aa r X i v : . [ m a t h . AG ] D ec ALGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC
ROBERT LATERVEERA
BSTRACT . Motivated by the Beauville–Voisin conjecture about Chow rings of powers of K surfaces, we consider a similar conjecture for Chow rings of powers of EPW sextics. We provepart of this conjecture for the very special EPW sextic studied by Donten–Bury et alii. We alsoprove some other results concerning the Chow groups of this very special EPW sextic, and ofcertain related hyperk¨ahler fourfolds.
1. I
NTRODUCTION
For a smooth projective variety X over C , let A i ( X ) = CH i ( X ) Q denote the Chow groupof codimension i algebraic cycles modulo rational equivalence with Q –coefficients. Intersectionproduct defines a ring structure on A ∗ ( X ) = ⊕ i A i ( X ) . In the case of K surfaces, this ringstructure has an interesting property: Theorem 1.1 (Beauville–Voisin [8]) . Let S be a K surface. Let D i , D ′ i ∈ A ( S ) be a finitenumber of divisors. Then ∑ i D i ⋅ D ′ i = in A ( S ) ⇔ ∑ i D i ⋅ D ′ i = in H ( S, Q ) . Conjecturally, a similar property holds for self–products of K surfaces: Conjecture 1.2 (Beauville–Voisin) . Let S be a K surface. For r ≥ , let D ∗ ( S r ) ⊂ A ∗ ( S r ) bethe Q –subalgebra generated by (the pullbacks of) divisors and the diagonal of S . The restrictionof the cycle class map induces an injection D i ( S r ) → H i ( S r , Q ) for all i and all r . (cf. [53], [54], [56], [58] for extensions and partial results concerning conjecture 1.2.)Beauville has asked which varieties have behaviour similar to theorem 1.1 and conjecture 1.2.This is the problem of determining which varieties verify the “weak splitting property” of [7].We briefly state this problem here as follows: Mathematics Subject Classification.
Primary 14C15, 14C25, 14C30. Secondary 14J32, 14J35, 14J70,14K99.
Key words and phrases.
Algebraic cycles, Chow groups, motives, finite–dimensional motives, weak split-ting property, weak Lefschetz conjecture for Chow groups, multiplicative Chow–K¨unneth decomposition, Bloch–Beilinson filtration, EPW sextics, hyperk¨ahler varieties, K3 surfaces, abelian varieties, Calabi–Yau varieties.
Problem 1.3 (Beauville [7]) . Find a nice class C of varieties (containing K surfaces andabelian varieties), such that for any X ∈ C , the Chow ring of X admits a multiplicative bi-grading A ∗ ( ∗ ) ( X ) , with A i ( X ) = ⨁ j ≥ A i ( j ) ( X ) for all i . This bigrading should split the conjectural Bloch–Beilinson filtration, in particular A ihom ( X ) = ⨁ j ≥ A i ( j ) ( X ) . It has been conjectured that hyperk¨ahler varieties are in C [7, Introduction]. Also, not allCalabi–Yau varieties can be in C [7, Example 1.7(b)]. An interesting novel approach of problem1.3 (as well as a reinterpretation of theorem 1.1) is provided by the concept of multiplicativeChow–K¨unneth decomposition (cf. [43], [50], [44] and subsection 2.3 below).In this note, we ask whether EPW sextics might be in C . An EPW sextic is a special sextic X ⊂ P ( C ) constructed in [18]. EPW sextics are not smooth; however, a generic EPW sextic is aquotient X = X /( σ ) , where X is a smooth hyperk¨ahler variety (called a double EPW sextic)and σ is an anti–symplectic involution [35, Theorem 1.1], [36]. Quotient varieties behave likesmooth varieties with respect to intersection theory with rational coefficients, so the followingconjecture makes sense: Conjecture 1.4.
Let X be an EPW sextic, and assume X is a quotient variety X = X / G with X smooth and G ⊂ Aut ( X ) a finite group. Then X ∈ C . There are two reasons why conjecture 1.4 is likely to be true: first, because an EPW sextic is aCalabi–Yau hypersurface (and these are probably in C ); secondly, because the hyperk¨ahler variety X should be in C , and the involution σ should behave nicely with respect to the bigrading on A ∗ ( ∗ ) ( X ) . Let us optimistically suppose conjecture 1.4 is true, and see what consequences thisentails for the Chow ring of EPW sextics. We recall that Chow groups are expected to satisfy aweak Lefschetz property, according to a long–standing conjecture: Conjecture 1.5 (Hartshorne [24]) . Let X ⊂ P n + ( C ) be a smooth hypersurface of dimension n ≥ . Then the cycle class map A ( X ) → H ( X, Q ) is injective. Conjecture 1.5 is notoriously open for all hypersurfaces of degree d ≥ n + . Since quotientvarieties behave in many ways like smooth varieties, it seems reasonable to expect that conjecture1.5 extends to hypersurfaces that are quotient varieties. This would imply that an EPW sextic X as in conjecture 1.4 has A hom ( X ) = . That is, conjecturally we have that A i ( X ) = A i ( ) ( X ) for all i ≤ . For any r ≥ , let us now define E ∗ ( X r ) ⊂ A ∗ ( X r ) LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 3 as the Q –subalgebra generated by (pullbacks of) elements of A ( X ) and A ( X ) and the class ofthe diagonal of X . The above remarks imply a conjectural inclusion E ∗ ( X r ) ⊂ A ∗ ( ) ( X r ) = A ∗ ( X r )/ A ∗ hom ( X r ) . We thus arrive at the following concrete, falsifiable conjecture:
Conjecture 1.6.
Let X be an EPW sextic as in conjecture 1.4. Then restriction of the cycle classmap E i ( X r ) → H i ( X r , Q ) is injective for all i and all r . Conjecture 1.6 is the analogon of conjecture 1.2 for EPW sextics; the role of divisors on the K surface is played by (the hyperplane section and) codimension cycles on the sextic. Themain result in this note provides some evidence for conjecture 1.6: we can prove it is true for –cycles and –cycles on one very special EPW sextic: Theorem (=theorem 4.7) . Let X be the very special EPW sextic of [16] . Let r ∈ N . Therestriction of the cycle class map E i ( X r ) → H i ( X r , Q ) is injective for i ≥ r − . The very special EPW sextic of [16] (cf. section 2.7 below for a definition) is not smooth,but it is a “Calabi–Yau variety with quotient singularities”. The very special EPW sextic X isvery symmetric; it is also remarkable for providing the only example known so far of a completefamily of pairwise incident planes in P ( C ) [16]. As resumed in theorem 2.28 below, thevery special EPW sextic X is related to hyperk¨ahler varieties in two different ways: (a) X isrationally dominated via a degree map by the Hilbert scheme S [ ] where S is a K surface ofPicard number ; (b) X admits a double cover that is the quotient of an abelian variety by afinite group of group automorphisms, and this quotient admits a hyperk¨ahler resolution X .To prove theorem 4.7, we first prove (proposition 3.3) that the very special EPW sextic X hasa multiplicative Chow–K¨unneth decomposition, in the sense of Shen–Vial [43], and so the Chowring of X has a bigrading. Next, we establish (proposition 3.8) that(1) A ( X ) = A ( ) ( X ) . Both these facts are proven using description (b), via the theory of symmetrically distinguishedcycles [37].Note that equality (1) might be considered as evidence for conjecture 1.5 for X . In order toprove conjecture 1.5 for the very special EPW sextic X , it remains to prove that A ( ) ( X ) ∩ A hom ( X ) ?? = . Likewise, in order to prove the full conjecture 1.6 for the very special EPW sextic X , it remainsto prove that A i ( ) ( X r ) ∩ A ihom ( X r ) ?? = for all i, r . ROBERT LATERVEER
We are not able to prove these equalities outside of the range i ≥ r − ; this is related to someof the open cases of Beauville’s conjecture on Chow rings of abelian varieties (remarks 4.4 and4.8).On the positive side, we establish a precise relation between the Chow ring of the very specialEPW sextic X and the Chow ring of the hyperk¨ahler fourfold X mentioned in description (b)(theorem 4.9). This relation provides an alternative description of the splitting of the Chow ringof X coming from a multiplicative Chow–K¨unneth decomposition (corollary 4.10). In provingthis relation, we exploit description (a); a key ingredient in the proof is a strong version of thegeneralized Hodge conjecture for X and X (proposition 3.1), which crucially relies on the factthat the K surface S has maximal Picard number.We also obtain some results concerning Bloch’s conjecture (subsection 5.1), as well as a con-jecture of Voisin (subsection 5.2), for the very special EPW sextic. The application to Bloch’sconjecture relies on description (b) (via the theory of symmetrically distinguished cycles), butalso on description (a) (via the surjectivity result proposition 3.12).We end this introduction with a challenge: can one prove theorem 4.7 for other (not veryspecial) EPW sextics ? Conventions.
In this note, 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 denote by A j X the Chow group of j –dimensional cycles on X with Q –coefficients; for X smooth of dimension n the notations A j X and A n − j X will be used interchangeably.The notations A jhom ( X ) , A jnum ( X ) , A jAJ ( X ) will be used to indicate the subgroups of homo-logically trivial, resp. numerically trivial, resp. Abel–Jacobi trivial cycles. The contravariantcategory of Chow motives (i.e., pure motives with respect to rational equivalence as in [42] , [34] )will be denoted M rat .We will write H j ( X ) and H j ( X ) to indicate singular cohomology H j ( X, Q ) , resp. Borel–Moore homology H j ( X, Q ) .
2. P
RELIMINARY MATERIAL
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. Proposition 2.2 (Fulton [22]) . 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 [22, Example 17.4.10]. (cid:3)
LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 5
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 [22, 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 ∣ ∑ g ∈ G Γ g . Finite–dimensionality.
We refer to [32], [4], [34], [26], [30] for basics on the notion offinite–dimensional motive. An essential property of varieties with finite–dimensional motive isembodied by the nilpotence theorem:
Theorem 2.4 (Kimura [32]) . 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 = ∈ A n ( X × X ) . Actually, the nilpotence property (for all powers of X ) could serve as an alternative definitionof finite–dimensional motive, as shown by a result of Jannsen [30, Corollary 3.9]. Conjecturally,all smooth projective varieties have finite–dimensional motive [32]. We are still far from knowingthis, but at least there are quite a few non–trivial examples: Remark 2.5.
The following varieties have finite–dimensional motive: abelian varieties, vari-eties dominated by products of curves [32] , K surfaces with Picard number or [38] ,surfaces not of general type with p g = [23, Theorem 2.11] , certain surfaces of general typewith p g = [23] , [40] , [55] , Hilbert schemes of surfaces known to have finite–dimensional mo-tive [13] , generalized Kummer varieties [57, Remark 2.9(ii)] , [21] , threefolds with nef tangentbundle [27] , [47, Example 3.16] , fourfolds with nef tangent bundle [28] , log–homogeneous va-rieties in the sense of [12] (this follows from [28, Theorem 4.4] ), certain threefolds of generaltype [49, Section 8] , varieties of dimension ≤ rationally dominated by products of curves [47,Example 3.15] , varieties X with A iAJ ( X ) = for all i [46, Theorem 4] , products of varietieswith finite–dimensional motive [32] . Remark 2.6.
It is an embarassing fact that up till now, all examples of finite-dimensional motiveshappen to lie in the tensor subcategory generated by Chow motives of curves, i.e. they are“motives of abelian type” in the sense of [47] . On the other hand, there exist many motives thatlie outside this subcategory, e.g. the motive of a very general quintic hypersurface in P [14,7.6] . The notion of finite–dimensionality is easily extended to quotient varieties:
Definition 2.7.
Let X = Y / G be a projective quotient variety. We say that X has finite–dimensional motive if the motive h ( Y ) G ∶ = ( Y, ∆ GY , ) ∈ M rat is finite–dimensional. (Here, ∆ GY denotes the idempotent ∣ G ∣ ∑ g ∈ G Γ g ∈ A n ( Y × Y ) .) ROBERT LATERVEER
Clearly, if Y has finite–dimensional motive then also X = Y / G has finite–dimensional mo-tive. The nilpotence theorem extends to this set–up: Proposition 2.8.
Let X = Y / G be a projective quotient variety of dimension n , and assume X has finite–dimensional motive. Let Γ ∈ A nnum ( X × X ) . Then there is N ∈ N such that Γ ◦ N = ∈ A n ( X × X ) . Proof.
Let p ∶ Y → X denote the quotient morphism. We associate to Γ a correspondence Γ Y ∈ A n ( Y × Y ) defined as Γ Y ∶ = t Γ p ◦ Γ ◦ Γ p ∈ A n ( Y × Y ) . By Lieberman’s lemma [47, Lemma 3.3], there is equality Γ Y = ( p × p ) ∗ Γ in A n ( Y × Y ) , and so Γ Y is G × G –invariant: ∆ GY ◦ Γ Y ◦ ∆ GY = Γ Y in A n ( Y × Y ) . This implies that Γ Y ∈ ∆ GY ◦ A n ( Y × Y ) ◦ ∆ GY , and so Γ Y ∈ End M rat ( h ( Y ) G ) . Since clearly Γ Y is numerically trivial, and h ( Y ) G is finite–dimensional (by assumption), thereexists N ∈ N such that ( Γ Y ) ◦ N = t Γ p ◦ Γ ◦ Γ p ◦ t Γ p ◦ ⋯ ◦ Γ p = in A n ( Y × Y ) . Using the relation Γ p ◦ t Γ p = d ∆ X , this boils down to d N − t Γ p ◦ Γ ◦ N ◦ Γ p = in A n ( Y × Y ) . From this, we deduce that also Γ ◦ N = d N + Γ p ◦ ( d N − t Γ p ◦ Γ ◦ N ◦ Γ p ) ◦ t Γ p = in A n ( X × X ) . (cid:3) MCK decomposition.Definition 2.9 (Murre [33]) . 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 ) . Remark 2.10.
The existence of a CK decomposition for any smooth projective variety is part ofMurre’s conjectures [33] , [29] . If a quotient variety X has finite–dimensional motive, and theK¨unneth components are algebraic, then X has a CK decomposition (this can be proven just as [29] , where this is stated for smooth X ). LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 7
Definition 2.11 (Shen–Vial [43]) . Let X be a projective quotient variety of dimension n . Let ∆ Xsm ∈ A n ( X × X × X ) be the class of the small diagonal ∆ Xsm ∶ = {( x, x, x ) ∣ x ∈ X } ⊂ X × X × X .
An MCK decomposition of X is a CK decomposition { Π i } of X that is multiplicative , i.e. itsatisfies Π k ◦ ∆ Xsm ◦ ( Π i × Π j ) = in A n ( X × X × X ) for all i + j / = k . (NB: the acronym “MCK” is shorthand for “multiplicative Chow–K¨unneth”.) Remark 2.12.
The small diagonal (seen as a correspondence from X × X to X ) induces the multiplication morphism ∆ Xsm ∶ h ( X ) ⊗ h ( X ) → h ( X ) in M rat . Suppose X has a CK decomposition h ( X ) = n ⨁ i = h i ( X ) in M rat . By definition, this decomposition is multiplicative if for any i, j the composition h i ( X ) ⊗ h j ( X ) → h ( X ) ⊗ h ( X ) ∆ Xsm −−− → h ( X ) in M rat factors through h i + j ( X ) .The property of having an MCK decomposition is severely restrictive, and is closely related toBeauville’s “weak splitting property” [7] . For more ample discussion, and examples of varietieswith an MCK decomposition, we refer to [43, Section 8] and also [50] , [44] , [21] . Lemma 2.13.
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 [50, Introduction]; the idea is that Rieß’s result [41] implies that X and X ′ have isomorphic Chow motives and the isomorphism is compatible with the multiplicativestructure.More precisely: let γ ∶ X ⇢ X ′ be a birational map between hyperk¨ahler varieties of dimension n , and suppose { Π Xi } is an MCK decomposition for X . Let ∆ Xsm , ∆ X ′ sm denote the small diagonalof X resp. X ′ . As explained in [43, Section 6], the argument of [41] gives the equality Γ γ ◦ ∆ Xsm ◦ t Γ γ × γ = ∆ X ′ sm in A n ( X ′ × X ′ × X ′ ) . The prescription Π X ′ i ∶ = Γ γ ◦ π Xi ◦ t Γ γ ∈ A n ( X ′ × X ′ ) defines a CK decomposition for F ′ . (The Π X ′ i are orthogonal idempotents thanks to Rieß’s resultthat Γ γ ◦ t Γ γ = ∆ X ′ and t Γ γ ◦ Γ γ = ∆ X [41].) ROBERT LATERVEER
To see this CK decomposition { Π X ′ i } is multiplicative, let us consider integers i, j, k such that i + j / = k . It follows from the above equalities that Π X ′ k ◦ ∆ X ′ sm ◦ ( Π X ′ i × Π X ′ j ) = Γ γ ◦ Π Xk ◦ t Γ γ ◦ Γ γ ◦ ∆ Xsm ◦ t Γ γ × γ ◦ Γ γ × γ ◦ ( Π Xi × Π Xj ) ◦ t Γ γ = Γ γ ◦ Π Xk ◦ ∆ Xsm ◦ ( Π Xi × Π Xj ) ◦ t Γ γ = in A n ( X ′ × X ′ ) . (Here we have again used Rieß’s result that Γ γ ◦ t Γ γ = ∆ X ′ and t Γ γ ◦ Γ γ = ∆ X .) (cid:3) Niveau filtration.Definition 2.14 (Coniveau filtration [10]) . Let X be a quasi–projective variety. The coniveaufiltration on cohomology and on homology is defined as N c H i ( X, Q ) = ∑ Im ( H iY ( X, Q ) → H i ( X, Q )) ; N c H i ( X, Q ) = ∑ Im ( H i ( Z, Q ) → H i ( X, Q )) , where Y runs over codimension ≥ c subvarieties of X , and Z over dimension ≤ i − c subvarieties. Vial introduced the following variant of the coniveau filtration:
Definition 2.15 (Niveau filtration [48]) . Let X be a smooth projective variety. The niveau filtra-tion on homology is defined as ̃ N j H i ( X ) = ∑ Γ ∈ A i − j ( Z × X ) Im ( H i − j ( Z ) → H i ( X )) , where the union runs over all smooth projective varieties Z of dimension i − j , and all corre-spondences Γ ∈ A i − j ( Z × X ) . The niveau filtration on cohomology is defined as ̃ N c H i X ∶ = ̃ N c − i + n H n − i X .
Remark 2.16.
The niveau filtration is included in the coniveau filtration: ̃ N j H i ( X ) ⊂ N j H i ( X ) . These two filtrations are expected to coincide; indeed, Vial shows this is true if and only if theLefschetz standard conjecture is true for all varieties [48, Proposition 1.1] .Using the truth of the Lefschetz standard conjecture in degree ≤ , it can be checked [48, page415 ”Properties”] that the two filtrations coincide in a certain range: ̃ N j H i ( X ) = N j H i X for all j ≥ i − . LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 9
Refined CK decomposition.Theorem 2.17 (Vial [48]) . Let X be a smooth projective variety of dimension n ≤ . Assumethe Lefschetz standard conjecture B ( X ) holds (in particular, the K¨unneth components π i ∈ H n ( X × X ) are algebraic). Then there is a splitting into mutually orthogonal idempotents π i = ∑ j π i,j ∈ H n ( X × X ) , such that ( π i,j ) ∗ H ∗ ( X ) = gr j ̃ N H i ( X ) . (Here, the graded gr j ̃ N H i ( X ) can be identified with a Hodge substructure of H i ( X ) using thepolarization.) In particular, ( π , ) ∗ H j ( X ) = H ( X ) ∩ F , ( π , ) ∗ H j ( X ) = H tr ( X ) . (Here F ∗ denotes the Hodge filtration, and H tr ( X ) is the orthogonal complement to H ( X ) ∩ F under the pairing H ( X ) ⊗ H ( X ) → Q ,a ⊗ b ↦ a ∪ h n − ∪ b . ) Proof.
This is [48, Theorem 1]. (cid:3)
Theorem 2.18 (Vial [48]) . Let X be as in theorem 2.17. Assume in addition X has finite–dimensional motive. Then there exists a CK decomposition Π i ∈ A n ( X × X ) , and a splittinginto mutually orthogonal idempotents Π i = ∑ j Π i,j ∈ A n ( X × X ) , such that Π i,j = π i,j in H n ( X × X ) , and ( Π i,i ) ∗ A k ( X ) = for all k / = i . The motive h i, ( X ) = ( X, Π i, , ) ∈ M rat is well–defined up to isomorphism.Proof. This is [48, Theorem 2]. The last statement follows from [48, Proposition 1.8] combinedwith [31, Theorem 7.7.3]. (cid:3)
Remark 2.19.
In case X is a surface with finite–dimensional motive, there is equality h , ( X ) = t ( X ) in M rat , where t ( X ) is the “transcendental part of the motive” constructed for any surface (not neces-sarily with finite–dimensional motive) in [31] . Lemma 2.20.
Let X be a smooth projective variety as in theorem 2.18, and assume dim H ( X, O X ) = . Then the motive h , ( X ) ∈ M rat is indecomposable , i.e. any non–zero submotive M ⊂ h , ( X ) is equal to h , ( X ) .Proof. (This kind of argument is well–known, cf. for instance [55, Corollary 3.11] or [39, Corol-lary 2.10] where this is proven for K surfaces with finite–dimensional motive.) The idea isthat there are no non–zero Hodge substructures strictly contained in H tr ( X ) . Since the motive M ⊂ h , ( X ) defines a Hodge substructure H ∗ ( M ) ⊂ H tr ( X ) , we must have H ∗ ( M ) = H tr ( X ) and thus an equality of homological motives M = h , ( X ) in M hom . Using finite–dimensionality of X , it follows there is an equality of Chow motives M = h , ( X ) in M rat . (cid:3) Lemma 2.21.
Let X , X be two projective quotient varieties of dimension . Assume X , X have finite–dimensional motive, verify the Lefschetz standard conjecture and N H H ( X j ) = ̃ N H ( X j ) for j = , , where N ∗ H is the Hodge coniveau filtration. Let Γ ∈ A ( X × X ) and Ψ ∈ A ( X × X ) . Thefollowing are equivalent:(i) Γ ∗ ∶ H , ( X ) → H , ( X ) is an isomorphism, with inverse Ψ ∗ ;(ii) Γ ∗ ∶ H tr ( X ) → H tr ( X ) is an isomorphism, with inverse Ψ ∗ ;(iii) Γ ∶ h , ( X ) → h , ( X ) in M rat is an isomorphism, with inverse Ψ .Proof. Assume (i), i.e. Ψ ∗ Γ ∗ = id ∶ H , ( X ) → H , ( X ) . Using the hypothesis N H = ̃ N , this implies Ψ ∗ Γ ∗ = id ∶ H ( X )/ ̃ N → H ( X )/ ̃ N , and so(2) ( Ψ ◦ Γ ◦ Π X , ) ∗ = ( Π X , ) ∗ ∶ H ∗ ( X ) → H ∗ ( X ) . LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 11
Considering the action on H tr ( X ) , this implies Ψ ∗ Γ ∗ = id ∶ H tr ( X ) → H tr ( X ) . Switching the roles of X and X , one finds that likewise Γ ∗ Ψ ∗ = id on H tr ( X ) , and so theisomorphism of (ii) is proven.Next, we note that it formally follows from equality (2) that Ψ is left–inverse to Γ ∶ h , ( X ) → h , ( X ) in M hom . Switching roles of X and X , one finds Ψ is also right–inverse to Γ and so Γ ∶ h , ( X ) → h , ( X ) in M hom is an isomorphism, with inverse Ψ . By finite–dimensionality, the same holds in M rat , establish-ing (iii). (cid:3) Remark 2.22.
The equality N H H ( X j ) = ̃ N H ( X j ) in the hypothesis of lemma 2.21 is the conjunction of the generalized Hodge conjecture N H = N and Vial’s conjecture N = ̃ N . Symmetrically distinguished cycles on abelian varieties.Definition 2.23 (O’Sullivan [37]) . Let A be an abelian variety. Let a ∈ A ∗ ( A ) be a cycle. For m ≥ , let V m ( a ) ⊂ A ∗ ( A m ) denote the Q –vector space generated by elements p ∗ (( p ) ∗ ( a r ) ⋅ ( p ) ∗ ( a r ) ⋅ . . . ⋅ ( p n ) ∗ ( a r n )) ∈ A ∗ ( A m ) . Here n ≤ m , and r j ∈ N , and p i ∶ A n → A denotes projection on the i –th factor, and p ∶ A n → A m is a closed immersion with each component A n → A being either a projection or the compositeof a projection with [ − ] ∶ A → A .The cycle a ∈ A ∗ ( A ) is said to be symmetrically distinguished if for every m ∈ N thecomposition V m ( a ) ⊂ A ∗ ( A m ) → A ∗ ( A m )/ A ∗ hom ( A m ) is injective. Theorem 2.24 (O’Sullivan [37]) . The symmetrically distinguished cycles form a Q –subalgebra A ∗ sym ( A ) ⊂ A ∗ ( A ) , and the composition A ∗ sym ( A ) ⊂ A ∗ ( A ) → A ∗ ( A )/ A ∗ hom ( A ) is an isomorphism. Symmetrically distinguished cycles are stable under pushforward and pull-back of homomorphisms of abelian varieties. Remark 2.25.
For discussion and applications of the notion of symmetrically distinguished cy-cles, in addition to [37] we refer to [43, Section 7] , [50] , [3] , [20] . Lemma 2.26.
Let A be an abelian variety of dimension g .(i) There exists an MCK decomposition { Π Ai } that is self–dual and consists of symmetricallydistinguished cycles.(ii) Assume g ≤ , and let { Π Ai } be as in (i). There exists a further splitting Π A = Π A , + Π A , in A g ( A × A ) , where the Π A ,i are symmetrically distinguished and Π A ,i = π A ,i in H g ( A × A ) .Proof. (i) An explicit formula for { Π Ai } is given in [43, Section 7 Formula (45)].(ii) The point is that Π A , is (by construction) a cycle of type ∑ j C j × D j in A g ( A × A ) , where D j ⊂ A is a symmetric divisor and C j ⊂ A is a curve obtained by intersecting a symmetricdivisor with hyperplanes. This implies Π A , is symmetrically distinguished. By assumption, Π A is symmetrically distinguished and hence so is Π A , . (cid:3) The very special EPW sextic.
This subsection introduces the main actor of this tale: thevery symmetric EPW sextic discovered in [16].
Definition 2.27 ([5]) . A hyperk¨ahler variety is a simply–connected smooth projective variety X such that H ( X, Ω X ) is spanned by a nowhere degenerate holomorphic –form. Theorem 2.28 (Donten–Bury et alii [16]) . Let X ⊂ P ( C ) be defined by the equation x + x + x + x + x + x + ( x x + x x + ⋯ + x x ) + ( x x x + x x x + ⋯ + x x x ) + x x x x x x = . (Note that the parentheses are symmetric functions in the variables x , . . . , x .)(i) The hypersurface X is an EPW sextic (in the sense of [18] , [35] ).(ii) Let S be the K surface obtained from a certain Del Pezzo surface in [51] , and let S [ ] denotethe Hilbert scheme of points on S . Then there is a rational map (of degree ) φ ∶ S [ ] ⇢ X .
There exists a commutative diagram S [ ] flops S [ ] − → X ′ ∶ = E /( G ′ ) ← − X φ g ↙ g X Here all horizontal arrows are birational maps. E is an elliptic curve and X ′ ∶ = E /( G ′ ) is a quotient variety, and X is a hyperk¨ahler variety with b ( X ) = which is a symplecticresolution of X ′ . The morphism g is a double cover; X is a projective quotient variety X = E / G where G = ( G ′ , i ) with i ∈ G ′ . The groups G ′ and G consist of automorphisms that are grouphomomorphisms.(iii) S [ ] and X have finite–dimensional motive and a multiplicative CK decomposition. LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 13
Proof. (i) [16, Proposition 2.6].(ii) This is a combination of [16, Proposition 1.1] and [16, Sections 5 and 6]. (Caveat: the groupthat we denote G ′ is written G in [16].)(iii) Vinberg’s K surface has Picard number ; as such, it is a Kummer surface and has finite–dimensional motive. This implies (using [13]) that S [ ] has finite–dimensional motive. As bi-rational hyperk¨ahler varieties have isomorphic Chow motives [41], X has finite–dimensionalmotive. The Hilbert scheme S [ ] of any K surface S has an MCK decomposition [43, Theo-rem 13.4]. As the isomorphism of [41] is an isomorphism of algebras in the category of Chowmotives, X also has an MCK decomposition (lemma 2.13). (cid:3) Remark 2.29.
The singular locus of the very special EPW sextic X consists of planes. Amongthese 60 planes, there is a subset of 20 planes which form a complete family of pairwise incidentplanes in P ( C ) [16] . This is the maximal number of elements in a complete family of pairwiseincident planes, and this seems to be the only known example of a complete family of 20 pairwiseincident planes. Remark 2.30.
The variety X is not unique. In [17, Section 6] , it is shown there exist symplectic resolutions of E /( G ′ ) (some of them non–projective). One noteworthy consequenceof theorem 2.28 is that the varieties X are of K [ ] type (this was not a priori clear from [17] ). Remark 2.31.
For a generic
EPW sextic X , there exists a hyperk¨ahler fourfold X (called a“double EPW sextic”) equipped with an anti–symplectic involution σ such that X = X /( σ ) [35, Theorem 1.1 (2)] . For the very special EPW sextic X , I don’t know whether such X exists.(For this, one would need to show that the Lagrangian subspace A defining the very special EPWsextic is in the Zariski open LG ( ∧ V ) ⊂ LG ( ∧ V ) defined in [35, page 3] .)
3. S
OME INTERMEDIATE STEPS
A strong version of the generalized Hodge conjecture.
For later use, we record here aproposition, stating that the very special EPW sextic, as well as some related varieties, satisfythe hypothesis of lemma 2.21:
Proposition 3.1.
Let X be any hyperk¨ahler variety as in theorem 2.28 (i.e., X is a symplecticresolution of E /( G ′ ) ). Then N H H ( X ) = ̃ N H ( X ) . (Here N ∗ H denotes the Hodge coniveau filtration and ̃ N ∗ denotes the niveau filtration (definition2.15).)The same holds for X ′ ∶ = E /( G ′ ) and for the very special EPW sextic X : N H H ( X ′ ) = ̃ N H ( X ′ ) ,N H H ( X ) = ̃ N H ( X ) . Proof.
The point is that Vinberg’s K surface S has Picard number , and so the correspondingstatement is easily proven for S [ ] : Lemma 3.2.
Let S be a smooth projective surface with q = and p g ( S ) = . Assume S is ρ –maximal (i.e. dim H tr ( S ) = ). Then N H H ( S [ ] ) = ̃ N H ( S [ ] ) . Proof.
Let ̃ S × S → S × S denote the blow–up of the diagonal. As is well–known, there areisomorphisms of homological motives h ( S [ ] ) ≅ h ( ̃ S × S ) S ,h ( ̃ S × S ) ≅ h ( S × S ) ⊕ h ( S )( ) in M hom , where S denotes the symmetric group on elements acting by permutation. It follows there isa correspondence–induced injection H ( S [ ] ) ↪ H ( S × S ) ⊕ H ( S ) . It thus suffices to prove the statement for S × S . Let us write H ( S ) = N ⊕ T ∶ = N S ( S ) ⊕ H tr ( S ) . We have N H H ( S × S ) = H ( S × S ) ∩ F = H ( S ) ⊗ H ( S ) ⊕ H ( S ) ⊗ H ( S ) ⊕ N ⊗ N ⊕ N ⊗ T ⊕ T ⊗ N ⊕ ( T ⊗ T ) ∩ F . All but the last summand are obviously in ̃ N . As to the last summand, we have that ( T ⊗ T ) ∩ F = ( T ⊗ T ) ∩ F . Since the Hodge conjecture is true for S × S (indeed, S is a Kummer surface and the Hodgeconjecture is known for powers of abelian surfaces [1, 7.2.2], [2, 8.1(2)]), there is an inclusion ( T ⊗ T ) ∩ F ⊂ N H ( S × S ) = ̃ N H ( S × S ) , and so the lemma is proven. (cid:3) Since birational hyperk¨ahler varieties have isomorphic cohomology rings [25, Corollary 2.7],and the isomorphism (being given by a correspondence) respects Hodge structures, this provesthe result for X . Since X dominates X ′ and X , the result for X ′ and X follows. Proposition3.1 is now proven. (cid:3) MCK for quotients of abelian varieties.Proposition 3.3.
Let A be an abelian variety of dimension n , and let G ⊂ Aut Z ( A ) be a finitegroup of automorphisms of A that are group homomorphisms. The quotient X = A / G has a self–dual MCK decomposition. LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 15
Proof.
A first step is to show there exists a self–dual CK decomposition for X induced by a CKdecomposition on A : Claim 3.4.
Let A and X be as in proposition 3.3, and let p ∶ A → X denote the quotient mor-phism. Let { Π Ai } be a CK decomposition as in lemma 2.26(i). Then Π Xi ∶ = d Γ p ◦ Π Ai ◦ t Γ p ∈ A n ( X × X ) , i = , . . . , n defines a self–dual CK decomposition for X . To prove the claim, we remark that clearly the given Π Xi lift the K¨unneth components of X ,and their sum is the diagonal of X . We will make use of the following property: Lemma 3.5.
Let A be an abelian variety of dimension n , and let { Π Ai } be an MCK decompositionas in lemma 2.26(i). For any g ∈ Aut Z ( A ) , we have Π Ai ◦ Γ g = Γ g ◦ Π Ai in A n ( A × A ) . Proof.
Because g ∗ H i ( A ) ⊂ H i ( A ) , we have a homological equivalence Π Ai ◦ Γ g − Γ g ◦ Π Ai = in H n ( A × A ) . But the left–hand side is a symmetrically distinguished cycle, and so it is rationally trivial. (cid:3)
To see that Π Xi is idempotent, we note that Π Xi ◦ Π Xi = d Γ p ◦ Π Ai ◦ t Γ p ◦ Γ p ◦ Π Ai ◦ t Γ p = d Γ p ◦ Π Ai ◦ ( ∑ g ∈ G Γ g ) ◦ Π Ai ◦ t Γ p = d Γ p ◦ Π Ai ◦ Π Ai ◦ ( ∑ g ∈ G Γ g ) ◦ t Γ p = d Γ p ◦ Π Ai ◦ ( ∑ g ∈ G Γ g ) ◦ t Γ p = d Γ p ◦ Π Ai ◦ t Γ p ◦ Γ p ◦ t Γ p = d Γ p ◦ Π Ai ◦ t Γ p ◦ d ∆ X = Γ p ◦ Π Ai ◦ t Γ p = Π Xi in A n ( X × X ) . (Here, the third equality is an application of lemma 3.5, and the fourth equality is because Π Ai is idempotent.) The fact that the Π Xi are mutually orthogonal is proven similarly; one needs toreplace Π Xi ◦ Π Xi by Π Xi ◦ Π Xj in the above argument. This proves claim 3.4.Now, it only remains to see that the CK decomposition { Π Xi } of claim 3.4 is multiplicative. Claim 3.6.
The CK decomposition { Π Xi } given by claim 3.4 is an MCK decomposition. To prove claim 3.6, let us consider the composition Π Xk ◦ ∆ Xsm ◦ ( Π Xi × Π Xj ) ∈ A n ( X × X ) , where we suppose i + j / = k . There are equalities Π Xk ◦ ∆ Xsm ◦ ( Π Xi × Π Xj ) = d Γ p ◦ Π Ak ◦ t Γ p ◦ ∆ Xsm ◦ Γ p × p ◦ ( Π Ai × Π Aj ) ◦ t Γ p × p = d Γ p ◦ Π Ak ◦ ∆ GA ◦ ∆ Asm ◦ ( ∆ GA × ∆ GA ) ◦ ( Π Ai × Π Aj ) ◦ t Γ p × p = d Γ p ◦ ∆ GA ◦ Π Ak ◦ ∆ Asm ◦ ( Π Ai × Π Aj ) ◦ ( ∆ GA × ∆ GA ) ◦ t Γ p × p = in A n ( X × X × X ) . Here, the first equality is by definition of the Π Xi , the second equality is lemma 3.7 below, thethird equality follows from lemma 3.5, and the fourth equality is the fact that { Π Ai } is an MCKdecomposition for A (lemma 2.26). Lemma 3.7.
There is equality t Γ p ◦ ∆ Xsm ◦ Γ p × p = d ( ∑ g ∈ G Γ g ) ◦ ∆ Asm ◦ (( ∑ g ∈ G Γ g ) × ( ∑ g ∈ G Γ g )) = d ∆ GA ◦ ∆ Asm ◦ ( ∆ GA × ∆ GA ) in A n ( A × A × A ) . Proof.
The second equality is just the definition of ∆ GA . As to the first equality, we first note that ∆ Xsm = d ( p × p × p ) ∗ ( ∆ Asm ) = d Γ p ◦ ∆ Asm ◦ t Γ p × p in A n ( X × X × X ) . This implies that t Γ p ◦ ∆ Xsm ◦ Γ p × p = d t Γ p ◦ Γ p ◦ ∆ Asm ◦ t Γ p × p ◦ Γ p × p . But t Γ p ◦ Γ p = ∑ g ∈ G Γ g , and thus t Γ p ◦ ∆ Xsm ◦ Γ p × p = d ( ∑ g ∈ G Γ g ) ◦ ∆ Asm ◦ (( ∑ g ∈ G Γ g ) × ( ∑ g ∈ G Γ g )) in A n ( A × A × A ) , as claimed. (cid:3) This ends the proof of proposition 3.3. (cid:3)
In the set–up of proposition 3.3, one can actually say more about certain pieces A i ( j ) ( X ) : Proposition 3.8.
Let X = A / G be as in proposition 3.3. Assume n = dim X ≤ and H ( X, O X ) = . Assume also there exists X ′ = A /( G ′ ) where G = ( G ′ , i ) with i ∈ G ′ , LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 17 and the action of i on H ( X ′ , O X ′ ) is minus the identity. Then any CK decomposition { Π i } of X verifies ( Π ) ∗ A j ( X ) = for all j / = , ( Π ) ∗ A j ( X ) = for all j / = . Proof.
It suffices to prove this for one particular CK decomposition, in view of the followinglemma:
Lemma 3.9.
Let X = A / G be as in proposition 3.3. Let Π , Π ′ ∈ A n ( X × X ) be idempotents,and assume Π − Π ′ = in H n ( X × X ) . Then ( Π ) ∗ A i ( X ) = ⇔ ( Π ′ ) ∗ A i ( X ) = . Proof.
This follows from [48, Lemma 1.14]. Alternatively, here is a direct proof. Let p ∶ A → X denote the quotient morphism, and let d ∶ = ∣ G ∣ . One defines Π A ∶ = d t Γ p ◦ Π ◦ Γ p ∈ A n ( A × A ) , Π ′ A ∶ = d t Γ p ◦ Π ′ ◦ Γ p ∈ A n ( A × A ) . It is readily checked Π A , Π ′ A are idempotents, and they are homologically equivalent.Let us assume ( Π ) ∗ A i ( X ) = for a certain i . Then also ( Π A ) ∗ p ∗ A i ( X ) = ( d t Γ p ◦ Π ◦ Γ p ◦ t Γ p ) ∗ A i ( X ) = ( t Γ p ◦ Π ) ∗ A i ( X ) = . By finite–dimensionality of A , the difference Π A − Π ′ A ∈ A nhom ( A × A ) is nilpotent, i.e. thereexists N ∈ N such that ( Π A − Π ′ A ) ◦ N = in A n ( A × A ) . Upon developing, this implies Π ′ A = ( Π ′ A ) ◦ N = Q + ⋯ + Q N in A n ( A × A ) , where each Q j is a composition Q j = Q j ◦ Q j ◦ ⋯ ◦ Q Nj , with Q kj ∈ { Π A , Π ′ A } , and at least one Q kj is Π A . Since by assumption ( Π A ) ∗ p ∗ A i ( X ) = , itfollows that ( Q j ) ∗ = ( something ) ∗ ( Π A ) ∗ (( Π ′ A ) ◦ r ) ∗ = ∶ p ∗ A i ( X ) → p ∗ A i ( X ) for all j . But then also ( Π ′ A ) ∗ p ∗ A i ( X ) = ( Q + ⋯ + Q N ) ∗ p ∗ A i ( X ) = . (cid:3) Now, let us take a projector for A of the form Π A = Π A , + Π A , ∈ A n ( A × A ) , where Π A , , Π A , are as in lemma 2.26. Lemma 3.10.
Let A be an abelian variety of dimension n ≤ , and let G ⊂ Aut Z ( A ) be a finitesubgroup. Let Π A , be as in lemma 2.26. Then Π A , ◦ ∆ GA = ∆ GA ◦ Π A , ∈ A n ( A × A ) is idempotent. (Here, as before, we write ∆ GA ∶ = ∣ G ∣ ∑ g ∈ G Γ g ∈ A n ( A × A ) .)Proof. For any g ∈ G , we have the commutativity Π A , ◦ Γ g = Γ g ◦ Π A , in A n ( A × A ) , for all g ∈ G , established in lemma 2.26(ii). (Indeed, these cycles are symmetrically distinguished by lemma2.26(ii), and their difference is homologically trivial because an automorphism g ∈ G respectsthe niveau filtration.)This commutativity clearly implies the equality Π A , ◦ ∆ GA = ∆ GA ◦ Π A , ∈ A n ( A × A ) . To check that Π A , ◦ ∆ GA is idempotent, we note that Π A , ◦ ∆ GA ◦ Π A , ◦ ∆ GA = Π A , ◦ Π A , ◦ ∆ GA ◦ ∆ GA = Π A , ◦ ∆ GA in A n ( A × A ) . (cid:3) Let us write G = G ′ × { , i } . Since by assumption, i ∗ = − id on H , ( X ′ ) , we have equality ( Π A , ◦ ∆ G ′ A + Π A , ◦ ∆ G ′ A ◦ Γ i ) = in H n ( A × A ) . On the other hand, the left–hand side is equal to the idempotent Π A , ◦ ∆ GA . By finite–dimensionality,it follows that Π A , ◦ ∆ GA = in A n ( A × A ) . Using Poincar´e duality, we also have i ∗ = − id on H , ( X ′ ) , and so (defining Π A , as the trans-pose of Π A , ) there is also an equality Π A , ◦ ∆ GA = ( Π A , ◦ ∆ G ′ A + Π A , ◦ ∆ G ′ A ◦ Γ i ) = in H n ( A × A ) , and hence, by finite–dimensionality Π A , ◦ ∆ GA = in A n ( A × A ) . Since Π A , does not act on A j ( A ) for j / = (theorem 2.18), we find in particular that ( Π A ) ∗ = ∶ A j ( A ) G → A j ( A ) G for all j / = . LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 19
Likewise, since Π A , = t Π A , does not act on A j ( A ) for j / = (theorem 2.18), we also find that ( Π A ) ∗ = ∶ A j ( A ) G → A j ( A ) G for all j / = . We now consider the CK decomposition for X defined as in lemma 3.4: Π Xi ∶ = d Γ p ◦ Π Ai ◦ t Γ p ∈ A n ( X × X ) . This CK decomposition has the required behaviour: ( Π X ) ∗ A j ( X ) = ( d Γ p ◦ Π A ◦ t Γ p ) ∗ A j ( X ) = ( d Γ p ) ∗ ( Π A ) ∗ p ∗ A j ( X ) = ( d Γ p ) ∗ ( Π A ) ∗ A j ( A ) G = for all j / = , and likewise ( Π X ) ∗ A j ( X ) = for all j / = . This proves proposition 3.8. (cid:3)
For later use, we record here a corollary of the proof of proposition 3.8:
Corollary 3.11.
Let A be an abelian variety of dimension n ≤ , and let Π A , , Π A , be as inlemma 2.26(ii). Let p ∶ A → X = A / G be a quotient variety with G ⊂ Aut Z ( A ) . The prescription Π X ,i ∶ = Γ p ◦ Π A ,i ◦ t Γ p in A n ( X × X ) defines a decomposition in orthogonal idempotents Π X = Π X , + Π X , in A n ( X × X ) . The Π X ,i verify the properties of the refined CK decomposition of theorem 2.18.Proof. One needs to check the Π X ,i are idempotent and orthogonal. This easily follows from thefact that the Π A ,i commute with Γ g for g ∈ G (lemma 3.10). (cid:3) A surjectivity statement.Proposition 3.12.
Let X be a hyperk¨ahler fourfold as in theorem 2.28. Let A ∗ ( ∗ ) ( X ) be thebigrading defined by the MCK decomposition. Then the intersection product map A ( ) ( X ) ⊗ A ( ) ( X ) → A ( ) ( X ) is surjective.The same holds for X ′ ∶ = E /( G ′ ) as in theorem 2.28: X ′ has an MCK decomposition, andthe intersection product map A ( ) ( X ′ ) ⊗ A ( ) ( X ′ ) → A ( ) ( X ′ ) is surjective. Proof.
The result of Rieß [41] implies there is an isomorphism of bigraded rings A ∗ ( ∗ ) ( S [ ] ) ≅ − → A ∗ ( ∗ ) ( X ) . For the Hilbert scheme of any K surface S , the intersection product map A ( ) ( S [ ] ) ⊗ A ( ) ( S [ ] ) → A ( ) ( S [ ] ) is known to be surjective [43, Theorem 3]. This proves the first statement.For the second statement, the existence of an MCK decomposition for X ′ is a special caseof proposition 3.3. To prove the surjectivity statement for X ′ , we note that φ ∶ X → X ′ is asymplectic resolution and so there are isomorphisms φ ∗ ∶ H p, ( X ′ ) ≅ − → H p, ( X ) ( p = , ) . Using lemma 2.21 (which is possible thanks to proposition 3.1), this implies there are isomor-phisms φ ∗ ∶ H ptr ( X ′ ) ≅ − → H ptr ( X ) ( p = , ) . This means there is an isomorphism of homological motives t Γ φ ∶ h p, ( X ′ ) ≅ − → h p, ( X ) in M hom ( p = , ) . By finite–dimensionality, there are isomorphisms of Chow motives t Γ φ ∶ h p, ( X ′ ) ≅ − → h p, ( X ) in M rat ( p = , ) . Taking Chow groups, this implies there are isomorphisms(3) ( Π X p ◦ t Γ φ ◦ Π X ′ p ) ∗ ∶ ( Π X ′ p ) ∗ A i ( X ′ ) → ( Π X p ) ∗ A i ( X ) ( p = , ) . Let us now consider the diagram A ( ) ( X ) ⊗ A ( ) ( X ) → A ( ) ( X ) ↑ ↑ A ( X ) ⊗ A ( X ) → A ( X ) ↑ ↑ A ( ) ( X ′ ) ⊗ A ( ) ( X ′ ) → A ( ) ( X ′ ) Here, the vertical arrows in the upper square are given by projecting to direct summand; thevertical arrows in the lower square are given by φ ∗ . Since pullback and intersection productcommute, the lower square commutes. Since A ∗ ( ∗ ) ( X ) is a bigraded ring, the upper squarecommutes.The composition of vertical arrows is an isomorphism by (3). The statement for X ′ nowfollows from the statement for X . (cid:3) LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 21
4. M
AIN RESULTS
Splitting of A ∗ ( X ) .Theorem 4.1. Let X be the very special EPW sextic of theorem 2.28. The Chow ring of X is abigraded ring A ∗ ( X ) = A ∗ ( ∗ ) ( X ) , where A ( X ) = A ( ) ( X ) = Q ,A ( X ) = A ( ) ( X ) ,A ( X ) = A ( ) ( X ) ⊕ A ( ) ( X ) = Q ⊕ A hom ( X ) ,A ( X ) = A ( ) ( X ) ⊕ A ( ) ( X ) = Q ⊕ A hom ( X ) . Proof.
It follows from theorem 2.28 that X is a quotient variety X = E / G with G ⊂ Aut Z ( A ) .Moreover, there is another quotient variety X ′ = E /( G ′ ) where G = ( G ′ , i ) and i ∈ G ′ andsuch that i acts on H ( X ′ , O X ′ ) as − id. Applying proposition 3.3, it follows that X has an MCKdecomposition { Π Xi } . Applying proposition 3.8, it follows that ( Π X ) ∗ A j ( X ) = for all j / = , ( Π X ) ∗ A j ( X ) = for all j / = . The projectors Π Xi are for i odd. (Indeed, X has no odd cohomology so the Π Xi are homologi-cally trivial. Using finite–dimensionality, they are rationally trivial.)The projectors { Π Xi } define a multiplicative bigrading A ∗ ( X ) = A ∗ ( ∗ ) ( X ) , where A j ( i ) ( X ) ∶ = ( Π X j − i ) ∗ A j ( X ) . The fact that A j ( i ) ( X ) = for i < follows from thecorresponding property for abelian fourfolds [6]. Likewise, the fact that A j ( ) ( X ) ∩ A jhom ( X ) = for all j ≥ follows from the corresponding property for abelian fourfolds [6]. (cid:3) Corollary 4.2.
Let X be the very special EPW sextic. The intersection product maps A ( X ) ⊗ A ( X ) → A ( X ) ,A ( X ) ⊗ A ( X ) → A ( X ) have image of dimension . Remark 4.3.
It is instructive to note that for smooth Calabi–Yau hypersurfaces X ⊂ P n + ( C ) ,Voisin has proven that the intersection product map A j ( X ) ⊗ A n − j ( X ) → A n ( X ) has image of dimension , for any < j < n [54, Theorem 3.4] , [56, Theorem 5.25] (cf. also [19] for a generalization to generic complete intersections).In particular, the first statement of corollary 4.2 holds for any smooth sextic in P ( C ) . Thesecond statement of corollary 4.2, however, is not known (and maybe not true) for a generalsextic in P ( C ) . It might be that the second statement is specific to EPW sextics, and related tothe presence of a hyperk¨ahler fourfold X which is generically a double cover. Remark 4.4.
Let F ∗ be the filtration on A ∗ ( X ) defined as F i A j ( X ) = ⨁ ℓ ≥ i A j ( ℓ ) ( X ) . For this filtration to be of Bloch–Beilinson type, it remains to prove that F A ( X ) ?? = A hom ( X ) . This would imply the vanishing A hom ( X ) = (i.e. the truth of conjecture 1.5 for X ).Unfortunately, we cannot prove this. At least, it follows from the above description that theconjectural vanishing A hom ( X ) = would follow from the truth of Beauville’s conjecture A hom ( E ) ?? = A ( ) ( E ) ⊕ A ( ) ( E ) , where E is an elliptic curve. Splitting of A ∗ ( X r ) .Definition 4.5. Let X be a projective quotient variety. For any r ∈ N , and any ≤ i < j < k ≤ r ,let p j ∶ X r → X ,p ij ∶ X r → X × X ,p ijk ∶ X r → X × X × X denote projection on the j -th factor, resp. projection on the i -th and j -th factor, resp. projectionon the i -th and j -th and k -th factor.We define E ∗ ( X r ) ⊂ A ∗ ( X r ) as the Q –subalgebra generated by ( p j ) ∗ A ( X ) and ( p j ) ∗ A ( X ) and ( p ij ) ∗ ( ∆ X ) ∈ A ( X r ) and ( p ijk ) ∗ ( ∆ Xsm ) ∈ A ( X r ) . As explained in the introduction, the hypothesis that EPW sextics that are quotient varietiesare in the class C leads to the following concrete conjecture: Conjecture 4.6.
Let X ⊂ P ( C ) be an EPW sextic which is a projective quotient variety. Let r ∈ N . The restriction of the cycle class map E i ( X r ) → H i ( X r ) is injective for all i . For the very special EPW sextic, we can prove conjecture 4.6 for –cycles and –cycles: LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 23
Theorem 4.7.
Let X be the very special EPW sextic of definition 2.28. Let r ∈ N . The restrictionof the cycle class map E i ( X r ) → H i ( X r ) is injective for i ≥ r − .Proof. The product X r has an MCK decomposition (since X has one, and the property of havingan MCK decomposition is stable under taking products [43, Theorem 8.6]). Therefore, there isa bigrading on the Chow ring of X r . As we have seen (theorem 4.1), A ( X ) = A ( ) ( X ) and A ( X ) = A ( ) ( X ) . Also, it is readily checked that ∆ X ∈ A ( ) ( X × X ) . (Indeed, this follows from the fact that ∆ X = ∑ i = Π Xi = ∑ i = Π Xi ◦ ∆ X ◦ Π Xi = ∑ i = ( Π Xi × Π X − i ) ∗ ∆ X in A ( X × X ) , where we have used the fact that the CK decomposition is self–dual.) The fact that X has anMCK decomposition implies that ∆ Xsm ∈ A ( ) ( X × X × X ) [43, Proposition 8.4].Clearly, the pullbacks under the projections p i , p ij , p ijk respect the bigrading. (Indeed, suppose a ∈ A ℓ ( ) ( X ) , which means a = ( Π X ℓ ) ∗ ( a ) . Then the pullback ( p i ) ∗ ( a ) can be written as X × ⋯ × X × ( Π X ℓ ) ∗ ( a ) × X × ⋯ × X ∈ A ℓ ( X r ) , which is the same as ( Π X × ⋯ × Π X × Π X ℓ × Π X × ⋯ × Π X ) ∗ ( X × ⋯ × X × a × X × ⋯ × X ) . This implies that ( p i ) ∗ ( a ) ∈ ( Π X r ℓ ) ∗ A ℓ ( X r ) = A ℓ ( ) ( X r ) , where Π X r ∗ is the product CK decomposition. Another way to prove the fact that the projections p i , p ij , p ijk respect the bigrading is by invoking [44, Corollary 1.6].)It follows there is an inclusion E ∗ ( X r ) ⊂ A ∗ ( ) ( X r ) . The finite morphism p × r ∶ A r → X r induces a split injection ( p × r ) ∗ ∶ A i ( ) ( X r ) ∩ A ihom ( X r ) → A i ( ) ( A r ) ∩ A ihom ( A r ) for all i. But the right–hand side is known to be for i ≥ r − [6], and so E i ( X r ) ∩ A ihom ( X r ) ⊂ A i ( ) ( X r ) ∩ A ihom ( X r ) = for all i ≥ r − . (cid:3) Remark 4.8.
As is clear from the proof of theorem 4.7, there is a link with Beauville’s conjecturesfor abelian varieties: let E be an elliptic curve, and suppose one knows that A i ( ) ( E r ) ∩ A ihom ( E r ) = for all i and all r . Then conjecture 4.6 is true for the very special EPW sextic.
Relation with some hyperk¨ahler fourfolds.Theorem 4.9.
Let X be the very special EPW sextic of definition 2.28. Let X be one of the hy-perk¨ahler fourfolds of [17, Corollary 6.4] , and let f ∶ X → X be the generically ∶ morphismconstructed in [16] . Then X has an MCK decomposition, and there is an isomorphism f ∗ ∶ A hom ( X ) ≅ − → A ( ) ( X ) . Proof.
The MCK decomposition for X was established in theorem 2.28. The morphism f ∶ X → X of [16] is constructed as a composition f ∶ X φ − → X ′ ∶ = E /( G ′ ) g − → X , where φ is a symplectic resolution and g is the double cover associated to an anti–symplecticinvolution. This implies f induces an isomorphism f ∗ ∶ H , ( X ) ≅ − → H , ( X ′ ) ≅ − → H , ( X ) . In view of the strong form of the generalized Hodge conjecture (proposition 3.1), X and X ′ and X verify the hypotheses of lemma 2.21. Applying lemma 2.21, we find isomorphisms of Chowmotives t Γ f ∶ h , ( X ) ≅ − → h , ( X ′ ) ≅ − → h , ( X ) in M rat . Since ( Π X ,i ) ∗ A ( X ) = for i ≥ for dimension reasons, we have ( Π X ) ∗ A ( X ) = ( Π X , ) ∗ A ( X ) , and the same goes for X ′ and X . It follows that f ∗ ∶ A hom ( X ) = A ( h , ( X )) ≅ − → A ( h , ( X )) = ∶ A ( ) ( X ) . (cid:3) As a corollary, we obtain an alternative description of the splitting A ∗ ( ∗ ) ( X ) for the hy-perk¨ahler fourfolds X : LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 25
Corollary 4.10.
Let f ∶ X → X be as in theorem 4.9. The splitting A ∗ ( ∗ ) ( X ) (given by the MCKdecomposition of X ) verifies A ( X ) = A ( ) ( X ) ⊕ A ( ) ( X ) ⊕ A ( ) ( X ) = f ∗ A hom ( X ) ⊕ ker ( A ( X ) f ∗ − → A ( X )) ⊕ Q ; A ( X ) = A ( ) ( X ) ⊕ A ( ) ( X ) = A hom ( X ) ⊕ H , ( X ) ; A ( X ) = A ( ) ( X ) ⊕ A ( ) ( X ) = ker ( A hom ( X ) f ∗ − → A ( X )) ⊕ A ( ) ( X ) . Remark 4.11.
Just as we noted for the EPW sextic X (remark 4.4), for this filtration to be ofBloch–Beilinson type one would need to prove that A ( ) ( X ) ∩ A hom ( X ) ?? = , which I cannot prove. This situation is similar to that of the Fano varieties F of lines on avery general cubic fourfold: thanks to work of Shen–Vial [43] there is a multiplicative bigrading A ∗ ( ∗ ) ( F ) which has many good properties and interesting alternative descriptions. The mainopen problem is to prove that A ( ) ( F ) ∩ A hom ( F ) ?? = , which doesn’t seem to be known for any single F . Remark 4.12.
Conjecturally, the relations of corollary 4.10 should hold for any double EPWsextic X (with X being the quotient of X under the anti–symplectic involution). However,short of knowing X has finite–dimensional motive (as is the case here, thanks to the presence ofthe abelian variety E ), this seems difficult to prove. Note that at least, for a general double EPWsextic X , the relations of corollary 4.10 give a concrete description of a filtration on A ∗ ( X ) that should be the Bloch–Beilinson filtration.
5. F
URTHER RESULTS
Bloch conjecture.Conjecture 5.1 (Bloch [9]) . Let X be a smooth projective variety of dimension n . Let Γ ∈ A n ( X × X ) be a correspondence such that Γ ∗ = ∶ H p, ( X ) → H p, ( X ) for all p > . Then Γ ∗ = ∶ A nhom ( X ) → A nhom ( X ) . A weak version of conjecture 5.1 is true for the very special EPW sextic:
Proposition 5.2.
Let X be the very special EPW sextic. Let Γ ∈ A ( X × X ) be a correspondencesuch that Γ ∗ = ∶ H , ( X ) → H , ( X ) . Then there exists N ∈ N such that ( Γ ◦ N ) ∗ = ∶ A hom ( X ) → A hom ( X ) . Proof.
As is well–known, this follows from the fact that X has finite–dimensional motive; weinclude a proof for completeness’ sake.By assumption, we have Γ ∗ = ∶ H ( X, C )/ F → H ( X, C )/ F (where F ∗ is the Hodge filtration). Thanks to the “strong form of the generalized Hodge conjec-ture” (proposition 3.1), this implies that also Γ ∗ = ∶ H ( X, Q )/ ̃ N → H ( X, Q )/ ̃ N . Using Vial’s refined CK projectors (theorem 2.18), this means Γ ◦ Π X , = in H ( X × X ) , or, equivalently, Γ − ∑ ( k,ℓ )/ = ( , ) Γ ◦ Π Xk,ℓ = in H ( X × X ) . By finite–dimensionality, this implies there exists N ∈ N such that ( Γ − ∑ ( k,ℓ )/ = ( , ) Γ ◦ Π Xk,ℓ ) ◦ N = in A ( X × X ) . Upon developing, this gives an equality(4) Γ ◦ N = Q + ⋯ + Q N in A ( X × X ) , where each Q j is a composition of correspondences Q j = Q j ◦ Q j ◦ ⋯ ◦ Q rj ∈ A ( X × X ) , and for each j , at least one Q ij is equal to Π Xk,ℓ with ( k, ℓ ) / = ( , ) . Since (for dimension reasons) ( Π Xk,ℓ ) ∗ A hom ( X ) = for all ( k, ℓ ) / = ( , ) , it follows that ( Q j ) ∗ A hom ( X ) = for all j . In view of equality (4), we thus have ( Γ ◦ N ) ∗ = ∶ A hom ( X ) → A hom ( X ) . (cid:3) For special correspondences, one can do better:
LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 27
Proposition 5.3.
Let X be the very special EPW sextic. Let Γ ∈ A ( X × X ) be a correspondencesuch that Γ ∗ = ∶ H , ( X ) → H , ( X ) . Assume moreover that Γ can be written as Γ = r ∑ i = c i Γ σ i in A ( X × X ) , with c i ∈ Q and σ i ∈ Aut ( X ) induced by a G –equivariant automorphism σ Ei ∶ E → E , where X = E /( G ) and σ Ei is a group homomorphism. Then Γ ∗ = ∶ A hom ( X ) → A hom ( X ) . Proof.
Let us write A = E , and X ′ ∶ = A /( G ′ ) for the double cover of X with dim H , ( X ′ ) = . The projection g ∶ X ′ → X induces an isomorphism g ∗ ∶ H , ( X ) ≅ − → H , ( X ′ ) , with inverse given by d g ∗ . Let σ ′ i ∶ X ′ → X ′ ( i = , . . . , r ) be the automorphism induced by σ Ei .For each i = , . . . , r , there is a commutative diagram H , ( X ′ ) ( σ ′ i ) ∗ −−− → H , ( X ′ ) g ∗ ↑ ↓ g ∗ H , ( X ) ( σ i ) ∗ −−− → H , ( X ) Defining a correspondence Γ ′ = r ∑ i = c i Γ σ ′ i in A ( X ′ × X ′ ) , we thus get a commutative diagram H , ( X ′ ) ( Γ ′ ) ∗ −−− → H , ( X ′ ) g ∗ ↑ ↓ g ∗ H , ( X ) Γ ∗ − → H , ( X ) The assumption on Γ ∗ thus implies that ( Γ ′ ) ∗ = ∶ H , ( X ′ ) → H , ( X ′ ) . Since (by construction of X ′ ) the cup–product map H , ( X ′ ) ⊗ H , ( X ′ ) → H , ( X ′ ) is an isomorphism of –dimensional C –vector spaces, we must have that ( Γ ′ ) ∗ = ∶ H , ( X ′ ) → H , ( X ′ ) . It is readily seen this implies(5) t Γ ′ ◦ Π X ′ , = in H ( X ′ × X ′ ) . Let Γ A denote the correspondence Γ A ∶ = r ∑ i = c i Γ σ Ei in A ( A × A ) . Let p ′ ∶ A → X ′ = A /( G ′ ) denote the quotient morphism. There are relations t Γ σ ′ = ∣ G ′ ∣ Γ p ′ ◦ t Γ A ◦ t Γ p ′ in A ( X ′ × X ′ ) , Π X ′ , = ∣ G ′ ∣ Γ p ′ ◦ Π A , ◦ t Γ p ′ in A ( X ′ × X ′ ) (6)(the first relation is by construction of the automorphisms σ ′ i ; the second relation can be taken asdefinition, cf. corollary 3.11). Plugging in these relations in equality (5), one obtains Γ p ′ ◦ t Γ A ◦ t Γ p ′ ◦ Γ p ′ ◦ Π A , ◦ t Γ p ′ = in H ( X ′ × X ′ ) . Composing with t Γ p ′ on the left and Γ p ′ on the right, this implies in particular that t Γ p ′ ◦ Γ p ′ ◦ t Γ A ◦ t Γ p ′ ◦ Γ p ′ ◦ Π A , ◦ t Γ p ′ ◦ Γ p ′ = in H ( A × A ) . Using the standard relation t Γ p ′ ◦ Γ p ′ = ∣ G ′ ∣ ∑ g ∈ G ′ Γ g , this simplifies to ( ∑ g ∈ G ′ Γ g ) ◦ t Γ A ◦ ( ∑ g ∈ G ′ Γ g ) ◦ Π A , = in H ( A × A ) . The left–hand side is a symmetrically distinguished cycle which is homologically trivial, and soit is rationally trivial (theorem 2.24). That is, ( ∑ g ∈ G ′ Γ g ) ◦ t Γ A ◦ ( ∑ g ∈ G ′ Γ g ) ◦ Π A , = in A ( A × A ) , in other words t Γ p ′ ◦ Γ p ′ ◦ t Γ A ◦ t Γ p ′ ◦ Γ p ′ ◦ Π A , = in A ( A × A ) . Now we descend again to X ′ by composing some more on both sides: Γ p ′ ◦ t Γ p ′ ◦ Γ p ′ ◦ t Γ A ◦ t Γ p ′ ◦ Γ p ′ ◦ Π A , ◦ t Γ p ′ = in A ( X ′ × X ′ ) . Using the relations (6), this shimmers down to ( t Γ ′ ) ◦ Π X ′ , = in A ( X ′ × X ′ ) . This implies that ( Γ ′ ) ∗ = ∶ A hom ( X ′ ) → A hom ( X ′ ) . Since A ( ) ( X ′ ) equals the image of the intersection product A hom ( X ′ ) ⊗ A hom ( X ′ ) → A ( X ′ ) (proposition 3.12), we also have that ( Γ ′ ) ∗ = ∶ A ( ) ( X ′ ) → A ( ) ( X ′ ) . LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 29
The commutative diagram A ( ) ( X ′ ) ( Γ ′ ) ∗ −−− → A ( ) ( X ′ ) g ∗ ↑ ↑ g ∗ A hom ( X ) Γ ∗ − → A hom ( X ) , in which vertical arrows are isomorphisms (proof of theorem 4.9), now implies that Γ ∗ = ∶ A hom ( X ) → A hom ( X ) . (cid:3) Voisin conjecture.
Motivated by the Bloch–Beilinson conjectures, Voisin formulated thefollowing conjecture:
Conjecture 5.4 (Voisin [52]) . Let X be a smooth Calabi–Yau variety of dimension n . Let a, a ′ ∈ A nhom ( X ) be two –cycles of degree . Then a × a ′ = ( − ) n a ′ × a in A n ( X × X ) . It seems reasonable to expect this conjecture to go through for Calabi–Yau’s that are quo-tient varieties. In particular, conjecture 5.4 should be true for all EPW sextics that are quotientvarieties. We can prove this for the very special EPW sextic:
Proposition 5.5.
Let X be the very special EPW sextic. Let a, a ′ ∈ A hom ( X ) . Then a × a ′ = a ′ × a in A ( X × X ) . Proof.
As we have seen, there is a finite morphism p ∶ A → X , where A is an abelian fourfoldand p ∗ ∶ A hom ( X ) → A ( ) ( A ) = ( Π A ) ∗ A ( A ) is a split injection. (The inverse to p ∗ is given by a multiple of p ∗ .) Proposition 5.5 now followsfrom the following fact: any c, c ′ ∈ A ( ) ( A ) verify c × c ′ = c ′ × c in A ( A × A ) ; this is [56, Example 4.40]. (cid:3) Acknowledgements .
The ideas developed in this note grew into being during the Strasbourg2014—2015 groupe de travail based on the monograph [56] . Thanks to all the participants ofthis groupe de travail for a stimulating atmosphere. I am grateful to Bert van Geemen and tothe referee for helpful comments, and to Charles Vial for making me appreciate [37] , which is anessential ingredient in this note.Many thanks to Yasuyo, Kai and Len for hospitably receiving me in the Schiltigheim Math.Research Institute, where this note was written. R EFERENCES [1] S. Abdulali, Filtrations on the cohomology of abelian varieties, in: The arithmetic and geometry of alge-braic cycles, Banff 1998 (B. Brent Gordon et al., eds.), CRM Proceedings and Lecture Notes, AmericanMathematical Society Providence 2000,[2] S. Abdulali, Tate twists of Hodge structures arising from abelian varieties, in: Recent Advances in HodgeTheory: Period Domains, Algebraic Cycles, and Arithmetic (M. Kerr et al., eds.), London MathematicalSociety Lecture Note Series 427, Cambridge University Press, Cambridge 2016,[3] G. Ancona, D´ecomposition de motifs ab´eliens, Manuscripta Math. 146 (3) (2015), 307—328,[4] Y. Andr´e, Motifs de dimension finie (d’apr`es S.-I. Kimura, P. O’Sullivan,...), S´eminaire Bourbaki2003/2004, Ast´erisque 299 Exp. No. 929, viii, 115—145,[5] A. Beauville, Vari´et´es K¨ahleriennes dont la premi`ere classe de Chern est nulle, J. Differential Geom. 18(4) (1984), 755—782,[6] A. Beauville, Sur l’anneau de Chow d’une vari´et´e ab´elienne, Math. Ann. 273 (1986), 647—651,[7] A. Beauville, On the splitting of the Bloch–Beilinson filtration, in: Algebraic cycles and motives (J. Nagelet alii, eds.), London Math. Soc. Lecture Notes 344, Cambridge University Press 2007,[8] A. Beauville and C. Voisin, On the Chow ring of a K surface, J. Alg. Geom. 13 (2004), 417—426,[9] S. Bloch, Lectures on algebraic cycles, Duke Univ. Press Durham 1980,[10] S. Bloch and A. Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci. Ecole Norm. Sup. 4(1974), 181—202,[11] S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, American Journal of Math-ematics Vol. 105, No 5 (1983), 1235—1253,[12] M. Brion, Log homogeneous varieties, in: Actas del XVI Coloquio Latinoamericano de Algebra, RevistaMatem´atica Iberoamericana, Madrid 2007, arXiv: math/0609669,[13] 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,[14] P. Deligne, La conjecture de Weil pour les surfaces K , Invent. Math. 15 (1972), 206—226,[15] C. Deninger and J. Murre, Motivic decomposition of abelian schemes and the Fourier transform. J. Reineu. Angew. Math. 422 (1991), 201—219,[16] M. Donten–Bury, B. van Geemen, G. Kapustka, M. Kapustka and J. Wi´sniewski, A very special EPWsextic and two IHS fourfolds, arXiv:1509.06214v2, to appear in Geometry & Topology,[17] M. Donten–Bury and J. Wi´sniewski, On symplectic resolutions of a –dimensional quotient by a groupof order , arXiv:1409.4204,[18] D. Eisenbud, S. Popescu and C. Walter, Lagrangian subbundles and codimension subcanonical sub-schemes, Duke Math. J. 107(3) (2001), 427—467,[19] L. Fu, Decomposition of small diagonals and Chow rings of hypersurfaces and Calabi–Yau completeintersections, Advances in Mathematics (2013), 894—924,[20] L. Fu, Beauville-Voisin conjecture for generalized Kummer varieties, International Mathematics ResearchNotices 12 (2015), 3878—3898,[21] L. Fu, Z. Tian and C. Vial, Motivic hyperk¨ahler resolution conjecture for generalized Kummer varieties,arXiv:1608.04968,[22] W. Fulton, Intersection theory, Springer–Verlag Ergebnisse der Mathematik, Berlin Heidelberg New YorkTokyo 1984,[23] V. Guletski˘ı and C. Pedrini, The Chow motive of the Godeaux surface, in: Algebraic Geometry, a volumein memory of Paolo Francia (M. Beltrametti et alii, eds.), Walter de Gruyter, Berlin New York, 2002,[24] 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,[25] D. Huybrechts, The K¨ahler cone of a compact hyperk¨ahler manifold, Math. Ann. 326 (2003), 499—513, LGEBRAIC CYCLES ON A VERY SPECIAL EPW SEXTIC 31 [26] F. Ivorra, Finite dimensional motives and applications (following S.-I. Kimura, P. O’Sullivan and others),in: Autour des motifs, Asian-French summer school on algebraic geometry and number theory, VolumeIII, Panoramas et synth`eses, Soci´et´e math´ematique de France 2011,[27] J. Iyer, Murre’s conjectures and explicit Chow–K¨unneth projectors for varieties with a nef tangent bundle,Transactions of the Amer. Math. Soc. 361 (2008), 1667—1681,[28] J. Iyer, Absolute Chow–K¨unneth decomposition for rational homogeneous bundles and for log homoge-neous varieties, Michigan Math. Journal Vol.60, 1 (2011), 79—91,[29] 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,[30] U. Jannsen, On finite–dimensional motives and Murre’s conjecture, in: Algebraic cycles and motives (J.Nagel et alii, eds.), Cambridge University Press, Cambridge 2007,[31] B. Kahn, J. P. Murre and C. Pedrini, On the transcendental part of the motive of a surface, in: Algebraiccycles and motives (J. Nagel et alii, eds.), Cambridge University Press, Cambridge 2007,[32] S. Kimura, Chow groups are finite dimensional, in some sense, Math. Ann. 331 (2005), 173—201,[33] J. Murre, On a conjectural filtration on the Chow groups of an algebraic variety, parts I and II, Indag.Math. 4 (1993), 177—201,[34] J. Murre, J. Nagel and C. Peters, Lectures on the theory of pure motives, Amer. Math. Soc. UniversityLecture Series 61, Providence 2013,[35] K. O’Grady, Irreducible symplectic –folds and Eisenbud–Popescu–Walter sextics, Duke Math. J. 134(1)(2006), 99—137,[36] K. O’Grady, Double covers of EPW–sextics, Michigan Math. J. 62 (2013), 143—184,[37] P. O’Sullivan, Algebraic cycles on an abelian variety, J. f. Reine u. Angew. Math. 654 (2011), 1—81,[38] C. Pedrini, On the finite dimensionality of a K surface, Manuscripta Mathematica 138 (2012), 59—72,[39] C. Pedrini, Bloch’s conjecture and valences of correspondences for K surfaces, arXiv:1510.05832v1,[40] C. Pedrini and C. Weibel, Some surfaces of general type for which Bloch’s conjecture holds, to appearin: Period Domains, Algebraic Cycles, and Arithmetic, Cambridge Univ. Press, 2015,[41] U. Rieß, On the Chow ring of birational irreducible symplectic varieties, Manuscripta Math. 145 (2014),473—501,[42] T. Scholl, Classical motives, in: Motives (U. Jannsen et alii, eds.), Proceedings of Symposia in PureMathematics Vol. 55 (1994), Part 1,[43] M. Shen and C. Vial, The Fourier transform for certain hyperK¨ahler fourfolds, Memoirs of the AMS 240(2016), no.1139,[44] M. Shen and C. Vial, The motive of the Hilbert cube X [ ] , Forum Math. Sigma 4 (2016),[45] C. Vial, Algebraic cycles and fibrations, Documenta Math. 18 (2013), 1521—1553,[46] C. Vial, Projectors on the intermediate algebraic Jacobians, New York J. Math. 19 (2013), 793—822,[47] C. Vial, Remarks on motives of abelian type, to appear in Tohoku Math. J.,[48] C. Vial, Niveau and coniveau filtrations on cohomology groups and Chow groups, Proceedings of theLMS 106(2) (2013), 410—444,[49] C. Vial, Chow–K¨unneth decomposition for – and –folds fibred by varieties with trivial Chow group ofzero–cycles, J. Alg. Geom. 24 (2015), 51—80,[50] C. Vial, On the motive of some hyperk¨ahler varieties, to appear in J. f. Reine u. Angew. Math.,[51] E. Vinberg, The two most algebraic K surfaces, Math. Ann. 265 (1) (1983), 1—21,[52] C. Voisin, Remarks on zero–cycles of self–products of varieties, in: Moduli of vector bundles, Proceed-ings of the Taniguchi Congress (M. Maruyama, ed.), Marcel Dekker New York Basel Hong Kong 1994,[53] C. Voisin, On the Chow ring of certain algebraic hyper–K¨ahler manifolds, Pure and Applied Math. Quar-terly 4 no 3 (2008), 613—649,[54] C. Voisin, Chow rings and decomposition theorems for K surfaces and Calabi—Yau hypersurfaces,Geometry & Topology 16 (2012), 433—473, [55] C. Voisin, Bloch’s conjecture for Catanese and Barlow surfaces, J. Differential Geometry 97 (2014),149—175,[56] C. Voisin, Chow Rings, Decomposition of the Diagonal, and the Topology of Families, Princeton Univer-sity Press, Princeton and Oxford, 2014,[57] Z. Xu, Algebraic cycles on a generalized Kummer variety, arXiv:1506.04297v1,[58] Q. Yin, Finite–dimensionality and cycles on powers of K surfaces, Comment. Math. 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