aa r X i v : . [ m a t h . AG ] D ec ALGEBRAIC CYCLES AND EPW CUBES
ROBERT LATERVEERA
BSTRACT . Let X be a hyperk¨ahler variety with an anti–symplectic involution ι . According toBeauville’s conjectural “splitting property”, the Chow groups of X should split in a finite numberof pieces such that the Chow ring has a bigrading. The Bloch–Beilinson conjectures predict how ι should act on certain of these pieces of the Chow groups. We verify part of this conjecture fora –dimensional family of hyperk¨ahler sixfolds that are “double EPW cubes” (in the sense ofIliev–Kapustka–Kapustka–Ranestad). This has interesting consequences for the Chow ring of thequotient X/ι , which is an “EPW cube” (in the sense of Iliev–Kapustka–Kapustka–Ranestad).
1. I
NTRODUCTION
For a smooth projective variety X over C , let us write A i ( X ) := CH i ( X ) ⊗ Q to denote the Chow groups of X (i.e. codimension i algebraic cycles on X modulo rationalequivalence), with Q –coefficients. As is well–known (and explained for instance in [17], [44],[29]), the Bloch–Beilinson conjectures form a powerful and coherent heuristic guide, useful informulating concrete predictions about Chow groups and their relation to cohomology. This noteis about one instance of such a prediction, concerning non–symplectic involutions on hyperk¨ahlervarieties.Let X be a hyperk¨ahler variety (i.e., a projective irreducible holomorphic symplectic manifold,cf. [1], [2]), and suppose X has an anti–symplectic involution ι . The action of ι on the subring H ∗ , ( X ) is well–understood: we have ι ∗ = − id : H i, ( X ) → H i, ( X ) for i odd ,ι ∗ = id : H i, ( X ) → H i, ( X ) for i even . The action of ι on the Chow ring A ∗ ( X ) is more mysterious. To state the conjectural behaviour,we will now assume the Chow ring of X has a bigraded ring structure A ∗ ( ∗ ) ( X ) , where each A i ( X ) splits into pieces A i ( X ) = M j A i ( j ) ( X ) , Mathematics Subject Classification.
Primary 14C15, 14C25, 14C30.
Key words and phrases.
Algebraic cycles, Chow groups, motives, Bloch’s conjecture, Bloch–Beilinson filtration,hyperk¨ahler varieties, (double) EPW cubes, K surfaces, non–symplectic involution, multiplicative Chow–K¨unnethdecomposition, splitting property. and the piece A i ( j ) ( X ) is isomorphic to the graded Gr jF A i ( X ) for the Bloch–Beilinson filtrationthat conjecturally exists for all smooth projective varieties. (Such a bigrading A ∗ ( ∗ ) ( − ) is expectedto exist for all hyperk¨ahler varieties; this is Beauville’s conjectural “splitting property” [3].)Since the pieces A i ( i ) ( X ) and A dim Xi ( X ) should only depend on the subring H ∗ , ( X ) , we areled to the following conjecture: Conjecture 1.1.
Let X be a hyperk¨ahler variety of dimension m , and let ι ∈ Aut( X ) be ananti–symplectic involution. Then ι ∗ = ( − i id : A i (2 i ) ( X ) → A i ( X ) ,ι ∗ = ( − i id : A m (2 i ) ( X ) → A m ( X ) . This conjecture is studied, and proven in some particular cases, in [22], [24], [23], [25], [26].The aim of this note is to provide some more examples where conjecture 1.1 is verified, byconsidering “double EPW cubes” in the sense of [15] (cf. also subsection 2.7 below). A doubleEPW cube is a –dimensional hyperk¨ahler variety X A , constructed as double cover X A → D A , where D A is a slightly singular subvariety of a Grassmannian (the variety D A is called an “EPWcube”). As shown in [15], double EPW cubes correspond to a –dimensional irreducible (andunirational) component of the moduli space of hyperk¨ahler sixfolds. A double EPW cube X A comes equipped with the covering involution ι A : X A → X A which is anti–symplectic (remark 2.24).The main result of this note is a partial verification of conjecture 1.1 for a –dimensionalfamily of double EPW cubes: Theorem (=theorem 4.1) . Let X be a double EPW cube, and assume X = X A for A ∈ ∆ general (where ∆ ⊂ LG ν is the divisor of theorem 2.23). Let ι = ι A ∈ Aut( X ) be the anti–symplectic involution. Then ι ∗ = − id : A ( X ) → A ( X ) , (Π X ) ∗ ι ∗ = − id : A ( X ) → A ( X ) . The divisor ∆ is such that for A ∈ ∆ general, the double EPW cube X A is birational toa Hilbert scheme ( S A ) [3] , where S A is a degree K surface. Since Hilbert schemes S [ m ] of K surfaces S have a multiplicative Chow–K¨unneth decomposition [38], double EPW cubes X = X A as in theorem 4.1 have a bigraded Chow ring A ∗ ( ∗ ) ( X ) (cf. corollary 2.26 below). Thecorrespondence Π X is a projector on A ( X ) .To prove theorem 4.1, we employ the method of “spread” of algebraic cycles as developed byVoisin [41], [42]. Theorem 4.1 has some rather striking consequences for the Chow ring of theEPW cubes in the –dimensional family under consideration (these consequences exploit theexistence of a multiplicative Chow–K¨unneth decomposition for X as in theorem 4.1): LGEBRAIC CYCLES AND EPW CUBES 3
Corollary (=corollary 5.1) . Let D = D A be an EPW cube for A ∈ ∆ general.(i) Let a ∈ A ( D ) be a –cycle which is either in the image of the intersection product map A ( D ) ⊗ A ( D ) ⊗ A ( D ) → A ( D ) , or in the image of the intersection product map A ( D ) ⊗ A ( D ) ⊗ A ( D ) → A ( D ) . Then a is rationally trivial if and only if a has degree .(ii) Let a ∈ A ( D ) be a –cycle which is in the image of the intersection product map A ( D ) ⊗ A ( D ) ⊗ A ( D ) → A ( D ) . Then a is rationally trivial if and only if a is homologically trivial. (NB: the EPW cube D is not smooth, but it is a quotient of a smooth variety; as such, theChow groups of D still have a ring structure, cf. subsection 2.1 below.)Corollary 5.1 is similar to multiplicative results in the Chow ring of K surfaces [4], in theChow ring of Hilbert schemes of K surfaces and of abelian surfaces [38], and in the Chowring of Calabi–Yau complete intersections [40], [10]. A more general version of corollary 5.1,concerning certain product varieties, can be proven similarly (corollary 5.5).It is my hope this note will stimulate further research on this topic. For one thing, it would beinteresting to prove theorem 4.1 for all double EPW cubes, and corollary 5.1 for all EPW cubes.
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 ) := CH j ( X ) ⊗ Q the Chow group of j –dimensional cycles on X with Q –coefficients. For X smooth of dimension n we will write A i ( X ) := A n − i ( X ) . The notations A ihom ( X ) , A iAJ ( X ) , A ialg ( X ) will be used to indicate the subgroups of homo-logically trivial, resp. Abel–Jacobi trivial, resp. algebraically trivial cycles. For a morphism f : X → Y , we will write Γ f ∈ A ∗ ( X × Y ) for the graph of f . The contravariant category of Chow motives (i.e., pure motives with respectto rational equivalence as in [34] , [29] ) will be denoted M rat .We will use 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 X = Y /G , where Y is a smooth projective variety and G ⊂ Aut ( Y ) is a finite group. ROBERT LATERVEER
Proposition 2.2 (Fulton [12]) . Let X be a projective quotient variety of dimension n . Let A ∗ ( X ) denote the operational Chow cohomology ring. The natural map A i ( X ) → A n − i ( X ) is an isomorphism for all i .Proof. This is [12, 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 [12, Example 16.1.13] ). Wecan thus consider motives ( X, p, ∈ M rat , where X is a projective quotient variety and p ∈ A n ( X × X ) is a projector. For a projective quotient variety X = Y /G , one readily proves (usingManin’s identity principle) that there is an isomorphism h ( X ) ∼ = h ( Y ) G := ( Y, ∆ GY , in M rat , where ∆ GY denotes the idempotent | G | P g ∈ G Γ g . MCK decomposition.Definition 2.4 (Murre [28]) . Let X be a smooth projective 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 in A n ( X × X ) and ( π i ) ∗ H ∗ ( X ) = H i ( X ) .(NB: “CK decomposition” is shorthand for “Chow–K¨unneth decomposition”.) Remark 2.5.
The existence of a CK decomposition for any smooth projective variety is part ofMurre’s conjectures [28] , [17] , [19] . Definition 2.6 (Shen–Vial [35]) . Let X be a smooth projective 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 2.7.
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 AND EPW CUBES 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 [28] .The property of having an MCK decomposition is severely restrictive, and is closely relatedto Beauville’s “(weak) splitting property” [3] . For more ample discussion, and examples ofvarieties with an MCK decomposition, we refer to [35, Section 8] , as well as [38] , [36] , [11] . Lemma 2.8.
Let
X, X ′ be birational hyperk¨ahler varieties. Then X has an MCK decompositionif and only if X ′ has one.Proof. This is noted in [38, Introduction]; the idea is that Rieß’s result [33] implies that X and X ′ have isomorphic Chow motives and the isomorphism is compatible with the multiplicativestructure. (For a detailed proof, cf. [22, Lemma 2.13].) (cid:3) MCK for S [ m ] .Theorem 2.9 (Vial [38]) . Let S be a projective K surface, and let X = S [ m ] be the Hilbertscheme of length m subschemes of S . Then X has a self–dual MCK decomposition { Π Xi } . Inparticular, A ∗ ( X ) = A ∗ ( ∗ ) ( X ) is a bigraded ring, where A i ( X ) = i M j =2 i − n A i ( j ) ( X ) , and A i ( j ) ( X ) = 0 for j odd.Proof. This is [38, Theorems 1 and 2]. (cid:3)
Remark 2.10.
Let X be as in theorem 2.9 and suppose m = 2 (i.e. X = S [2] is a hyperk¨ahlerfourfold). Then the bigrading A ∗ ( ∗ ) ( X ) of theorem 2.9 has an interesting alternative descriptionin terms of a Fourier operator on Chow groups [35] . For m > , there is no such “Fourieroperator” description of the bigrading A ∗ ( ∗ ) ( S [ m ] ) ; the bigrading is defined exclusively by anMCK decomposition.Another point particular to m = 2 is that (thanks to [35] ) we know that A i ( j ) ( S [2] ) = 0 ∀ j < . This vanishing statement is (conjecturally true but) open for S [ m ] with m > . ROBERT LATERVEER
Any K surface S has an MCK decomposition [35, Example 8.17]. Since this property isstable under products [35, Theorem 8.6], S m also has an MCK decomposition. The followinglemma records a basic compatibility between the bigradings on A ∗ ( S [ m ] ) and on A ∗ ( S m ) : Lemma 2.11.
Let S be a K surface, and let X = S [ m ] . Let Φ ∈ A m ( X × S m ) be thecorrespondence coming from the diagram S [ m ] ←− f S mh ↓ ↓ S ( m ) g ←− S m (the arrow labelled h is the Hilbert–Chow morphism; the right vertical arrow is the blow–up ofthe diagonal). Then (Φ) ∗ R ( X ) ⊂ R ( S m ) , ( t Φ) ∗ R ( S m ) ⊂ R ( X ) , where R () = A m ( j ) () or A () .Proof. We first prove the statement for t Φ . By construction of the MCK decomposition for X ,there is a relation(1) Π Xk = 1 m t Φ ◦ Π S m k ◦ Φ +
Rest in A m ( X × X ) , ( k = 0 , , , . . . , m ) , where { Π S m k } is a product MCK decomposition for S m , and “Rest” is a term coming fromvarious partial diagonals. For dimension reasons, the term “Rest” does not act on A m ( X ) andon A AJ ( X ) . Since m t Φ ◦ Φ is the identity on A m ( X ) and on A hom ( X ) = A AJ ( X ) , we canwrite ( t Φ) ∗ (Π S m k ) ∗ = ( t Φ ◦ Π S m k ) ∗ = ( 1 m t Φ ◦ Φ ◦ t Φ ◦ Π S m k ) ∗ : T ( S m ) → T ( X ) , where T () is either A m () or A hom () . In view of sublemma 2.12 below, this implies ( t Φ) ∗ (Π S m k ) ∗ = ( 1 m t Φ ◦ Π S m k ◦ Φ ◦ t Φ) ∗ : T ( S m ) → T ( X ) . But then, plugging in relation (1), we find ( t Φ) ∗ (Π S m k ) ∗ T ( S m ) ⊂ (Π Xk ) ∗ T ( X ) . Taking k = 2 and T = A hom () , this proves ( t Φ) ∗ A ( S m ) ⊂ A ( X ) . Taking k = 4 m − j and T = A m () , this proves ( t Φ) ∗ A m ( j ) ( S m ) ⊂ A m ( j ) ( X ) . The proof of the first statement of lemma 2.11 is similar: equality (1) implies that Φ ∗ (Π Xk ) ∗ = 1 m (cid:0) Φ ◦ t Φ ◦ Π S m k ◦ Φ (cid:1) ∗ : T ( X ) → T ( S m ) . LGEBRAIC CYCLES AND EPW CUBES 7
Using sublemma 2.12, this slinks down to Φ ∗ (Π Xk ) ∗ = 1 m (cid:0) Π S m k ◦ Φ ◦ t Φ ◦ Φ (cid:1) ∗ = (Π S m k ◦ Φ) ∗ : T ( X ) → T ( S m ) . This proves the first statement of lemma 2.11.
Sublemma 2.12.
There is commutativity (cid:0) Φ ◦ t Φ ◦ Π S m k (cid:1) ∗ = (cid:0) Π S m k ◦ Φ ◦ t Φ (cid:1) ∗ A i ( S m ) → A i ( S m ) ∀ i , ∀ k . To prove the sublemma, we remark that h ∗ h ∗ = m id : A i ( S ( m ) ) → A i ( S ( m ) ) , and so(2) (Φ ◦ t Φ) ∗ = m g ∗ g ∗ = m ( X σ ∈ S m Γ σ ) ∗ : A i ( S m ) → A i ( S m ) , where the symmetric group S m acts in the natural way on the product S m . But { Π S m k } , being aproduct decomposition, is symmetric and hence Γ σ ◦ Π S m k ◦ Γ σ − = ( σ × σ ) ∗ Π S m k = Π S m k in A m ( S m × S m ) ∀ σ ∈ S m , ∀ k . This implies commutativity Γ σ ◦ Π S m k = Π S m k ◦ Γ σ in A m ( S m × S m ) ∀ σ ∈ S m , ∀ k . Combining with equation (2), this proves the sublemma. (cid:3)
Remark 2.13.
Lemma 2.11 is probably true for any ( i, j ) (i.e., the correspondence Φ should be“of pure grade ” in the language of [36, Definition 1.1] ). I have not been able to prove this. Relative MCK for S m .Notation 2.14. Let
S → B be a family (i.e., a smooth projective morphism). For r ∈ N , wewrite S r/B for the relative r –fold fibre product S r/B := S × B S × B · · · × B S ( r copies of S ). Proposition 2.15.
Let
S → B be a family of K surfaces. There exist relative correspondences Π S m/B j ∈ A m ( S m/B × S m/B ) ( j = 0 , , , . . . , m ) , such that for each b ∈ B , the restriction Π ( S b ) m j := Π S m/B j | ( S b ) m ∈ A (( S b ) m × ( S b ) m ) defines a self–dual MCK decomposition for ( S b ) m . ROBERT LATERVEER
Proof.
On any K surface S b , there is the distinguished –cycle o S b such that c ( S b ) = 24 o S b [4]. Let p i : S m/B → S , i = 1 , . . . , m , denote the projections to the two factors. Let T S /B denotethe relative tangent bundle. The assignment Π S := ( p ) ∗ (cid:0) c ( T S /B ) (cid:1) A ( S × B S ) , Π S := ( p ) ∗ (cid:0) c ( T S /B ) (cid:1) A ( S × B S ) , Π S := ∆ S − Π S − Π S defines (by restriction) an MCK decomposition for each fibre, i.e. Π S b j := Π S j | S b × S b ∈ A ( S b × S b ) ( j = 0 , , is an MCK decomposition for any b ∈ B [35, Example 8.17].Next, we consider the m –fold relative fibre product S m/B . Let p i,j : S m/B → S /B (1 ≤ i < j ≤ m ) denote projection to the i -th and j -th factor. We define Π S m/B j := X k + k + ··· + k m = j ( p ,m +1 ) ∗ (Π S k ) · ( p ,m +2 ) ∗ (Π S k ) · . . . · ( p m, m ) ∗ (Π S k m ) ∈ A m ( S m/B ) , ( j = 0 , , , . . . , m ) . By construction, the restriction to each fibre induces an MCK decomposition (the “product MCKdecomposition”) Π ( S b ) m j := Π S m/B j | ( S b ) m = X k + k + ··· + k m = j Π S b k × Π S b k × · · · × Π S b k m ∈ A m (( S b ) m ) , ( j = 0 , , , . . . , m ) . (cid:3) Proposition 2.16.
Let
S → B be a family of K surfaces. There exist relative correspondences Θ , . . . , Θ m ∈ A m ( S m/B × B S ) , Ξ , . . . , Ξ m ∈ A ( S × B S m/B ) such that for each b ∈ B , the composition A m (2) (cid:0) ( S b ) m (cid:1) ((Θ | ( Sb ) m +1 ) ∗ ,..., (Θ m | ( Sb ) m +1 ) ∗ ) −−−−−−−−−−−−−−−−−−−→ A ( S b ) ⊕ · · · ⊕ A ( S b ) ((Ξ + ... +Ξ m ) | ( Sb ) m +1 ) ∗ −−−−−−−−−−−−−−→ A m (cid:0) ( S b ) m (cid:1) is the identity.Proof. As before, let p i,j : S m/B → S /B (1 ≤ i < j ≤ m ) denote projection to the i -th and j -th factor, and let p i : S m/B → S (1 ≤ i ≤ m ) LGEBRAIC CYCLES AND EPW CUBES 9 denote projection to the i –th factor.We now claim that for each b ∈ B , there is equality (Π S m/B m − ) | ( S b ) m = 124 m − (cid:16) t Γ p ◦ Π S ◦ Γ p ◦ (cid:0) ( p ,m +1 ) ∗ (∆ S ) · Y ≤ j ≤ m ( p j ) ∗ c ( T S /B ) (cid:1) + . . . + t Γ p m ◦ Π S ◦ Γ p m ◦ (cid:0) ( p m, m ) ∗ (∆ S ) · Y ≤ j ≤ m − j = m ( p j ) ∗ c ( T S /B ) (cid:1)(cid:17) | ( S b ) m in A m (( S b ) m × ( S b ) m ) . (3)Indeed, using Lieberman’s lemma [12, 16.1.1], we find that ( t Γ p ◦ Π S ◦ Γ p ) | ( S b ) m = (cid:0) ( t Γ p ,m +1 ) ∗ (Π S ) (cid:1) | ( S b ) m = (cid:0) ( p ,m +1 ) ∗ (Π S ) (cid:1) | ( S b ) m , ... ( t Γ p m ◦ Π S ◦ Γ p m ) | ( S b ) m = (cid:0) ( t Γ p m, m ) ∗ (Π S ) (cid:1) | ( S b ) m = (cid:0) ( p m, m ) ∗ (Π S ) (cid:1) | ( S b ) m . Let us now (by way of example) consider the first summand of the right–hand–side of (3). Forbrevity, let P : ( S b ) m → ( S b ) m denote the projection on the first m and last m factors. Writing out the definition of compositionof correspondences, we find that m − (cid:16) t Γ p ◦ Π S ◦ Γ p ◦ (cid:0) ( p ,m +1 ) ∗ (∆ S ) · Y ≤ j ≤ m j = m +1 ( p j ) ∗ c ( T S /B ) (cid:1)(cid:17) | ( S b ) m =124 m − (cid:16)(cid:0) ( p ,m +1 ) ∗ (Π S b ) (cid:1) ◦ (cid:0) ( p ,m +1 ) ∗ (∆ S b ) · Y m +2 ≤ j ≤ m ( p j ) ∗ c ( T S b ) (cid:1)(cid:17) = P ∗ (cid:16)(cid:0) (∆ S b ) (1 ,m +1) × o S b × · · · × o S b × S b × · · · × S b (cid:1) · (cid:0) S b × · · · × S b × (Π S b ) ( m +1 , m +1) × S b × · · · × S b (cid:1)(cid:17) = P ∗ (cid:16)(cid:0) (∆ S b × S b ) · ( S b × Π S b ) (cid:1) (1 ,m +1 , m +1) × o S b × · · · × o S b × S b × · · · × S b (cid:17) =Π S b × Π S b × · · · × Π S b in A m (cid:0) ( S b ) m × ( S b ) m (cid:1) . (Here, we use the notation ( C ) ( i,j ) to indicate that the cycle C lies in the i th and j th factor, andlikewise for ( D ) ( i,j,k ) .)Doing the same for the other summands in (3), one convinces oneself that both sides of (3) areequal to the fibrewise product Chow–K¨unneth component Π ( S b ) m m − = Π S b × Π S b × · · · × Π S b + · · · + Π S b × · · · × Π S b × Π S b ∈ A m (( S b ) m × ( S b ) m ) , thus proving the claim. Let us now define Θ i := 124 m − Γ p i ◦ (cid:0) ( p i,m + i ) ∗ (∆ S ) · Y j ∈ [ m +2 , m ] j i,m + i } ( p j ) ∗ c ( T S /B ) (cid:1) ∈ A m (( S m/B ) × B S ) , Ξ i := t Γ p i ◦ Π S ∈ A ( S × B ( S m/B )) , where ≤ i ≤ m . It follows from equation (3) that there is equality (cid:16) (Ξ ◦ Θ + · · · + Ξ m ◦ Θ m ) | ( S b ) m (cid:17) ∗ = (cid:0) Π ( S b ) m m − (cid:1) ∗ : A i ( j ) (cid:0) ( S b ) m (cid:1) → A i ( j ) (cid:0) ( S b ) m (cid:1) ∀ b ∈ B ∀ ( i, j ) . (4)Taking ( i, j ) = (2 m, , this proves the proposition. (cid:3) The following is a version of proposition 2.16 for the group A (( S b ) m ) : Proposition 2.17.
Let
S → B be a family of K surfaces. There exist relative correspondences Θ ′ , . . . , Θ ′ m ∈ A m ( S × B ( S m/B )) , Ξ ′ , . . . , Ξ ′ m ∈ A (( S m/B ) × B S ) such that for each b ∈ B , the composition A (cid:0) ( S b ) m (cid:1) ((Ξ ′ | ( Sb ) m +1 ) ∗ ,..., (Ξ ′ m | ( Sb ) m +1 ) ∗ ) −−−−−−−−−−−−−−−−−−−→ A ( S b ) ⊕ · · · ⊕ A ( S b ) ((Θ ′ + ... +Θ ′ m ) | ( Sb ) m +1 ) ∗ −−−−−−−−−−−−−−→ A (cid:0) ( S b ) m (cid:1) is the identity.Proof. One may take Θ ′ i := t Θ i ∈ A m ( S × B ( S m/B )) , Ξ ′ i := t Ξ i A (( S m/B ) × B S ) ( i = 1 , . . . , m ) . By construction, the product MCK decomposition { Π ( S b ) m i } satisfies Π ( S b ) m = t (cid:0) Π ( S b ) m m − (cid:1) in A m (cid:0) ( S b ) m × ( S b ) m (cid:1) . Hence, the transpose of equation (4) gives the equality (cid:0) Π ( S b ) m (cid:1) ∗ = (cid:0) t (Π ( S b ) m m − ) (cid:1) ∗ = (cid:0) t Θ ◦ t Ξ + . . . + t Θ m ◦ t Ξ m (cid:1) ∗ : A i ( j ) (cid:0) ( S b ) m (cid:1) → A i ( j ) (cid:0) ( S b ) m (cid:1) ∀ b ∈ B ∀ ( i, j ) . Taking ( i, j ) = (2 , , this proves the proposition. (cid:3) LGEBRAIC CYCLES AND EPW CUBES 11
Spread.Lemma 2.18 (Voisin [41], [42]) . Let M be a smooth projective variety of dimension n + r , andlet L , . . . , L r be very ample line bundles on M . Let X → B denote the universal family of codimension r smooth complete intersections X b ⊂ M of type X b = M ∩ D ∩ · · · ∩ D r , D i ∈ | L i | , i = 1 , . . . r . (That is, B ⊂ | L | × · · · × | L r | is a Zariski open.) Let p : ^ X × B X → X × B X denote the blow–up of the relative diagonal. Then ^ X × B X is Zariski open in V , where V is a fibre bundle over ^ M × M , the blow–up of M × M along the diagonal, and the fibres of V → ^ M × M are products of projective spaces.Proof. This is [41, Proof of Proposition 3.13] or [42, Lemma 1.3]. The idea is to define V as V := n(cid:0) ( x, y, z ) , σ (cid:1) | σ | z = 0 o ⊂ ^ M × M × | L | . The very ampleness assumption ensures that V → ^ M × M is a projective bundle. (cid:3) This is used in the following key proposition:
Proposition 2.19 (Voisin [42]) . Let M be a smooth projective variety of dimension n + r , andsuppose that A ∗ hom ( M ) = 0 . Let L , . . . , L r be very ample line bundles on M , and let X → B be as in lemma 2.18.Assume Γ ∈ A n ( X × B X ) is such that the restriction Γ b := Γ | X b × X b ∈ A n ( X b × X b ) is homologically trivial, for very general b ∈ B . Then there exists δ ∈ A n ( M × M ) such that Γ b + δ b = 0 in A n ( X b × X b ) ∀ b ∈ B .
Proof.
This follows from [42, Proposition 1.6]. (NB: The result [42, Proposition 1.6] is statedonly for hypersurfaces, i.e. r = 1 . However, as noted in [42, Remark 0.7], the complete inter-section case follows from this.)In the special case n = 2 (which is the only case we will need in this note), proposition 2.19 isalready contained in [41]. Indeed, the Leray spectral sequence argument [41, Lemmas 3.11 and3.12] gives the existence of δ ∈ A ( M × M ) such that (after shrinking the base B ) Γ + δ | X × B X = 0 in H ( X × B X ) . But using lemma 2.18 (plus some basic properties of varieties with trivial Chow groups, cf. [41,Section 3.1]), one finds that A hom ( X × B X ) = 0 . Therefore, we must have
Γ + δ | X × B X = 0 in A ( X × B X ) . In particular, this implies that Γ b + δ b = 0 in A n ( X b × X b ) for general b ∈ B .
To obtain the result for all b ∈ B , one can invoke [44, Lemma 3.2]. (cid:3) Mukai models.Theorem 2.20 (Mukai [27]) . Let S be a general K surface of degree (i.e. genus g ( S ) = 6 ).Let G = G (2 , denote the Grassmannian of lines in P . Then S is isomorphic to the zero locusof a section of O G (1) ⊕ ⊕ O G (2) . Remark 2.21.
Let
S ⊂ G × B denote the universal family of smooth codimension complete intersections defined by O G (1) ⊕ ⊕O G (2) , where B ⊂ P H (cid:0) G, O G (1) (cid:1) × × P H (cid:0) G, O G (2) (cid:1) is the Zariski open parametrizing smooth surfaces S b ⊂ G . We will refer to the family S → B as the universal family of degree K surfaces . EPW cubes.Definition 2.22 (Iliev–Kapustka–Kapustka–Ranestad [15]) . Let W be a complex vector space ofdimension equipped with a skew–symmetric form ν : ∧ W × ∧ W → C . Let LG ν denote the variety of –dimensional subspaces in ∧ W that are Lagrangian withrespect to ν . For any –dimensional subspace U ∈ G (3 , W ) , the –dimensional subspace T U := ∧ U ∧ W ⊂ ∧ W is in LG ν .Given A ∈ LG ν and k ∈ N , define the degenerary locus D Ak := (cid:8) U ∈ G (3 , W ) | dim( A ∩ T U ) ≥ k (cid:9) ⊂ G (3 , W ) . The scheme D A is called an EPW cube . For A generic, the EPW cube D A is of dimension , andSing ( D A ) = D A is a smooth threefold. LGEBRAIC CYCLES AND EPW CUBES 13
Theorem 2.23 (Iliev–Kapustka–Kapustka–Ranestad [15]) . Notation as in definition 2.22.(i) There is a Zariski open LG ν ⊂ LG ν with the following property: for any A ∈ LG ν , thereexists a double cover Y A → D A branched along D A , and Y A is a hyperk¨ahler variety.(ii) There is a divisor ∆ ⊂ LG ν such that for general A ∈ ∆ , the variety Y A is birational tothe Hilbert scheme ( S A ) [3] for some degree K surface S A .(iii) Given a generic degree K surface S , there exists A ∈ ∆ such that S = S A .Proof. Point (i) is contained in [15, Theorem 1.1].Point (ii) is [15, Section 5]. (NB: the divisor that we denote ∆ is written as ∆ \ (Γ ∪ Σ) in[15].)For point (iii), we note that the construction of S A for general A ∈ ∆ in [15, Section 4] ismodelled on O’Grady’s construction in [32, Section 4.1]; point (iii) thus follows from O’Grady’sresult [32, Proposition 4.14]. (cid:3) Remark 2.24.
As noted in [15] , a noteworthy consequence of theorem 2.23(ii) is that doubleEPW cubes Y A are of K [3]3 type.Theorem 2.23(iii) implies that the covering involution ι A : Y A → Y A is anti–symplectic: indeed (as noted in [15] ), if it were symplectic the fixed–locus would bea symplectic subvariety, whereas the fixed–locus of ι A is the inverse image of D A which is ofdimension .Theorem 2.23(iii) implies that if S is a generic degree K surface, there exists an anti–symplectic birational involution ι : S [3] S [3] . I do not know whether there is a geometric interpretation of the involution ι , similar to thegeometric interpretation of the birational involution ι : S [2] S [2] related to double EPW sextics given in [30, Section 4.3] . We now translate some of the results of [15] into statements that will be convenient for thepurposes of this note:
Proposition 2.25.
Let ∆ ⊂ LG ν be the divisor of theorem 2.23. Let T → M be the universalgenus K surface over the moduli space M .(i) There exist projective morphisms X ∆ ρ −→ D ∆ π −→ ∆ , such that for each A ∈ ∆ , the fibre X A := ( π ◦ ρ ) − ( A ) is a double EPW cube, and the fibre D A := π − ( A ) is an EPW cube. (ii) Let T → M be the universal genus K surface over the moduli space M , and let T [3] → M denote the universal Hilbert cube. There exist Zariski opens ∆ , ⊂ ∆ , and M ⊂ M , and a generically rational map Ψ : T [3] / M E . Here, T [3] / M := ( T [3] ) × M M , and E is the quotient stack E := D ∆ , /P , where D ∆ , := D ∆ × ∆ ∆ , , and P := P GL ( W ) acts on ∆ , and on G = G (3 , W ) . Themap Ψ fits into a diagram S /B T [3] / M X ∆ , ֒ → Xց Ψ ↓ ↓↓ ↓ E := D ∆ , /P ← D ∆ , ֒ → D↓ ↓ ↓ B → M f → M ∆ , = ∆ , /P ← ∆ , ֒ → LG ν Here, M ∆ , is the image of ∆ , under the period map to the moduli space, and M ∆ , is ageometric quotient M ∆ , = ∆ , /P . The morphism f is an isomorphism. (And S → B is the universal family of remark 2.21, and B ⊂ B is a Zariski open).(iii) The quotient stack E is a Deligne–Mumford stack, and so A i ( E ) ∼ = A iP ( D ∆ , ) , where the right–hand side denotes equivariant Chow groups, in the sense of Edidin–Graham [8] .Proof. (i) There exists a tower of projective morphisms X → D → LG ν , where a fibre D A is an EPW cube, and a fibre X A is a double EPW cube, and X → LG ν issmooth [15, Section 5]. By base change, one obtains X ∆ → D ∆ → ∆ . (ii) First, we note that (as proven in [32]) for a given A ∈ ∆ , , the associated K surface S A iswell–defined up to projectivities, and so there is a map ∆ , → M . Conversely, given a generalgenus K surface S , the element A ∈ ∆ such that S = S A is well–defined up to the action of P = P GL ( W ) . This proves that f is an isomorphism on appropriate opens.To construct E , we note that D ∆ , is defined as D ∆ , := (cid:8) ( U, A ) | U ∈ D A (cid:9) ⊂ G × ∆ , , and so P acts naturally on D ∆ , .The map Ψ is defined by sending a generic point x ∈ ( S b ) [3] to ρ (cid:0) ( φ b )( x ) (cid:1) ∈ D f ( b )2 , LGEBRAIC CYCLES AND EPW CUBES 15 where φ b : ( S b ) [3] X f ( b ) is the birational map of theorem 2.23, and ρ : X f ( b ) → D f ( b )2 is thedouble cover.(iii) Let s : D ∆ , → ∆ , denote the projection. The stabilizer of a point e ∈ D ∆ , for the actionof P is contained in the stabilizer of s ( e ) for the P –action on ∆ , . This stabilizer is finite, since ∆ , is contained in LG ν , which is contained in the stable locus [31].The statement about the Chow group of E follows from this. (For any Deligne–Mumfordstack, Chow groups with rational coefficients have been defined [13], [39]. These Chow groupsagree with the equivariant Chow groups [8].) (cid:3) Corollary 2.26.
Let A ∈ ∆ be general, and let X = X A be the associated double EPW cube.Then X has an MCK decomposition, and the Chow ring of X has a bigrading A ∗ ( ∗ ) ( X ) with A i ( j ) ( X ) = 0 if j > i and A i ( j ) ( X ) = 0 if j is odd.Proof. The variety X is birational to a Hilbert cube ( S A ) [3] (theorem 2.23(ii)). Hilbert cubes of K surfaces have an MCK decomposition (theorem 2.9). It follows from lemma 2.8 that X hasan MCK decomposition, and that there is an isomorphism of bigraded rings A ∗ ( ∗ ) ( X ) ∼ = A ∗ ( ∗ ) (( S A ) [3] ) . The vanishing A i ( j ) ( X ) = 0 for j > i and for j odd follows from the corresponding property for ( S A ) [3] . (cid:3)
3. H
ARD L EFSCHETZ
In this section, we prove a “hard Lefschetz type” isomorphism for Chow groups of certain va-rieties. This hard Lefschetz result (and in particular, the version for double EPW cubes, corollary3.5) will be a crucial ingredient in the proof of the main result of this note (theorem 4.1).
Theorem 3.1.
Let
S → B be the universal family of K surfaces of degree (cf. remark 2.21).Let L ∈ A ( S m/B ) be a line bundle such that the restriction L b (to the fibre over b ∈ B ) is bigfor very general b ∈ B . Then · ( L b ) m − : A (( S b ) m ) → A m (2) (( S b ) m ) is an isomorphism for all b ∈ B .Proof. This is proven using the technique of spread as developed by Voisin [41], [42]. Let uswrite Γ L m − := ( p ) ∗ ( L m − ) · ∆ S m/B ∈ A m − (cid:0) ( S m/B ) × B ( S m/B ) (cid:1) , where ∆ S m/B ⊂ ( S m/B ) × B ( S m/B ) is the relative diagonal, and p : ( S m/B ) × B ( S m/B ) → S m/B is projection on the first factor. The relative correspondence Γ L m − acts on Chow groups asmultiplication by L m − .As “input”, we will make use of the following result: Proposition 3.2 (L. Fu [9]) . Let X be a smooth projective variety of dimension n verifying theLefschetz standard conjecture B ( X ) . Let L ∈ A ( X ) be a big line bundle. Then ∪ L n − : H ( X ) /N H ( X ) → H n − ( X ) /N n − H n − ( X ) is an isomorphism. (Here N ∗ denotes the coniveau filtration [6] , so N i H i ( X ) is the image ofthe cycle class map.) Moreover, there is a correspondence C ∈ A ( X × X ) inducing the inverseisomorphism.Proof. This follows from the proof of [9, Theorem 4.11]. Alternatively, here is an explicit argu-ment: it follows from [9, Lemma 3.3] that ∪ L n − : H ( X ) /N H ( X ) → H n − ( X ) /N n − H n − ( X ) is an isomorphism. Since the category of motives for numerical equivalence M num is semisimple[16], it follows that there is an isomorphism of motives h ( X ) ⊕ M i L ( m i ) ∼ = h n − ( X )( n − ⊕ M j L ( m j ) in M num , where the arrow from h ( X ) to h n − ( X )( n − is given by Γ L n − ∈ A n − ( X × X ) , and L denotes the Lefschetz motive. Since homological and numerical equivalence coincide for X andfor L , this implies there is also an isomorphism h ( X ) ⊕ M i L ( m i ) ∼ = h n − ( X )( n − ⊕ M j L ( m j ) in M hom , with the arrow from h ( X ) to h n − ( X )( n − being given by Γ L n − . It follows that there existsa correspondence C as required. (cid:3) Any fibre ( S b ) m of the family S m/B → B verifies the Lefschetz standard conjecture (theLefschetz standard conjecture is known for products of surfaces). Applying proposition 3.2, thismeans that for all b ∈ B there exists a correspondence C b ∈ A (cid:0) ( S b ) m × ( S b ) m (cid:1) with the property that the compositions H (cid:0) ( S b ) m (cid:1) /N · ( L b ) m − −−−−−→ H m − (cid:0) ( S b ) m (cid:1) /N m − C b ) ∗ −−−→ H (cid:0) ( S b ) m (cid:1) /N and H m − (cid:0) ( S b ) m (cid:1) /N m − C b ) ∗ −−−→ H (cid:0) ( S b ) m (cid:1) /N · ( L b ) m − −−−−−→ H m − (cid:0) ( S b ) m (cid:1) /N m − are the identity. In other words, for all b ∈ B there exist γ b , γ ′ b ∈ A m (cid:0) ( S b ) m × ( S b ) m (cid:1) supported on D b × D b ⊂ ( S b ) m × ( S b ) m for some divisor D b ⊂ ( S b ) m and such that Π S m/B | ( S b ) m ◦ C b ◦ (cid:0) (Π S m/B m − ◦ Γ L m − ◦ Π S m/B ) | ( S b ) m (cid:1) = Π S m/B | ( S b ) m + γ b , Π S m/B m − | ( S b ) m ◦ (cid:0) Γ L m − ◦ (Π S m/B (cid:1) | ( S b ) m ◦ C b ◦ (cid:0) Π S m/B m − ) | ( S b ) m (cid:1) = Π S m/B m − | ( S b ) m + γ ′ b in H m (cid:0) ( S b ) m × ( S b ) m (cid:1) . LGEBRAIC CYCLES AND EPW CUBES 17
Applying a Hilbert schemes argument as in [41, Proposition 3.7] (cf. also [21, Proposition 2.10]),we can find a relative correspondence
C ∈ A (cid:0) ( S m/B ) × B ( S m/B ) (cid:1) doing the same job as the various C b , i.e. such that for all b ∈ B one has (Π S m/B ◦ C ◦ Π S m/B m − ◦ Γ L m − ◦ Π S m/B ) | ( S b ) m = Π S m/B | ( S b ) m + γ b , (Π S m/B m − ◦ Γ L m − ◦ Π S m/B ◦ C ◦ Π S m/B m − ) | ( S b ) m = Π S m/B m − | ( S b ) m + γ ′ b in H m (cid:0) ( S b ) m × ( S b ) m (cid:1) . Applying once more the same Hilbert schemes argument [41, Proposition 3.7], we can also finda divisor
D ⊂ S m/B and relative correspondences γ , γ ′ ∈ A m (cid:0) S m/B × B S m/B (cid:1) supported on D × B D and doing the same job as the various γ b , resp. γ ′ b . That is, γ and γ ′ aresuch that for all b ∈ B one has (Π S m/B ◦ C ◦ Π S m/B m − ◦ Γ L m − ◦ Π S m/B ) | ( S b ) m = (Π S m/B + γ ) | ( S b ) m , (Π S m/B m − ◦ Γ L m − ◦ Π S m/B ◦ C ◦ Π S m/B m − ) | ( S b ) m = (Π S m/B m − + γ ′ ) | ( S b ) m in H m (cid:0) ( S b ) m × ( S b ) m (cid:1) . We now make an effort to rewrite this more compactly: the relative correspondences definedas
Γ := Π S m/B ◦ C ◦ Π S m/B m − ◦ Γ L m − ◦ Π S m/B − Π S m/B − γ , Γ ′ := Π S m/B m − ◦ Γ L m − ◦ Π S m/B ◦ C ◦ Π S m/B m − − Π S m/B m − − γ ′ ∈ A m (cid:0) ( S m/B ) × B ( S m/B ) (cid:1) (5)have the property that their restriction to any fibre is homologically trivial. That is, writing Γ b := Γ | ( S b ) m × ( S b ) m Γ ′ b := (Γ ′ ) | ( S b ) m × ( S b ) m ∈ A m (cid:0) ( S b ) m × ( S b ) m (cid:1) for the restriction to a fibre, we have that(6) Γ b , Γ ′ b ∈ A mhom (cid:0) ( S b ) m × ( S b ) m (cid:1) ∀ b ∈ B ,
Let us now define the modified relative correspondences Γ := Π S m/B ◦ Γ ◦ Π S m/B , Γ ′ := Π S m/B m − ◦ Γ ′ ◦ Π S m/B m − ∈ A m (cid:0) S m/B × B S m/B (cid:1) . This modification does not essentially modify the fibrewise rational equivalence class: wehave (Γ ) b = Γ b + ( γ ) b , (Γ ′ ) b = (Γ ′ ) b + ( γ ′ ) b in A m (cid:0) ( S b ) m × ( S b ) m (cid:1) , (7) where γ , γ ′ ∈ A m (cid:0) S m/B × B S m/B (cid:1) are relative correspondences supported on D × B D . (In-deed, this is true because (Π ( S b ) m i ) ◦ = Π ( S b ) m i for all i , and the relative correspondences Π S m/B ◦ γ ◦ Π S m/B , Π S m/B m − ◦ γ ′ ◦ Π S m/B m − are still supported on D × B D .)As Γ and Γ ′ were fibrewise homologically trivial (equation (6)), the same is true for Γ and Γ ′ :(8) (Γ ) b , (Γ ′ ) b ∈ A mhom (cid:0) ( S b ) m × ( S b ) m (cid:1) ∀ b ∈ B ,
We now proceed to upgrade (8) to a statement concerning the action on Chow groups:
Claim 3.3.
We have (cid:0) (Γ ) b (cid:1) ∗ = 0 : A ihom (cid:0) ( S b ) m (cid:1) → A ihom (cid:0) ( S b ) m (cid:1) ∀ b ∈ B , (cid:0) (Γ ′ ) b (cid:1) ∗ = 0 : A ihom (cid:0) ( S b ) m (cid:1) → A ihom (cid:0) ( S b ) m (cid:1) ∀ b ∈ B .
Let us prove claim 3.3 for Γ (the argument for Γ ′ is only notationally different). Usingproposition 2.17, one finds there is a fibrewise equality modulo rational equivalence(9) (Γ ) b = (cid:16) ( m X i =1 Ξ i ◦ Θ i ) ◦ Γ ◦ ( m X i =1 Ξ i ◦ Θ i ) (cid:17) b in A m (cid:0) ( S b ) m × ( S b ) m (cid:1) ∀ b ∈ B .
To rewrite this, let us define relative correspondences Γ k,ℓ := Θ k ◦ Γ ◦ Ξ ℓ ∈ A (cid:0) S × B S (cid:1) (1 ≤ k, ℓ ≤ m ) . With this notation, equality (9) becomes the equality(10) (Γ ) b = (cid:16) m X k =1 m X ℓ =1 Ξ k ◦ Γ k,ℓ ◦ Θ ℓ (cid:17) b in A m (cid:0) ( S b ) m × ( S b ) m (cid:1) ∀ b ∈ B . As Γ is fibrewise homologically trivial (equation (6)), the same is true for the various Γ k,ℓ : (Γ k,ℓ ) b ∈ A hom ( S b × S b ) ∀ b ∈ B (1 ≤ k, ℓ ≤ m ) . This means that we can apply Voisin’s key result, proposition 2.19, to the relative correspondence Γ k,ℓ . The conclusion is that for each ≤ k, ℓ ≤ m , there exists a cycle δ k,ℓ ∈ A ( G × G ) (where G = G (2 , is the Grassmannian as in theorem 2.20) such that (Γ k,ℓ ) b + ( δ k,ℓ ) b = 0 in A ( S b × S b ) ∀ b ∈ B .
Since a Grassmannian has trivial Chow groups, this implies in particular that (cid:0) (Γ k,ℓ ) b (cid:1) ∗ = 0 : A ihom ( S b ) → A ihom ( S b ) ∀ b ∈ B .
In view of equality (10), this implies (cid:0) (Γ ) b (cid:1) ∗ : = 0 A ihom (cid:0) ( S b ) m (cid:1) → A ihom (cid:0) ( S b ) m (cid:1) ∀ b ∈ B , as claimed.
LGEBRAIC CYCLES AND EPW CUBES 19 (The argument for Γ ′ is the same; it suffices to replace the use of proposition 2.17 by proposi-tion 2.16.) Claim 3.3 is now proven.It is high time to wrap up the proof of theorem 4.1. For b ∈ B general, the restrictions ( γ ) b , ( γ ′ ) b of equation (7) will be supported on D b × D b ⊂ ( S b ) m × ( S b ) m , where D b ⊂ ( S b ) m is a divisor. As such, the action (cid:0) ( γ ) b (cid:1) ∗ : R (cid:0) ( S b ) m (cid:1) → R (cid:0) ( S b ) m (cid:1) , (cid:0) ( γ ′ ) b (cid:1) ∗ : R (cid:0) ( S b ) m (cid:1) → R (cid:0) ( S b ) m (cid:1) , is for general b ∈ B , where R is either A hom or A m . Combining this observation with equation(7) and claim (3.3), we find that (Γ b ) ∗ = 0 : R (cid:0) ( S b ) m (cid:1) → R (cid:0) ( S b ) m (cid:1) , (Γ ′ b ) ∗ = 0 : R (cid:0) ( S b ) m (cid:1) → R (cid:0) ( S b ) m (cid:1) (where, once more, R is either A hom or A m ).In view of the definition (5) of Γ , Γ ′ (and using that the cycles γ b , γ ′ b occuring in (5) are sup-ported in codimension for b ∈ B general, and so act trivially on A hom () and on A m () ), itfollows that (cid:16) Π ( S b ) m ◦ C b ◦ Π ( S b ) m m − ◦ (Γ L m − ) b ◦ Π ( S b ) m − Π ( S b ) m (cid:17) ∗ = 0 : A hom (cid:0) ( S b ) m (cid:1) → A hom (cid:0) ( S b ) m (cid:1) , (cid:16) Π ( S b ) m m − ◦ (Γ L m − ) b ◦ Π ( S b ) m ◦ C b ◦ Π ( S b ) m m − − Π ( S b ) m m − (cid:17) ∗ = 0 : A m (cid:0) ( S b ) m (cid:1) → A m (cid:0) ( S b ) m (cid:1) , (11)for general b ∈ B . Since Π ( S b ) m acts as the identity on A (( S b ) m ) , it follows from the first lineof (11) that (cid:16) Π ( S b ) m ◦ C b ◦ Π ( S b ) m m − ◦ (Γ L m − ) b (cid:17) ∗ = id : A (cid:0) ( S b ) m (cid:1) → A (cid:0) ( S b ) m (cid:1) ; in particular · L m − : A (cid:0) ( S b ) m (cid:1) → A m (2) (cid:0) ( S b ) m (cid:1) is injective for general b ∈ B . Likewise, it follows from the second line of (11) that (cid:16) Π ( S b ) m m − ◦ (Γ L m − ) b ◦ Π ( S b ) m ◦ C b (cid:17) ∗ = id : A m (2) (cid:0) ( S b ) m (cid:1) → A m (2) (cid:0) ( S b ) m (cid:1) for general b ∈ B . However, the image of A (cid:0) ( S b ) m (cid:1) · L m − −−−−→ A m (cid:0) ( S b ) m (cid:1) is contained in A m (2) (( S b ) m ) , since L ∈ A (( S b ) m ) = A (( S b ) m ) , and so this further simplifiesto (cid:16) (Γ L m − ) b ◦ Π ( S b ) m ◦ C b (cid:17) ∗ = id : A m (2) (cid:0) ( S b ) m (cid:1) → A m (2) (cid:0) ( S b ) m (cid:1) for general b ∈ B . In particular, · L m − : A (cid:0) ( S b ) m (cid:1) → A m (2) (cid:0) ( S b ) m (cid:1) is surjective for general b ∈ B . Theorem 3.1 is now proven for general b ∈ B (this suffices for the purposes of this note).To prove the theorem for all b ∈ B , one may observe that the above argument can be made towork “locally around a given b ∈ B ”, i.e. given b ∈ B one can find relative correspondences γ, γ ′ , . . . supported in codimension and in general position with respect to the fibre over b . (cid:3) Theorem 3.1 can be reformulated in terms of Hilbert schemes:
Corollary 3.4.
Let S b be a K surface of degree , and let X = ( S b ) [ m ] be the Hilbert schemeof length m subschemes of S . Let L ∈ A ( S m/B ) be a relatively big line bundle, and set L X := ( f b ) ∗ ( p b ) ∗ ( L b ) ∈ A ( X ) , where p b : ( S b ) m → ( S b ) ( m ) denotes the projection, and f b : ( S b ) [ m ] → ( S b ) ( m ) denotes theHilbert–Chow morphism. Then · ( L X ) m − : A ( X ) → A m (2) ( X ) is an isomorphism.Proof. Let the symmetric group S m act on S m/B by permuting the factors, and let p : S m/B → S m/B / S m denote the quotient morphism. Theorem 3.1 applies to the line bundle L ′ := p ∗ p ∗ ( L ) = X σ ∈ S σ ∗ ( L ) ∈ A ( S m/B ) . There is a commutative diagram A (( S b ) m ) S m · ( L ′ b ) m − −−−−−→ A m (2) (( S b ) m ) S m ( p b ) ∗ ↑ ∼ = ( p b ) ∗ ↑ ∼ = A (( S b ) ( m ) ) · (( p b ) ∗ ( L b )) m − −−−−−−−−−→ A m (2) (( S b ) ( m ) ) In view of theorem 3.1 (applied to L ′ ), the lower horizontal arrow is an isomorphism.It follows from the de Cataldo–Migliorini isomorphism of motives [7] that there is a correspondence–induced isomorphism A ( X ) ∼ = A (( S b ) (3) ) ⊕ A () ⊕ A () , and so in particular an isomorphism A AJ ( X ) ∼ = A AJ (( S b ) (3) ) . Since A () ⊂ A AJ () , and the de Cataldo–Migliorini respects the bigrading (by construction ofthe latter), this implies that f ∗ : A (( S b ) ( m ) ) → A ( X ) is an isomorphism.Similarly, there is an isomorphism f ∗ : A m (( S b ) ( m ) ) ∼ = −→ A m ( X ) LGEBRAIC CYCLES AND EPW CUBES 21 which respects the bigrading.Corollary 3.4 now follows from what we have said above, in view of the commutative diagram A ( X ) · ( L X ) m − −−−−−→ A m (2) ( X ) ( f b ) ∗ ↑ ∼ = ( f b ) ∗ ↑ ∼ = A (( S b ) ( m ) ) · ( p ∗ ( L b )) m − −−−−−−−→ A m (2) (( S b ) ( m ) ) (cid:3) One can also reformulate theorem 3.1 in terms of double EPW cubes; this will come in usefulwhen proving our main result (theorem 4.1).
Corollary 3.5.
Let X ∆ → ∆ be the family of double EPW cubes parametrized by the divisor ∆ ⊂ LG ν of theorem 2.23. Let L ∈ A ( X ∆ ) be a line bundle that is in the image of thepullback map A ( E ) h ∗ −→ A ( X ∆ ) (where h : X ∆ E is as in proposition 2.25). Assume L is relatively big. Then · ( L A ) : A ( X A ) → A ( X A ) is an isomorphism for general A ∈ ∆ .Proof. Let us write L = h ∗ ( L E ) , where L E ∈ A ( E ) is relatively big.Let S /B denote the family of third powers of degree K surfaces over the Zariski open B ⊂ B as in proposition 2.25. We have seen (theorem 2.23) that for a general A ∈ ∆ there is b ∈ B such that A = f ( b ) and there is a birational map ( S b ) [3] φ b X A . This fits into a commutative diagram ( S b ) b ( S b ) [3] φ b X A ց ւ ց Ψ b ւ h b ( S b ) (3) Ψ ′ b D A The pullback L S := Φ ∗ Ψ ∗ ( L E ) ∈ A ( S /B ) is relatively big, and so theorem 3.1 applies to L S . There is a commutative diagram A (cid:0) ( S b ) (cid:1) S · (( L S ) b ) −−−−−→ A (cid:0) ( S b ) (cid:1) S ↑ ∼ = ↑ ∼ = A (cid:0) ( S b ) (3) (cid:1) · ((Ψ ′ b ) ∗ (( L E ) A )) −−−−−−−−−→ A (cid:0) ( S b ) (3) (cid:1) ↓ ∼ = ↓ ∼ = A (cid:0) ( S b ) [3] (cid:1) · ((Ψ b ) ∗ (( L E ) A )) −−−−−−−−−→ A (cid:0) ( S b ) [3] (cid:1) ↑ ∼ = ↑ ∼ = A ( X A ) · ( L A ) −−−→ A ( X A ) (Here the lowest vertical arrows are isomorphisms thanks to Rieß’s isomorphism [33]. Thelowest square is commutative, because φ b is a codimension isomorphism, and the divisors L A = ( h b ) ∗ (( L E ) A )) and (Ψ b ) ∗ (( L E ) A ) coincide on the open where φ b is an isomorphism.)Theorem 3.1 implies the top horizontal arrow is an isomorphism. It follows that all horizontalarrows are isomorphisms, and corollary 3.5 is proven. (cid:3) Remark 3.6.
Looking at corollary 3.4, one might hope that a similar result is true more gener-ally.Let X be any hyperk¨ahler variety of dimension m , and suppose the Chow ring of X has abigraded ring structure A ∗ ( ∗ ) ( X ) . One can ask the following questions:(i) Let L ∈ A ( X ) be an ample line bundle. Is it true that there are isomorphisms · L m − i + j : A i ( j ) ( X ) ∼ = −→ A m − i + j ( j ) ( X ) for all ≤ i − j ≤ m ? (ii) Let L ∈ A ( X ) be a big line bundle. Is it true that there are isomorphisms · L m − i : A i ( i ) ( X ) ∼ = −→ A m ( i ) ( X ) for all ≤ i ≤ m ? The answer to the first question is “yes” for generalized Kummer varieties [20] . The answerto both questions is “I don’t know” for Hilbert schemes of K surfaces.(However, if the K surface S has small genus there exists a Mukai model, and presumablythe above proof can then be extended to settle questions (i) and (ii) affirmatively for A ( S [ m ] ) and line bundles L that exist relatively. The question for A i ( j ) ( S [ m ] ) with i > becomes morecomplicated, as one would need an analogon of proposition 2.19 for higher fibre products S m/B with m > .) Remark 3.7.
Let X be either S m or S [ m ] where S is a degree K surface. It follows from(the proof of) corollary 3.4 that A ( X ) ⊂ A alg ( X ) , where A ∗ alg () ⊂ A ∗ () denotes the subgroup of algebraically trivial cycles. This is in agreementwith a conjecture of Jannsen [18] , stipulating that for any smooth projective variety Z one shouldhave F i A i ( Z ) ⊂ A ialg ( Z ) , where F ∗ is the conjectural Bloch–Beilinson filtration. Remark 3.8.
Let X be either S m or S [ m ] where S is a degree K surface, and let L ∈ A ( X ) be a line bundle as in theorem 3.1 (resp. as in corollary 3.4). Provided L is sufficiently ample,there exists a smooth complete intersection surface Y ⊂ X defined by the linear system | L | .Theorem 3.1 (resp. corollary 3.4) then implies that A m (2) ( X ) is supported on Y , and that A ( X ) → A ( Y ) is injective. This injectivity statement is in agreement with Hartshorne’s “weak Lefschetz” con-jecture for Chow groups [14] (we recall that it is expected that A ( X ) = A hom ( X ) for these X ). LGEBRAIC CYCLES AND EPW CUBES 23
4. M
AIN RESULT
Theorem 4.1.
Let X be a double EPW cube, and assume X = X A for A ∈ ∆ general (where ∆ ⊂ LG ν is the divisor of theorem 2.23). Let ι = ι A ∈ Aut( X ) be the anti–symplecticinvolution given by the double cover X A → D A . Then ι ∗ = − id : A ( X ) → A ( X ) , (Π X ) ∗ ι ∗ = − id : A ( X ) → A ( X ) . Proof.
In a first reduction step, we show that it suffices to prove the first statement of theorem4.1. Let X ∆ , → D ∆ , → ∆ , be the families as in theorem 2.23, so a fibre D A of D ∆ , over A ∈ ∆ , is an EPW cube, anda fibre X A of X ∆ , over A is a double EPW cube birational to a Hilbert cube K [3] . Takingthe restriction of a P –invariant ample line bundle on the Grassmannian, one can find a relativelyample line bundle L E ∈ A ( E ) = A P ( E ) , where E = D ∆ , /P is as in proposition 2.25. Pullingback to X ∆ , , one obtains a ι –invariant relatively ample line bundle in A ( X ∆ , ) .Applying corollary 3.5 to X = X A for A ∈ ∆ general, one obtains an isomorphism(12) · ( L | X ) : A ( X ) ∼ = −→ A ( X ) . But L | X is ι –invariant by construction, and so ι ∗ (cid:0) ( L | X ) · b ) (cid:1) = ( L | X ) · ι ∗ ( b ) in A ( X ) ∀ b ∈ A ( X ) . Suppose now the first statement of theorem 4.1 holds true. Then we find that ( L | X ) · (cid:0) b + ι ∗ ( b ) (cid:1) = 0 in A ( X ) ∀ b ∈ A ( X ) . In view of the isomorphism (12), this implies ι ∗ ( b ) = − b + b in A ( X ) , where b ∈ A ( X ) (and actually b ∈ A ,hom ( X ) , which is conjecturally ). This proves thesecond statement of theorem 4.1. It remains to prove the first statement of theorem 4.1.In view of Rieß’s isomorphism, to prove the first statement it suffices to prove that(13) ( φ b ) ∗ ( ι b ) ∗ ( φ b ) ∗ = − id : A (cid:0) ( S b ) [3] (cid:1) → A (cid:0) ( S b ) [3] (cid:1) , where S b is a general degree K surface and φ b : ( S b ) [3] X is the birational map.Consider now the commutative square ( S b ) [3] ← ] ( S b ) ↓ ↓ ( S b ) (3) ← ( S b ) (where vertical arrows are a composition of blow–ups of various partial diagonals). This givesrise to a correspondence Φ b ∈ A (( S b ) [3] × ( S b ) ) , and the blow–up exact sequence implies that (Φ b ) ∗ (Φ b ) ∗ = id : A (cid:0) ( S b ) [3] (cid:1) → A (cid:0) ( S b ) [3] (cid:1) . Therefore, we can work with the self–product ( S b ) rather than the Hilbert cube ( S b ) [3] : toprove (13), it suffices to prove that(14) (Φ b ) ∗ ( φ b ) ∗ ι ∗ ( φ b ) ∗ (Φ b ) ∗ = − id : A (cid:0) ( S b ) (cid:1) → A (cid:0) ( S b ) (cid:1) , for general b ∈ B .Thanks to the following compatibility lemma, things further simplify: Lemma 4.2.
Let T [3] / M → M be the “universal Hilbert cube” as in proposition 2.25, and let ι T : T [3] / M T [3] / M be the birational involution induced by the generically rational map Ψ : T [3] / M E ofproposition 2.25. Let Γ ι S be the relative correspondence Γ ι S := t ¯Γ g ◦ ¯Γ ι T ◦ ¯Γ g ∈ A ( S /B × B S /B ) (where g : S /B T [3] / M is the natural rational map).Then there is equality (cid:0) (Γ ι S ) b (cid:1) ∗ = (Φ b ) ∗ ( φ b ) ∗ ι ∗ ( φ b ) ∗ (Φ b ) ∗ : A (cid:0) ( S b ) (cid:1) → A (cid:0) ( S b ) (cid:1) for general b ∈ B .Proof. One should remember that for general b ∈ B , there is a birational map φ b : ( S b ) [3] X := X f ( b ) , where f ( b ) ∈ M ∆ in the notation of proposition 2.25. Let Z p ւ ց q S b φ b X be an elimination of indeterminacy. Let ι Z : Z → Z be the birational involution induced by ι .There is a commutative diagram(15) ( S b ) [3] p ← Z q → X ↓ ι Sb ↓ ι Z ↓ ι ( S b ) [3] p ← Z q → X (here ι S b and ι Z are birational maps, not morphisms).For general b ∈ B , the restriction (¯Γ ι T ) b is just the closure of the graph of the rational involu-tion ι S b : S b S b (induced by ι ), and so(16) (cid:0) (Γ ι S ) b (cid:1) ∗ = (Φ b ) ∗ ( ι S b ) ∗ (Φ b ) ∗ = (Φ b ) ∗ p ∗ ( ι Z ) ∗ p ∗ (Φ b ) ∗ : A i (cid:0) ( S b ) (cid:1) → A i (cid:0) ( S b ) (cid:1) . As for the right–hand–side in lemma 4.2, since ι ∗ = q ∗ ( ι Z ) ∗ q ∗ and ( φ b ) ∗ = p ∗ q ∗ (and likewise ( φ b ) ∗ = q ∗ p ∗ ), we find that (Φ b ) ∗ ( φ b ) ∗ ι ∗ ( φ b ) ∗ (Φ b ) ∗ = (Φ b ) ∗ p ∗ q ∗ q ∗ ( ι Z ) ∗ q ∗ q ∗ p ∗ (Φ b ) ∗ : A i (cid:0) ( S b ) (cid:1) → A i (cid:0) ( S b ) (cid:1) . LGEBRAIC CYCLES AND EPW CUBES 25
Since q : Z → X is birational, we have that q ∗ q ∗ = id : A ( Z ) → A ( Z ) , and so for i = 6 theabove boils down to(17) (Φ b ) ∗ ( φ b ) ∗ ι ∗ ( φ b ) ∗ (Φ b ) ∗ = (Φ b ) ∗ p ∗ ( ι Z ) ∗ p ∗ (Φ b ) ∗ : A (cid:0) ( S b ) (cid:1) → A (cid:0) ( S b ) (cid:1) . Comparing equations (16) and (17), we ascertain that we have proven the lemma. (cid:3)
Thanks to lemma 4.2, we conclude that in order to prove (14), it suffices to prove that(18) (cid:0) (Γ ι S ) b (cid:1) ∗ = − id : A (cid:0) ( S b ) (cid:1) → A (cid:0) ( S b ) (cid:1) , for general b ∈ B .We now introduce one further reduction step: we claim that in order to prove statement (18),it suffices to prove that(19) (Π ( S b ) ) ∗ (cid:0) (Γ ι S ) b (cid:1) ∗ = − id : A (cid:0) ( S b ) (cid:1) → A (cid:0) ( S b ) (cid:1) , for general b ∈ B .To prove this claim, we observe that equation (19) implies (by composing on both sides) that ( φ b ) ∗ (Φ b ) ∗ (Π ( S b ) ) ∗ (cid:0) (Γ ι S ) b (cid:1) ∗ (Φ b ) ∗ ( φ b ) ∗ = − id : A ( X ) → A ( X ) , for general b ∈ B . Here X = X A is the double EPW cube such that φ b : ( S b ) [3] X is birational. Using lemma 4.2, this implies that also ( φ b ) ∗ (Φ b ) ∗ (Π ( S b ) ) ∗ (Φ b ) ∗ ( φ b ) ∗ ι ∗ ( φ b ) ∗ (Φ b ) ∗ (Φ b ) ∗ ( φ b ) ∗ = − id : A ( X ) → A ( X ) , for general X = X A with A ∈ ∆ . This simplifies to(20) ( φ b ) ∗ (Φ b ) ∗ (Π ( S b ) ) ∗ (Φ b ) ∗ ( φ b ) ∗ ι ∗ = − id : A ( X ) → A ( X ) . But (Φ b ) ∗ (Π ( S b ) ) ∗ = (Π ( S b ) [3] ) ∗ (Φ b ) ∗ : A (( S b ) ) → A (( S b ) [3] ) (lemma 2.11 ), and ( φ b ) ∗ (Π ( S b ) [3] ) ∗ = (Π X ) ∗ ( φ b ) ∗ : A i (( S b ) [3] ) → A i ( X ) (since Rieß’s isomorphism is an isomorphism of bigraded rings, cf. lemma 2.8). Therefore,equation (20) further simplifies to (Π X ) ∗ ι ∗ = − id : A ( X ) → A ( X ) . This means that any b ∈ A ( X ) satisfies(21) ι ∗ ( b ) = − b + b + b in A ( X ) , where b j ∈ A j ) ( X ) (NB: ι ∗ ( b ) cannot have a component in A ( X ) since ι ∗ ( b ) ∈ A hom ( X ) .)On the other hand, using corollary 3.5 (just as at the beginning of this proof) we can write b = L · a where a ∈ A ( X ) and L is a ι –invariant ample line bundle. This implies that ι ∗ ( b ) = ι ∗ ( L · a ) = ι ∗ ( L ) · ι ∗ ( a ) = L · ι ∗ ( a ) in A ( X ) . But ι ∗ ( a ) ∈ A ( X ) = A ( X ) ⊕ A ( X ) and so (exploiting the fact that A ∗ ( ∗ ) ( X ) is a bigradedring, thanks to lemma 2.26) we find that(22) ι ∗ ( b ) ∈ A ( X ) ⊕ A ( X ) . Comparing equations (22) and (21), we see that we must have b = b = 0 , and so ι ∗ ( b ) = − b in A ( X ) ∀ b ∈ A ( X ) , as claimed. This proves the claim; it now remains to prove statement (19).In order to prove statement (19), we rely once again on the machinery of “spread” of cyclesin a family [41], [42]; this is very similar to the argument proving theorem 3.1. We consider thefamily S /B → B , where
S → B is (once more) the universal family of degree K surfaces (remark 2.21).Let us define a relative correspondence Γ := Π S /B ◦ (cid:0) Γ ι S + ∆ S /B (cid:1) ◦ Π S /B ∈ A (cid:0) S /B × B S /B (cid:1) . Clearly, statement (19) that we want to prove is equivalent to the statement(23) (cid:0) Γ b (cid:1) ∗ = 0 : A hom (cid:0) ( S b ) (cid:1) → A (cid:0) ( S b ) (cid:1) for general b ∈ B . (Here, as before, for any relative correspondence Γ we use the notation Γ b to indicate the restric-tion of Γ to the fibre over b ∈ B .)The homological input that we have at our disposition is that the involution ι = ι A of X = X A (and hence the induced involution of ( S b ) [3] ) is anti–symplectic (remark 2.24), and so (cid:0) Γ b (cid:1) ∗ = 0 : H , (cid:0) ( S b ) (cid:1) → H , (cid:0) ( S b ) (cid:1) for general b ∈ B .
Using the Lefschetz (1 , –theorem, this implies that for general b ∈ B , there exist a curve V b and a divisor W b inside ( S b ) , and a cycle γ b ∈ A ( W b × V b ) such that Γ b + γ b = 0 in H (cid:0) ( S b ) × ( S b ) (cid:1) . Applying the Hilbert schemes argument [41, Proposition 3.7], one can find a curve V and adivisor W inside S /B , and a cycle γ supported on W × B V such that(24) (cid:0) Γ + γ (cid:1) b = 0 in H (cid:0) ( S b ) × ( S b ) (cid:1) ∀ b ∈ B .
Let us now consider a modified relative correspondence Γ := Π S /B ◦ (Γ + γ ) ◦ Π S /B ∈ A (cid:0) S /B × B S /B (cid:1) . Since Π ( S b ) is idempotent for all b ∈ B , there is a fibrewise equality (Γ ) b = (Γ + γ ′ ) b in A (cid:0) ( S b ) × ( S b ) (cid:1) ∀ b ∈ B , where γ ′ is (just like γ ) a cycle supported on W × B V . For a general fibre, the restriction ( γ ′ ) b will be supported on (divisor) × (curve) and as such will not act on A (( S b ) ) . It follows that(25) (cid:0) (Γ ) b (cid:1) ∗ = (Γ b ) ∗ : A (cid:0) ( S b ) (cid:1) → A (cid:0) ( S b ) (cid:1) for b ∈ B general . LGEBRAIC CYCLES AND EPW CUBES 27
On the other hand, in view of proposition 2.16, there is a fibrewise equality of action (cid:0) (Γ ) b (cid:1) ∗ = (cid:16)(cid:0) ( X k =1 Ξ k ◦ Θ k ) ◦ (Γ + γ ) ◦ ( X ℓ =1 Ξ ℓ ◦ Θ ℓ ) (cid:1) b (cid:17) ∗ : A (cid:0) ( S b ) (cid:1) → A (cid:0) ( S b ) (cid:1) ∀ b ∈ B .
That is, we have equality(26) (cid:0) (Γ ) b (cid:1) ∗ = (cid:16)(cid:0) X k =1 3 X ℓ =1 Ξ k ◦ Γ k,ℓ ◦ Θ ℓ (cid:1) b (cid:17) ∗ : A (cid:0) ( S b ) (cid:1) → A (cid:0) ( S b ) (cid:1) ∀ b ∈ B, where we have defined Γ k,ℓ := Θ k ◦ (Γ + γ ) ◦ Ξ ℓ ∈ A ( S × B S ) 1 ≤ k, ℓ ≤ . We observe that equation (24) implies that (Γ k,ℓ ) b ∈ A hom ( S b × S b ) ∀ b ∈ B , ≤ k, ℓ ≤ . But then, applying proposition 2.19 to the relative correspondence Γ k,ℓ we may conclude thereexists δ k,ℓ ∈ A ( G × G ) (where G is the Grassmannian of lines in P ) such that (Γ k,ℓ ) b + ( δ k,ℓ ) b = 0 ∈ A ( S b × S b ) ∀ b ∈ B .
Since the Grassmannian has trivial Chow groups, the correspondence ( δ k,ℓ ) b acts trivially on A ∗ hom ( S b ) , and so (cid:0) (Γ k,ℓ ) b (cid:1) ∗ = 0 : A ∗ hom ( S b ) → A ∗ ( S b ) ∀ b ∈ B , ≤ k, ℓ ≤ . Plugging this in equation (26), we find that (cid:0) (Γ ) b (cid:1) ∗ = 0 : A hom (cid:0) ( S b ) (cid:1) → A (cid:0) ( S b ) (cid:1) ∀ b ∈ B .
Returning to equality (25), this implies that (Γ b ) ∗ = 0 : A hom (cid:0) ( S b ) (cid:1) → A (cid:0) ( S b ) (cid:1) for general b ∈ B , which is exactly statement (23) that we needed to prove. The proof of theorem 4.1 is nowcomplete. (cid:3)
5. S
OME COROLLARIES
Corollary 5.1.
Let D = D A be an EPW cube for A ∈ ∆ general (where ∆ ⊂ LG ν is thedivisor of theorem 2.23).(i) Let a ∈ A ( D ) be a –cycle which is either in the image of the intersection product map A ( D ) ⊗ A ( D ) ⊗ A ( D ) → A ( D ) , or in the image of the intersection product map A ( D ) ⊗ A ( D ) ⊗ A ( D ) → A ( D ) . Then a is rationally trivial if and only if a has degree .(ii) Let a ∈ A ( D ) be a –cycle which is in the image of the intersection product map A ( D ) ⊗ A ( D ) ⊗ A ( D ) → A ( D ) . Then a is rationally trivial if and only if a is homologically trivial.Proof. We first establish some lemmas:
Lemma 5.2.
Let A ∈ ∆ be general, and let X = X A be the corresponding double EPW cube.Let ι = ι A be the covering involution. Then ι ∗ = id : A ( X ) → A ( X ) . Proof.
The subgroup A ( X ) is generated by L , where L is any ample divisor. Taking L anample divisor of the form L = p ∗ ( L D ) where L D is ample on D , we see that the lemma must betrue. (cid:3) Lemma 5.3.
Let A ∈ ∆ be general, and let X = X A and D = D A be the corresponding doubleEPW cube, resp. EPW cube. Let p : X → D be the quotient morphism. We have p ∗ A ( D ) ⊂ A ( X ) . Proof.
By construction, there is an inclusion p ∗ A ( D ) ⊂ A ( X ) ι , where ι = ι A ∈ Aut( X ) is the covering involution.Given b ∈ A ( D ) , let us write p ∗ ( b ) = c + c ∈ A ( X ′ ) ⊕ A ( X ′ ) . Applying ι , we find(27) ι ∗ p ∗ ( b ) = p ∗ ( b ) = c + c ∈ A ( X ′ ) ⊕ A ( X ′ ) . On the other hand, we have(28) ι ∗ p ∗ ( b ) = ι ∗ ( c ) + ι ∗ ( c ) = ι ∗ ( c ) + d − c ∈ A ( X ) ⊕ A ( X ) , where we have used sublemma 5.4 below to obtain that ι ∗ ( c ) ∈ A ( X ) , and theorem 4.1 toobtain that ι ∗ ( c ) = − c + d for some d ∈ A ( X ) . Comparing expressions (27) and (28), wefind ι ∗ ( c ) + d = c in A ( X ) , − c = c in A ( X ) , proving lemma 5.3. Sublemma 5.4.
Set–up as above. Let b ∈ A ( D ) , and write p ∗ ( b ) = c + c ∈ A ( X ) ⊕ A ( X ) . Then ι ∗ ( c ) ∈ A ( X ) . Proof.
Suppose ι ∗ ( c ) = d + d in A ( X ) , with d ∈ A ( X ) and d ∈ A ( X ) . LGEBRAIC CYCLES AND EPW CUBES 29
Let L ∈ A ( X ) be a ι –invariant ample divisor as in the proof of theorem 4.1. The –cycle c · L is in A ( X ) , and so (using lemma 5.2) we have(29) ι ∗ ( c · L ) = c · L in A ( X ) On the other hand, we have(30) ι ∗ ( c · L ) = ι ∗ ( c ) · ι ∗ ( L ) = ( d + d ) · L = d · L + d · L in A ( X ) . Since d · L ∈ A ( X ) and d · L ∈ A ( X ) , comparing expressions (30) and (29), we seethat we must have d · L = c · L in A ( X ) , d · L = 0 in A ( X ) . Using the injectivity part of corollary 3.5, this implies that d = 0 in A ( X ) . This proves sublemma 5.4. (cid:3)(cid:3)
Let us now prove corollary 5.1(i). Suppose first a ∈ A ( D ) is a –cycle in the image of A ( D ) ⊗ A ( D ) ⊗ A ( D ) → A ( D ) . Then p ∗ ( a ) ∈ A ( X ) is in the image of p ∗ A ( X ) ⊗ p ∗ A ( X ) ⊗ p ∗ A ( X ) → A ( X ) . In view of lemma 5.3, this is contained in the image of A ( X ) ⊗ A ( X ) ⊗ A ( X ) → A ( X ) , which is A ( X ) . It follows that p ∗ ( a ) is rationally trivial if and only if p ∗ ( a ) has degree . Since a = 2 p ∗ p ∗ ( a ) , the statement for a follows.Next, suppose a ∈ A ( D ) is a –cycle in the image of A ( D ) ⊗ A ( D ) ⊗ A ( D ) → A ( D ) . Then p ∗ ( a ) ∈ A ( X ) is in the image of p ∗ A ( X ) ⊗ p ∗ A ( X ) ⊗ p ∗ A ( X ) → A ( X ) . In view of lemma 5.3 and corollary 2.26, this is contained in the image of (cid:0) A ( X ) ⊕ A ( X ) (cid:1) ⊗ A ( X ) ⊗ A ( X ) → A ( X ) , and so we find that p ∗ ( a ) ∈ A ( X ) ⊕ A ( X ) . On the other hand, p ∗ ( a ) is ι –invariant, and we have (cid:16) A ( X ) ⊕ A ( X ) (cid:17) ∩ A ( X ) ι = A ( X ) (in view of lemma 5.2 and theorem 4.1). Therefore we must have p ∗ ( a ) ∈ A ( X ) , from which the conclusion follows as above.The proof of corollary 5.1(ii) is similar: let a ∈ A ( D ) be a –cycle in the image of A ( D ) ⊗ A ( D ) ⊗ A ( D ) → A ( D ) . Then p ∗ ( a ) is in the image of A ( X ) ⊗ A ( X ) ⊗ A ( X ) → A ( X ) , which is contained in A ( X ) . But A ( X ) injects into cohomology (this follows from Rieß’sisomorphism [33], combined with the corresponding statement for A ( S [3] ) which is noted in[38, Introduction]). (cid:3) The argument proving corollary 5.1 actually proves a more general statement:
Corollary 5.5.
Let X be a variety of dimension m of the form X = D × · · · × D r × K × · · · × K s × X × · · · × X t , where each D j is an EPW cube D A j for A j ∈ ∆ , , and each K j is a generalized Kummervariety, and each X j is a Hilbert scheme ( S j ) [ m j ] where S j is a K surface.Let E ∗ ( X ) ⊂ A ∗ ( X ) be the subring generated by (pullbacks of) A ( D j ) , A ( D j ) , A ( K j ) , c r ( K j ) , A ( X j ) , c r ( X j ) , where c r () ∈ A r () denote the Chern classes. Then the cycle class map E i ( X ) → H i ( X ) is injective for i ≥ m − .Proof. Let us consider the variety Y := Y × · · · × Y r × K × · · · × K s × X × · · · × X t , where p j : Y j → D j is the double cover from the double EPW cube Y j to the EPW cube D j , andthe finite morphism p : Y → X .
The variety Y has an MCK decomposition. (Indeed, the varieties Y j , K j and X j have an MCKdecomposition, thanks to corollary 2.26, resp. [11], resp. [38]). As the property of having anMCK decomposition is stable under products [35, Theorem 8.6], the statement for the variety Y follows.)There is an inclusion p ∗ E ∗ ( X ) ⊂ A ∗ (0) ( Y ) . (Indeed, we have seen in corollary 5.1 that ( p j ) ∗ A ( D j ) ⊂ A ( Y j ) . Furthermore, it is knownthat c r ( K j ) ∈ A r (0) ( K j ) , c r ( X j ) ∈ A r (0) ( X j ) [11, Proposition 7.13], resp. [38, Theorem 2]. Let π denote projection from Y to any of thefactors Y j or K j or X j . Then π is “of pure grade ”, in the sense of [36, Definition 1.1], whichmeans that π ∗ preserves the bigrading [36, Corollary 1.6]. This proves the stated inclusion.) LGEBRAIC CYCLES AND EPW CUBES 31
Since A i (0) ( Y ) → H i ( Y ) is injective for i ≥ m − , and p ∗ : A i ( X ) → A i ( Y ) is injective for all i , this proves the corollary. (cid:3) Acknowledgements.
It is a pleasure to thank Len for numerous shared readings of ”Het huisvan Barbapapa”. R EFERENCES [1] A. Beauville, Some remarks on K¨ahler manifolds with c = 0 , in: Classification of algebraic and analyticmanifolds (Katata, 1982), Birkh¨auser Boston, Boston 1983,[2] A. Beauville, Vari´et´es K¨ahleriennes dont la premi`ere classe de Chern est nulle, J. Differential Geom. 18no. 4 (1983), 755—782,[3] A. Beauville, On the splitting of the Bloch–Beilinson filtration, in: Algebraic cycles and motives (J. Nageland C. Peters, editors), London Math. Soc. Lecture Notes 344, Cambridge University Press 2007,[4] A. Beauville and C. Voisin, On the Chow ring of a K surface, J. Alg. Geom. 13 (2004), 417—426,[5] S. Bloch, Lectures on algebraic cycles, Duke Univ. Press Durham 1980,[6] S. Bloch and A. Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci. Ecole Norm. Sup. 4(1974), 181—202,[7] M. de Cataldo and L. Migliorini, The Chow groups and the motive of the Hilbert scheme of points on asurface, Journal of Algebra 251 no. 2 (2002), 824—848,[8] D. Edidin and W. Graham, Equivariant intersection theory, Invent. Math. 131 no. 3 (1998), 595—634,[9] L. Fu, On the coniveau of certain sub–Hodge structures, Math. Res. Lett. 19 (2012), 1097—1116,[10] L. Fu, Decomposition of small diagonals and Chow rings of hypersurfaces and Calabi–Yau completeintersections, Advances in Mathematics (2013), 894—924,[11] L. Fu, Z. Tian and C. Vial, Motivic hyperk¨ahler resolution conjecture for generalized Kummer varieties,arXiv:1608.04968,[12] W. Fulton, Intersection theory, Springer–Verlag Ergebnisse der Mathematik, Berlin Heidelberg New YorkTokyo 1984,[13] H. Gillet, Intersection theory on algebraic stacks and Q –varieties, in: Algebraic K –theory, Luminy 1983,[14] R. Hartshorne, Equivalence relations on algebraic cycles and subvarieties of small codimension, in: Al-gebraic geometry, Arcata 1974, Proc. Symp. Pure Math. Vol. 29, Amer. Math. Soc., Providence 1975,[15] A. Iliev, G. Kapustka, M. Kapustka and K. Ranestad, EPW cubes, arXiv:1505.02389v2, to appear in J. f.Reine u. Angew. Math.,[16] U. Jannsen, Motives, numerical equivalence, and semi-simplicity, Invent. Math. 107(3) (1992), 447—452,[17] U. Jannsen, Motivic sheaves and filtrations on Chow groups, in: Motives (U. Jannsen et alii, eds.), Pro-ceedings of Symposia in Pure Mathematics Vol. 55 (1994), Part 1,[18] U. Jannsen, Equivalence relations on algebraic cycles, in: The arithmetic and geometry of algebraiccycles, Proceedings of the Banff Conference 1998 (B. Gordon et alii, eds.), Kluwer,[19] U. Jannsen, On finite–dimensional motives and Murre’s conjecture, in: Algebraic cycles and motives (J.Nagel and C. Peters, eds.), Cambridge University Press, Cambridge 2007,[20] R. Laterveer, Hard Lefschetz for Chow groups of generalized Kummer varieties, Abh. Math. Semin. Univ.Hamburg 87 no. 1 (2017), 135—144, [21] R. Laterveer, A family of cubic fourfolds with finite–dimensional motive, to appear in Journal Math. Soc.Japan,[22] R. Laterveer, Algebraic cycles on a very special EPW sextic, Rend. Sem. Mat. Univ. Padova,[23] R. Laterveer, Bloch’s conjecture for certain hyperk¨ahler fourfolds, and EPW sextics, submitted,[24] R. Laterveer, About Chow groups of certain hyperk¨ahler varieties with non–symplectic automorphisms,Vietnam J. Math.,[25] R. Laterveer, On the Chow groups of certain EPW sextics, submitted,[26] R. Laterveer, On the Chow groups of some hyperk¨ahler fourfolds with a non–symplectic involution,International Journal of Math.,[27] S. Mukai, Curves, K3 surfaces and Fano 3-folds of genus ≤ , in: Algebraic Geometry and CommutativeAlgebra I (H.Hijikata et al., eds.), Kinokuniya Tokyo 1988,[28] J. Murre, On a conjectural filtration on the Chow groups of an algebraic variety, parts I and II, Indag.Math. 4 (1993), 177—201,[29] J. Murre, J. Nagel and C. Peters, Lectures on the theory of pure motives, Amer. Math. Soc. UniversityLecture Series 61, Providence 2013,[30] K. O’Grady, Involutions and linear systems on holomorphic symplectic manifolds, Geom. Funct. Anal.15 no 6 (2005), 1223—1274,[31] K. O’Grady, Irreducible symplectic –folds and Eisenbud–Popescu–Walter sextics, Duke Math. J. 134(1)(2006), 99—137,[32] K. O’Grady, Double covers of EPW–sextics, Michigan Math. J. 62 (2013), 143—184,[33] U. Rieß, On the Chow ring of birational irreducible symplectic varieties, Manuscripta Math. 145 (2014),473—501,[34] T. Scholl, Classical motives, in: Motives (U. Jannsen et alii, eds.), Proceedings of Symposia in PureMathematics Vol. 55 (1994), Part 1,[35] M. Shen and C. Vial, The Fourier transform for certain hyperK¨ahler fourfolds, Memoirs of the AMS 240(2016), no.1139,[36] M. Shen and C. Vial, The motive of the Hilbert cube X [3] , Forum Math. Sigma 4 (2016),[37] C. Vial, Remarks on motives of abelian type, to appear in Tohoku Math. J.,[38] C. Vial, On the motive of some hyperk¨ahler varieties, to appear in J. f¨ur Reine u. Angew. Math.,[39] A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math. 97 no. 3(1989), 613—670,[40] C. Voisin, Chow rings and decomposition theorems for K surfaces and Calabi–Yau hypersurfaces,Geom. Topol. 16 (2012), 433—473,[41] C. Voisin, The generalized Hodge and Bloch conjectures are equivalent for general complete intersections,Ann. Sci. Ecole Norm. Sup. 46, fascicule 3 (2013), 449—475,[42] C. Voisin, The generalized Hodge and Bloch conjectures are equivalent for general complete intersections,II, J. Math. Sci. Univ. Tokyo 22 (2015), 491—517,[43] C. Voisin, Bloch’s conjecture for Catanese and Barlow surfaces, J. Differential Geometry 97 (2014),149—175,[44] C. Voisin, Chow Rings, Decomposition of the Diagonal, and the Topology of Families, Princeton Univer-sity Press, Princeton and Oxford, 2014,[45] C. Voisin, Remarks and questions on coisotropic subvarieties and –cycles of hyper–K¨ahler varieties,in: K3 Surfaces and Their Moduli, Proceedings of the Schiermonnikoog conference 2014 (C. Faber, G.Farkas, G. van der Geer, editors), Progress in Maths 315, Birkh¨auser 2016.I NSTITUT DE R ECHERCHE M ATH ´ EMATIQUE A VANC ´ EE , CNRS – U NIVERSIT ´ E DE S TRASBOURG , 7 R UE R EN ´ E D ESCARTES , 67084 S
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