ) ?? ⊂ A ( X ) (cf. [24, Section 22.3] for discussion). Refined CK decomposition.Theorem 2.13.
Let X be a smooth projective hyperk¨ahler fourfold of K [2] –type. Assume that X has finite–dimensional motive. Then X has a CK decomposition { π Xi } . Moreover, there existsa further splitting π X = π X , + π X , in A ( X × X ) , where π X , and π X , are orthogonal idempotents, and π X , is supported on C × D ⊂ X × X ,where C and D are a curve, resp. a divisor on X . The action on cohomology verifies ( π X , ) ∗ H ∗ ( X ) = H tr ( X ) , where H tr ( X ) ⊂ H ( X ) is defined as the orthogonal complement of N S ( X ) with respect to theBeauville–Bogomolov form. The action on Chow groups verifies ( π X , ) ∗ A ( X ) = ( π X ) ∗ A ( X ) . Proof.
It is known [9] that X verifies the Lefschetz standard conjecture B ( X ) . Combined withfinite–dimensionality, this implies the existence of a CK decomposition [15, Lemma 5.4].For the “moreover” statement, one observes that X verifies conditions (*) and (**) of Vial’s[28], and so [28, Theorems 1 and 2] apply. This gives the existence of refined CK projectors π Xi,j ,which act on cohomology as projectors on gradeds for the “niveau filtration” e N ∗ of loc. cit. Inparticular, π X , acts as projector on N S ( X ) , and π X , acts as projector on H tr ( X ) . The projector π X , , being supported on C × D , acts trivially on A ( X ) for dimension reasons; this proves thelast equality. (cid:3)
3. M
AIN RESULT
Theorem 3.1.
Let Y ⊂ P ( C ) be a smooth cubic fourfold defined by an equation f ( X , X , X , X ) + g ( X , X ) = 0 , where f and g are homogeneous polynomials of degree . Let X = F ( Y ) be the Fano variety oflines in Y . Let σ ∈ Aut( X ) be the order non–symplectic automorphism induced by σ P : P ( C ) → P ( C ) , [ X : . . . : X ] [ X : X : X : X : νX : νX ] , where ν is a primitive rd root of unity.Then (id + σ + σ ) ∗ A i ( j ) ( X ) = 0 for ( i, j ) ∈ { (2 , , (4 , , (4 , } . In particular, (id + σ + σ ) ∗ A hom ( X ) = 0 . LGEBRAIC CYCLES ON CERTAIN HK FOURFOLDS 7
Proof. (NB: the family of Fano varieties of theorem 3.1 is described in [8, Example 6.5], fromwhich I learned that the automorphism σ is non–symplectic.)The last phrase of the theorem follows from the one–but–last phrase, since A hom ( X ) = A ( X ) ⊕ A ( X ) [24, Theorem 4].In a first step of the proof, let us show that the automorphism σ respects (most of) the Fourierdecomposition of the Chow ring: Proposition 3.2.
Let X and σ be as in theorem 3.1. Let A ∗ ( ∗ ) ( X ) be the Fourier decomposition(theorem 2.11). Then σ ∗ A i ( j ) ( X ) ⊂ A i ( j ) ( X ) ∀ ( i, j ) = (2 , . Proof.
Here, the alternative description of the Fourier decomposition A ∗ ( ∗ ) ( X ) in terms of a cer-tain rational map φ : X X comes in handy.Let Y ⊂ P ( C ) be any smooth cubic fourfold (not necessarily with automorphisms), and let X = F ( Y ) be the Fano variety of lines in Y . There exists a degree rational map φ : X X [30], [24, Section 18]. The map φ is defined as follows: Let x ∈ X be a point, and let ℓ ⊂ Y bethe line corresponding to x . For a general point x ∈ X , there is a unique plane H ⊂ P that istangent to Y along ℓ . Then φ ( x ) ∈ X is defined as the point corresponding to ℓ ′ ⊂ Y , where H ∩ Y = 2 ℓ + ℓ ′ . As in [24, Definition 21.8], for any λ ∈ Q let us consider the eigenspaces V iλ := (cid:8) c ∈ A i ( X ) | φ ∗ ( c ) = λ · c (cid:9) . These eigenspaces are related to the Fourier decomposition of the Chow ring: indeed, Shen–Vial show [24, Theorem 21.9 and Proposition 21.10] that there is a decomposition(1) A i ( j ) ( X ) = V iλ ⊕ · · · ⊕ V iλ r ∀ ( i, j ) = (2 , . Let us now return to X and σ as in theorem 3.1, and let us prove proposition 3.2. In view ofthe decomposition (1), we see that to prove proposition 3.2, it suffices to prove the following: Claim 3.3.
Let X and σ be as in theorem 3.1. Then φ ∗ σ ∗ = σ ∗ φ ∗ : A i ( X ) → A i ( X ) . In order to prove the claim, we first establish a little lemma:
Lemma 3.4.
Set–up as above. There is an equality of rational maps φ ◦ σ = σ ◦ φ : X X .
Proof.
Let x ∈ X be a point outside of the indeterminacy locus of φ , and let H ⊂ P be theplane tangent to Y along the line ℓ corresponding to x . By definition, φ ( x ) ∈ X is the pointcorresponding to ℓ ′ ⊂ Y , where H ∩ Y = 2 ℓ + ℓ ′ . ROBERT LATERVEER
Let σ P : P → P denote the linear transformation inducing the automorphism σ . The plane σ P ( H ) is tangent to Y along σ P ( ℓ ) , and σ P ( H ) ∩ Y = 2 σ P ( ℓ ) + σ P ( ℓ ′ ) . It follows that φ ( σ ( x )) = σ ( φ ( x )) . (cid:3) Lemma 3.4 furnishes a commutative diagram X φ X ↓ σ ↓ σ X φ X .
This can be “resolved” by a commutative diagram X p ′ ← Z ′ q ′ → X ↓ σ ↓ σ Z ↓ σ X p ← Z q → X , where horizontal arrows are birational morphisms such that φ ◦ p ′ = q ′ and φ ◦ p = q (and so φ ∗ = p ∗ q ∗ = ( p ′ ) ∗ ( q ′ ) ∗ : A i ( X ) → A i ( X ) .).Let us now prove claim 3.3. We have equalities φ ∗ σ ∗ = ( p ′ ) ∗ ( q ′ ) ∗ σ ∗ = ( p ′ ) ∗ ( σ Z ) ∗ q ∗ = σ ∗ p ∗ q ∗ = σ ∗ φ ∗ : A i ( X ) → A i ( X ) . Here, in the third equality we have used the following:
Sublemma 3.5.
Set–up as above. There is equality ( p ′ ) ∗ ( σ Z ) ∗ = σ ∗ p ∗ : A i ( Z ) → A i ( X ) . Proof.
Since σ Z is a birational morphism, there is equality σ ∗ ( p ′ ) ∗ ( σ Z ) ∗ = p ∗ ( σ Z ) ∗ ( σ Z ) ∗ = p ∗ : A i ( Z ) → A i ( X ) . Composing on the left with σ ∗ , this implies σ ∗ σ ∗ ( p ′ ) ∗ ( σ Z ) ∗ = σ ∗ p ∗ : A i ( Z ) → A i ( X ) . But σ ∗ = ( σ ) ∗ and so the left–hand side simplifies to ( p ′ ) ∗ ( σ Z ) ∗ , proving the sublemma. (cid:3)(cid:3) For later use, we recast proposition 3.2 as follows:
Corollary 3.6.
Set–up as above. Let { Π Xj } be a CK decomposition as in theorem 2.11. Then σ ∗ (Π Xj ) ∗ = (Π Xj ) ∗ σ ∗ (Π Xj ) ∗ : A i ( X ) → A i ( X ) ∀ ( i, j ) = (2 , . LGEBRAIC CYCLES ON CERTAIN HK FOURFOLDS 9
Proof.
This is just a translation of proposition 3.2, using the fact that Π Xj acts on A i ( X ) asprojector on A i (2 i − j ) ( X ) . (cid:3) The second step of the proof is to ascertain that X has finite–dimensional motive: Proposition 3.7.
Let Y ⊂ P ( C ) and X = F ( Y ) be as in theorem 3.1. Then Y and X havefinite–dimensional motive.Proof. To establish finite–dimensionality of Y is an easy exercice in using what is known as the“Shioda inductive structure” [26], [17]. Indeed, applying [17, Remark 1.10], we find there existsa dominant rational map φ : Y × Y Y , where Y ⊂ P ( C ) is the smooth cubic threefold defined as f ( X , X , X , X ) + V = 0 , and Y ⊂ P ( C ) is the smooth cubic curve defined as g ( X , X ) + W = 0 . The indeterminacy locus of φ is resolved by blowing up the locus S × P ⊂ Y × Y , where S ⊂ Y is a cubic surface, and P ⊂ Y is a set of points. Let us call this blow–up ˆ Y . Using theorems 2.4and 2.5 and an induction on the dimension, we find that ˆ Y has finite–dimensional motive. Since ˆ Y dominates Y , it follows from [18, Proposition 6.9] that the cubic Y has finite–dimensionalmotive.Finally, [19, Theorem 4] states that for any cubic Y with finite–dimensional motive, the Fanovariety X = F ( Y ) also has finite–dimensional motive. (cid:3) The third step of the proof is to show the desired statement for A ( X ) , i.e. we now provethat(2) (id + σ + σ ) ∗ A ( X ) = 0 . In order to do so, let us abbreviate ∆ G := 13 (cid:0) ∆ X + Γ σ + Γ σ ◦ σ (cid:1) ∈ A ( X × X ) . Since the action of σ is non–symplectic [8, Example 6.5 and Lemma 6.2], we have that (∆ G ) ∗ = 0 : H ( X, O X ) → H ( X, O X ) . Using the Lefschetz (1 , –theorem, we see that ∆ G ◦ Π X = γ in H ( X × X ) , where γ is some cycle supported on D × D ⊂ X × X , for some divisor D ⊂ X . In other words,the correspondence Γ := ∆ G ◦ Π X − γ ∈ A ( X × X ) is homologically trivial. But then (since X has finite-dimensional motive) there exists N ∈ N such that Γ ◦ N = 0 in A ( X × X ) . Upon developing this expression, one finds an equality Γ ◦ N = (∆ G ◦ Π X ) ◦ N + γ ′ = 0 in A ( X × X ) , where γ ′ is supported on D × D ⊂ X × X . In particular, γ ′ acts trivially on A ( X ) ⊂ A AJ ( X ) ,and so (cid:0) (∆ G ◦ Π X ) ◦ N (cid:1) ∗ = 0 : A ( X ) → A ( X ) . Corollary 3.6 (combined with the fact that ∆ G and Π X are idempotents) implies that (cid:0) (∆ G ◦ Π X ) ◦ N (cid:1) ∗ = (∆ G ◦ Π X ) ∗ : A i ( X ) → A i ( X ) , and so we find that (cid:0) ∆ G ◦ Π X (cid:1) ∗ = (∆ G ) ∗ = 0 : A ( X ) → A ( X ) . This proves equality (2).The argument for A ( X ) is similar: the correspondence Γ being homologically trivial, itstranspose t Γ = Π X ◦ ∆ G − γ ′′ ∈ A ( X × X ) is also homologically trivial (where γ ′′ is supported on D × D ). Using nilpotence and lemma3.6, this implies (just as above) that(3) (Π X ◦ ∆ G ) ∗ = (cid:0) ∆ G ◦ Π X (cid:1) ∗ = (∆ G ) ∗ = 0 : A ( X ) → A ( X ) . In the final step of the proof, it remains to consider the action on A ( X ) . Ideally, one wouldlike to use Vial’s projector π X , of [28] (mentioned in the proof of theorem 2.13). Unfortunately,this approach runs into problems (cf. remark 3.10). We therefore proceed somewhat differently:to establish the statement for A ( X ) , we use the following proposition: Proposition 3.8.
Notation as above. One has ∆ G ◦ Π X − R = 0 in H ( X × X ) , where R ∈ A ( X × X ) is a correspondence with the property that R ∗ = 0 : A ( X ) → A ( X ) . Obviously, this proposition clinches the proof: using the nilpotence theorem, one sees thatthere exists N ∈ N such that (cid:0) ∆ G ◦ Π X + R (cid:1) ◦ N = 0 in A ( X × X ) . Developing, and applying the result to A ( X ) , one finds that (cid:0) (∆ G ◦ Π X ) ◦ N (cid:1) ∗ = 0 : A ( X ) → A ( X ) . Corollary 3.6 (combined with the fact that ∆ G and Π X are idempotents) implies that (cid:0) (∆ G ◦ Π X ) ◦ N (cid:1) ∗ = (∆ G ◦ Π X ) ∗ : A i ( X ) → A i ( X ) ∀ i = 2 . Therefore, we conclude that (cid:0) ∆ G ◦ Π X (cid:1) ∗ = (∆ G ) ∗ = 0 : A ( X ) → A ( X ) . LGEBRAIC CYCLES ON CERTAIN HK FOURFOLDS 11
It only remains to prove proposition 3.8. Here, we use the fact that X is of K [2] –type and sothere is an isomorphism H ( X ) = Sym H ( X ) [4, Proposition 3]. Using the truth of the standard conjectures for X [9, Theorem 1.1], and thesemi–simplicity of motives for numerical equivalence [14, Theorem 1], this means that the map ∆ sm : h ( X ) ⊗ h ( X ) → h ( X ) in M hom admits a right–inverse, where ∆ sm ∈ A (( X × X ) × X ) is as before the “small diagonal” (cf.definition 2.9). Let Ψ ∈ A ( X × ( X × X )) denote this right–inverse.Using the splitting π X = π X , + π X , in A ( X × X ) of theorem 2.13, one obtains a splittingmodulo homological equivalence of Π X in 4 components Π X = Π X ◦ ∆ sm ◦ ( π X × π X ) ◦ Ψ ◦ Π X = Π X ◦ ∆ sm ◦ (cid:0) ( π X , + π X , ) × ( π X , + π X , ) (cid:1) ◦ Ψ ◦ Π X = X k,ℓ ∈{ , } Π X ◦ ∆ sm ◦ ( π X ,k × π X ,ℓ ) ◦ Ψ ◦ Π X =: X k,ℓ ∈{ , } Π X ,k,ℓ in H ( X × X ) . We note that (by construction) Π X , , acts as a projector onSym H tr ( X ) ⊂ Sym H ( X ) = H ( X ) . Also, we recall that π X , is supported on C × D ⊂ X × X (theorem 2.13), which implies that Π X ,k,ℓ ∈ A ( X × X ) is supported on X × D for ( k, ℓ ) = (0 , .It will be convenient to consider the transpose decomposition Π X = t Π X = t Π X , , + t Π X , , + t Π X , , + t Π X , , in H ( X × X ) (where we have used that Π X is transpose–invariant, cf. theorem 2.11).This decomposition induces in particular a decomposition ∆ G ◦ Π X = ∆ G ◦ t Π X , , + ∆ G ◦ t Π X , , + t ∆ G ◦ Π X , , + t ∆ G ◦ Π X , , in H ( X × X ) . The last summands in this decomposition act trivially on A ( X ) (indeed, the correspondence t Π X ,k,ℓ is supported on D × X ⊂ X × X for ( k, ℓ ) = (0 , , and hence acts trivially on A ( X ) ).These last summands will form the correspondence called R in proposition 3.8. To proveproposition 3.8, it remains to establish that(4) ∆ G ◦ t Π X , , = 0 in H ( X × X ) . Taking transpose, one sees this is equivalent to proving that Π X , , ◦ ∆ G = 0 in H ( X × X ) , which in turn (since applying σ ∗ and projecting to Sym H tr ( X ) commute) is equivalent to prov-ing that ∆ G ◦ Π X , , = 0 in H ( X × X ) . Invoking Manin’s identity principle, it suffices to prove that σ ∗ ( c ∪ c ) + ( σ ) ∗ ( c ∪ c ) + c ∪ c = 0 in H ( X ) ∀ c , c ∈ H tr ( X ) . Thanks to the equality c ∪ c = 12 (cid:16) ( c + c ) ∪ ( c + c ) − c ∪ c − c ∪ c (cid:17) , it suffices to prove that(5) σ ∗ ( c ∪ c ) + ( σ ) ∗ ( c ∪ c ) + c ∪ c = 0 in H ( X ) ∀ c ∈ H tr ( X ) . We now make the following claim:
Claim 3.9.
Set–up as above. Let c ∈ H tr ( X ) . Then ( σ ∗ )( c ) ∪ ( σ ) ∗ ( c ) = c ∪ c in H ( X ) . It is readily checked that claim 3.9 implies equality (5) (and hence equality (4) and hence alsoproposition 3.8): We have σ ∗ ( c ∪ c ) + ( σ ) ∗ ( c ∪ c ) + c ∪ c = (cid:0) σ ∗ ( c ) + ( σ ) ∗ ( c ) (cid:1) ∪ − σ ∗ ( c ) ∪ ( σ ) ∗ ( c ) + c ∪ c = 2 c ∪ c − σ ∗ ( c ) ∪ ( σ ) ∗ ( c )= 0 in H ( X ) , proving equality (5). (Here, the second equality is because σ ∗ ( c ) + ( σ ) ∗ ( c ) = − c , and the thirdequality is the claim.)Let us now prove claim 3.9. The point is that the subgroup H := (cid:8) c ∈ H ( X ) | ( σ ∗ )( c ) ∪ ( σ ) ∗ ( c ) = c ∪ c in H ( X ) (cid:9) ⊂ H ( X ) , together with its complexification H C , defines a sub–Hodge structure of H ( X ) . Let ω ∈ H , ( X ) be a generator. Then ω is in H C (since σ ∗ ω = ν · ω , with ν = 1 , ν primitive).But H tr ( X ) ⊂ H ( X ) is the smallest sub–Hodge structure containing ω , and so we must have H tr ( X ) ⊂ H , which proves claim 3.9. (cid:3)
Remark 3.10.
To prove the statement for A ( X ) in the final step of the above proof, it wouldbe natural to try and use Vial’s projector π X , of [28, Theorems 1 and 2] (mentioned in the proofof theorem 2.13). However, this approach is difficult to put into practice: the problem is that itseems impossible to prove that ∆ G ◦ π X , = 0 in H ( X × X ) , short of knowing that (1) H ( X ) ∩ F = N H ( X ) , and (2) N H ( X ) = e N H ( X ) , where N ∗ is the usual coniveau filtration and e N ∗ is Vial’s niveau filtration. Both (1) and (2) seem difficult. LGEBRAIC CYCLES ON CERTAIN HK FOURFOLDS 13
4. S
OME COROLLARIES
Corollary 4.1.
Let X and σ be as in theorem 3.1. Let Z := X/ < σ > be the quotient. Then A ( Z ) ∼ = Q . Proof.
We have a natural isomorphism A ( Z ) ∼ = A ( X ) σ . But theorem 3.1 (combined with thefact that σ ∗ A j ) ( X ) ⊂ A j ) ( X ) for all j , cf. proposition 3.2) implies that A ( X ) σ ⊂ A ( X ) . Since there exists a σ –invariant ample divisor L ∈ A ( X ) , and L generates the –dimensional Q –vector space A ( X ) , there is equality A ( X ) σ = A ( X ) . (cid:3) Corollary 4.2.
Let X and σ be as in theorem 3.1. Then the invariant part of cohomology H ( X ) σ ⊂ H ( X ) is supported on a divisor.Proof. This follows from theorem 3.1 by applying the Bloch–Srinivas “decomposition of thediagonal” argument [7]. For the benefit of readers not familiar with [7], we briefly resume thisargument.Let k ⊂ C be a subfield such that X and ∆ G are defined over k , and such that k is finitelygenerated over Q . Let k ( X ) denote the function field of X k . Since there is an embedding k ( X ) ⊂ C , there is a natural homomorphism A ∗ ( X k ( X ) ) → A ∗ ( X C ) that is injective [6, Appendix to Lecture 1]. In particular, there is an injective homomorphism A ( X k ( X ) ) σ ֒ → A ( X C ) σ . As the right–hand side has dimension (theorem 3.1), it follows that also dim A ( X k ( X ) ) σ = 1 . We now consider the image of ∆ G ∈ A ( X k × X k ) σ × σ under the restriction homomorphism A ( X k × X k ) σ × σ → lim −→ A ( X k × U ) σ ∼ = A ( X k ( X ) ) σ = Q (here the limit is over Zariski opens U ⊂ X k , and the isomorphism follows from [6, Appendixto Lecture 1]). This gives a decomposition ∆ G = x × X + γ in A ( X k × X k ) , where γ is supported on X × D for some divisor D ⊂ X . Considering this decomposition for X = X C , and looking at the action of correspondences on cohomology, we find that H ( X ) σ = (∆ G ) ∗ H ( X ) = γ ∗ H ( X ) , and thus H ( X ) σ is supported on the divisor D . (cid:3) Acknowledgements .
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