aa r X i v : . [ m a t h . AG ] A p r ON THE MOTIVE OF SOME HYPERK ¨AHLER VARIETIES
CHARLES VIAL
Abstract.
We show that the motive of the Hilbert scheme of length- n subschemes on a K3 surfaceor on an abelian surface admits a decomposition similar to the decomposition of the motive of anabelian variety obtained by Shermenev, Beauville, and Deninger and Murre. Introduction
In this work, we fix a field k and all varieties are defined over this field k . Chow groups are alwaysmeant with rational coefficients and H ∗ ( − , Q ) is Betti cohomology with rational coefficients. Up toreplacing Betti cohomology with a suitable Weil cohomology theory (for example ℓ -adic cohomology),we may and we will assume that k is a subfield of the complex numbers C . We use freely the languageof (Chow) motives as is described in [12].Work of Shermenev [19], Beauville [2], and Deninger and Murre [7] unravelled the structure of themotives of abelian varieties : Theorem (Beauville, Deninger–Murre, Shermenev) . Let A be an abelian variety of dimension g .Then the Chow motive h ( A ) of A splits as (1) h ( A ) = g M i =0 h i ( A ) with the following properties :(i) H ∗ ( h i ( A ) , Q ) = H i ( A, Q ) ;(ii) the multiplication h ( A ) ⊗ h ( A ) → h ( A ) ( cf. (5) ) factors through h i + j ( A ) when restricted to h i ( A ) ⊗ h j ( A ) ;(iii) the morphism [ n ] ∗ : h i ( A ) → h i ( A ) induced by the multiplication by n morphism [ n ] : A → A ismultiplication by n i . Here, n is an integer. In particular, h i ( A ) is an eigen-submotive for theaction of [ n ] . For an arbitrary smooth projective variety X , it is expected that a decomposition of the motive h ( X ) as in (1) satisfying (i) should exist ; see [14]. Such a decomposition is called a Chow–K¨unnethdecomposition . However, in general, there is no analogue of the multiplication by n morphisms, andthe existence of a Chow–K¨unneth decomposition of the motive of X satisfying (ii) (in that case, theChow–K¨unneth decomposition is said to be multiplicative ) is very restrictive. We refer to [17, Section8] for some discussion on the existence of such a multiplicative decomposition.Nonetheless, inspired by the seminal work of Beauville and Voisin [4], [3] and [20], we were led toask in [17] whether the motives of hyperK¨ahler varieties admit a multiplicative decomposition similarto that of the motive of abelian varieties as in the theorem of Beauville, Deninger and Murre, andShermenev. Here, by hyperK¨ahler variety we mean a simply connected smooth projective variety X whose space of global 2-forms H ( X, Ω X ) is spanned by a nowhere degenerate 2-form. When k = C ,a hyperK¨ahler variety is nothing but a projective irreducible holomorphic symplectic manifold [1]. Date : July 7, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Hyperk¨ahler manifolds, Irreducible holomorphic symplectic varieties, K3 surfaces, abelianvarieties, Hilbert schemes of points, Motives, Algebraic cycles, Chow ring, Chow–K¨unneth decomposition, Bloch–Beilinson filtration.The author is supported by EPSRC Early Career Fellowship number EP/K005545/1.
Conjecture 1.
Let X be a hyperK¨ahler variety of dimension n . Then the Chow motive h ( X ) of X splits as h ( X ) = n M i =0 h i ( X ) with the property that(i) H ∗ ( h i ( X ) , Q ) = H i ( X, Q ) ;(ii) the multiplication h ( X ) ⊗ h ( X ) → h ( X ) ( cf. (5) ) factors through h i + j ( X ) when restricted to h i ( X ) ⊗ h j ( X ) . An important class of hyperK¨ahler varieties is given by the Hilbert schemes S [ n ] of length- n subschemes on a K3 surface S ; see [1]. The following theorem shows in particular that the motiveof S [ n ] for S a K3 surface admits a decomposition with properties (i) and (ii) and thus answersaffirmatively the question raised in Conjecture 1 in that case. Theorem 1.
Let S be either a K3 surface or an abelian surface, and let n be a positive integer. Thenthe Chow motive h ( S [ n ] ) of S [ n ] splits as h ( S [ n ] ) = n M i =0 h i ( S [ n ] ) with the property that(i) H ∗ ( h i ( S [ n ] ) , Q ) = H i ( S [ n ] , Q ) ;(ii) the multiplication h ( S [ n ] ) ⊗ h ( S [ n ] ) → h ( S [ n ] ) factors through h i + j ( S [ n ] ) when restricted to h i ( S [ n ] ) ⊗ h j ( S [ n ] ) . Theorem 6 of [17] can then be improved by including the Hilbert schemes of length- n subschemeson K3 surfaces. Theorem 1 is due for S a K3 surface and n = 1 to Beauville and Voisin [4] (see [17,Proposition 8.14] for the link between the original statement of [4] (recalled in Theorem 3.4) and thestatement given here), and was established in [17] for n = 2. Its proof in full generality is given inSection 3. Note that, as explained in Section 1, the existence of a Chow–K¨unneth decomposition forthe Hilbert scheme S [ n ] of any smooth projective surface S goes back to de Cataldo and Migliorini[5] (the existence of such a decomposition for S is due to Murre [13]). Our main contribution is theclaim that by choosing the Beauville–Voisin decomposition of K3 surfaces [4], the induced Chow–K¨unneth decomposition of Hilbert schemes of K3 surfaces established by de Cataldo and Migliorini[5] is multiplicative, i.e. it satisfies (ii) .Let us then define for all i ≥ s ∈ Z CH i ( S [ n ] ) s := CH i ( h i − s ( S [ n ] )) . We have the following corollary to Theorem 1 :
Theorem 2.
The Chow ring CH ∗ ( S [ n ] ) admits a multiplicative bigrading CH ∗ ( S [ n ] ) = M i,s CH i ( S [ n ] ) s that is induced by a Chow–K¨unneth decomposition of the diagonal (as defined in § c i ( S [ n ] ) belong to the graded-zero part CH i ( S [ n ] ) of CH i ( S [ n ] ) . Theorem 2 answers partially a question raised by Beauville in [3] : the filtration F • defined byF l CH i ( X ) := L s ≥ l CH i ( X ) s is a filtration on the Chow ring CH ∗ ( S [ n ] ) that is split. Moreover, thisfiltration is expected to be the one predicted by Bloch and Beilinson (because it is induced by a Chow–K¨unneth decomposition – conjecturally all such filtrations coincide). For this filtration to be of Bloch–Beilinson type, one would need to establish Murre’s conjectures, namely that CH i ( S [ n ] ) s = 0 for s < L s> CH i ( S [ n ] ) s is exactly the kernel of the cycle class map CH i ( S [ n ] ) → H i ( S [ n ] , Q ).Note that for i = 0 , , n − n , it is indeed the case that CH i ( S [ n ] ) s = 0 for s < L s> CH i ( S [ n ] ) s = Ker { CH i ( S [ n ] ) → H i ( S [ n ] , Q ) } . Therefore, we have N THE MOTIVE OF SOME HYPERK¨AHLER VARIETIES 3
Corollary 1.
Let i , . . . , i m be positive integers such that i + · · · + i m = 2 n − or n , and let γ l be cycles in CH i l ( S [ n ] ) for l = 1 , . . . , m that sit in CH i l ( S [ n ] ) for the grading induced by thedecomposition of Theorem 1. Then, [ γ ] · [ γ ] · · · [ γ m ] = 0 in H ∗ ( S [ n ] , Q ) if and only if γ · γ · · · γ m = 0 in CH ∗ ( S [ n ] ) . Let us mention that Theorem 1 and Theorem 2 (and a fortiori Corollary 1) are also valid forhyperK¨ahler varieties that are birational to S [ n ] , for some K3 surface S . Indeed, Riess [15] showedthat birational hyperK¨ahler varieties have isomorphic Chow rings and isomorphic Chow motives (asalgebras in the category of Chow motives) ; see also [17, Section 6]. As for more evidence as whyConjecture 1 should be true, Mingmin Shen and I showed [17] that the variety of lines on a verygeneral cubic fourfold satisfies the conclusions of Theorem 2.Finally, we use the notion of multiplicative Chow–K¨unneth decomposition to obtain new decom-position results in the spirit of [21] ; see Theorem 4.3. Notations.
A morphism denoted pr r will always denote the projection on the r th factor and amorphism denoted pr s,t will always denote the projection on the product of the s th and t th factors.The context will usually make it clear which varieties are involved. Chow groups CH i are with rationalcoefficients. If X is a variety, the cycle class map sends a cycle σ ∈ CH i ( X ) to its cohomology class[ σ ] ∈ H i ( X, Q ). If Y is another variety and if γ is a correspondence in CH i ( X × Y ), its transpose t γ ∈ CH i ( Y × X ) is the image of γ under the action of the permutation map X × Y → Y × X . If γ , · · · , γ n are correspondences in CH ∗ ( X × Y ), then the correspondence γ ⊗· · ·⊗ γ n ∈ CH ∗ ( X n × Y n )is defined as γ ⊗ · · · ⊗ γ n := Q ni =1 ( pr i,n + i ) ∗ γ i .1. Chow–K¨unneth decompositions
A Chow motive M is said to have a Chow–K¨unneth decomposition if it splits as M = L i ∈ Z M i withH ∗ ( M i , Q ) = H i ( M, Q ). In other words, M admits a K¨unneth decomposition that lifts to rationalequivalence. Concretely, if M = ( X, p, n ) with X a smooth projective variety of pure dimension d and p ∈ CH d ( X × X ) an idempotent and n an integer, then M has a Chow–K¨unneth decomposition ifthere exist finitely many correspondences p i ∈ CH d ( X × X ), i ∈ Z , such that p = P i p i , p i ◦ p i = p i , p ◦ p i = p i ◦ p = p i , p i ◦ p j = 0 for all i ∈ Z and all j = i and such that p i ∗ H ∗ ( X, Q ) = p ∗ H i +2 n ( X, Q ).A smooth projective variety X of dimension d has a Chow–K¨unneth decomposition if its Chowmotive h ( X ) has a Chow–K¨unneth decomposition, that is, there exist correspondences π i ∈ CH d ( X × X ) such that ∆ X = P di =0 π i , with π i ◦ π i = π i , π i ◦ π j = 0 for i = j and π i ∗ H ∗ ( X, Q ) = H i ( X, Q ). AChow–K¨unneth decomposition { π i , ≤ i ≤ d } of X is said to be self-dual if π d − i = t π i for all i .If is the class of a rational point on X (or more generally a zero-cycle of degree 1 on X ), then π := pr ∗ = × X and π d = pr ∗ = X × define mutually orthogonal idempotents such that π ∗ H ∗ ( X, Q ) = H ( X, Q ) and π d ∗ H ∗ ( X, Q ) = H d ( X, Q ). Note that pairs of idempotents with theproperty above are certainly not unique : a different choice (modulo rational equivalence) of zero-cycle of degree 1 gives different idempotents in the ring of correspondences CH d ( X × X ). From theabove, one sees that every curve C admits a Chow–K¨unneth decomposition : one defines π and π as above and then π is simply given by ∆ C − π − π . It is a theorem of Murre [13] that everysmooth projective surface S admits a Chow–K¨unneth decomposition ∆ S = π S + π S + π S + π S + π S .The notion of Chow–K¨unneth decomposition is significant because when it exists it induces a fil-tration F l CH i ( X ) := L s ≥ l CH i ( X ) s on the Chow group CH ∗ ( X ) which should not depend on thechoice of the Chow–K¨unneth decomposition ∆ X = P di =0 π i and which should be of Bloch–Beilinsontype ; cf. [11, 14].Let now S [ n ] denote the Hilbert scheme of length- n subschemes on a smooth projective surface S .By Fogarty [9], the scheme S [ n ] is in fact a smooth projective variety, and it comes equipped with amorphism S [ n ] → S ( n ) to the n th symmetric product of S , called the Hilbert–Chow morphism . DeCataldo and Migliorini [5] have given an explicit description of the motive of S [ n ] . Let us introducesome notations related to this description. Let µ = { A , . . . , A l } be a partition of the set { , . . . , n } , CHARLES VIAL where all the A i ’s are non-empty. The integer l , also denoted l ( µ ), is the length of the partition µ .Let S µ ≃ S l ⊆ S n be the set { ( s , . . . , s n ) : s i = s j if i, j ∈ A k for some k } and let Γ µ := ( S µ × S ( n ) S [ n ] ) red ⊂ S µ × S [ n ] , where the subscript red means the underlying reduced scheme. It is known that Γ µ is irreducible ofdimension n + l ( µ ). The subgroup S µ of S n that acts on { , . . . , n } by permuting the A i ’s with samecardinality acts on the first factor of the product S µ × S [ n ] , and the correspondence Γ µ is invariantunder this action. We can therefore defineˆΓ µ := Γ µ / S µ ∈ CH ∗ ( S ( µ ) × S [ n ] ) = CH ∗ ( S µ × S [ n ] ) S µ , where S ( µ ) := S µ / S µ . Since for a variety X endowed with the action of a finite group G we haveCH ∗ ( X/G ) = CH ∗ ( X ) G (with rational coefficients), the calculus of correspondences and the theoryof motives in the setting of smooth projective varieties endowed with the action of a finite group issimilar in every way to the usual case of smooth projective varieties. We will therefore freely consideractions of correspondences and motives of quotient varieties by the action of a finite group.The symmetric groups S n acts naturally on the set of partitions of { , . . . , n } . By choosing oneelement in each orbit for the above action, we may define a subset B ( n ) of the set of partitions of { , . . . , n } . This set is isomorphic to the set of partitions of the integer n . Theorem 1.1 (de Cataldo and Migliorini [5]) . Let S be a smooth projective surface defined over anarbitrary field. The morphism (2) M µ ∈ B ( n ) t ˆΓ µ : h ( S [ n ] ) ≃ −→ M µ ∈ B ( n ) h ( S ( µ ) )( l ( µ ) − n ) is an isomorphism of Chow motives. Moreover, its inverse is given by the correspondence P µ ∈ B ( n ) 1 m µ ˆΓ µ for some non-zero rational numbers m µ . Let now ∆ S = π S + π S + π S + π S + π S be a Chow–K¨unneth decomposition of S . For all non-negativeintegers m , the correspondences(3) π iS m := X i + ... + i m = i π i S ⊗ · · · ⊗ π i m S in CH m ( S m × S m )define a Chow–K¨unneth decomposition of S m that is clearly S m -equivariant. Therefore, these corre-spondences can be seen as correspondences of CH m ( S ( m ) × S ( m ) ) and they do define a Chow–K¨unnethdecomposition of the m th symmetric product S ( m ) . Let us denote this decomposition∆ S ( m ) = π S ( m ) + · · · + π mS ( m ) in CH m ( S ( m ) × S ( m ) ) . Since S l is endowed with a S l -equivariant Chow–K¨unneth decomposition as above and since S µ is asubgroup of S l , S µ ≃ S l is endowed with a S µ -equivariant Chow–K¨unneth decomposition. Therefore S ( µ ) is endowed with a natural Chow–K¨unneth decomposition∆ S ( µ ) = π S ( µ ) + · · · + π lS ( µ ) in CH l ( S ( µ ) × S ( µ ) )coming from that of S . In particular, the isomorphism of de Cataldo and Migliorini gives a naturalChow–K¨unneth decomposition for the Hilbert scheme S [ n ] coming from that of S . Precisely, thisChow–K¨unneth decomposition is given by(4) π iS [ n ] = X µ ∈ B ( n ) m µ ˆΓ µ ◦ π i − n +2 l ( µ ) S ( µ ) ◦ t ˆΓ µ . Note that if the Chow–K¨unneth decomposition { π iS } of S is self-dual, then the Chow–K¨unnethdecomposition { π iS [ n ] } of S [ n ] is also self-dual.We will show that when S is either a K3 surface or an abelian surface the Chow–K¨unneth de-composition above induces a decomposition of the motive h ( S [ n ] ) that satisfies the conclusions ofTheorem 1 for an appropriate choice of Chow–K¨unneth decomposition for S . N THE MOTIVE OF SOME HYPERK¨AHLER VARIETIES 5 Multiplicative Chow–K¨unneth decompositions
Let X be a smooth projective variety of dimension d and let ∆ ∈ CH d ( X × X × X ) be the smalldiagonal, that is, the class of the subvariety { ( x, x, x ) : x ∈ X } ⊂ X × X × X. If we view ∆ as a correspondence from X × X to X , then ∆ induces the multiplication morphism (5) h ( X ) ⊗ h ( X ) → h ( X ) . Note that if α and β are cycles in CH ∗ ( X ), then (∆ ) ∗ ( α × β ) = α · β in CH ∗ ( X ).If X admits a Chow–K¨unneth decomposition(6) h ( X ) = d M i =0 h i ( X ) , then this decomposition is said to be multiplicative if the multiplication morphism h i ( X ) ⊗ h j ( X ) → h ( X ) factors through h i + j ( X ) for all i and j . For a variety to be endowed with a multiplicativeChow–K¨unneth decomposition is very restrictive ; we refer to [17], where this notion was introduced,for some discussions. Examples of varieties admitting a multiplicative Chow–K¨unneth decompositionare provided by [17, Theorem 6] and include hyperelliptic curves, K3 surfaces, abelian varieties, andtheir Hilbert squares.If one writes ∆ X = π X + . . . + π dX for the Chow–K¨unneth decomposition (6) of X , then by definitionthis decomposition is multiplicative if π kX ◦ ∆ ◦ ( π iX ⊗ π jX ) = 0 in CH d ( X ) for all k = i + j, or equivalently if ( t π iX ⊗ t π jX ⊗ π kX ) ∗ ∆ = 0 in CH d ( X ) for all k = i + j. If the Chow–K¨unneth decomposition { π iX } is self-dual, then it is multiplicative if( π iX ⊗ π jX ⊗ π kX ) ∗ ∆ = 0 in CH d ( X ) for all i + j + k = 4 d. Note that the above three relations always hold modulo homological equivalence.Given a multiplicative Chow–K¨unneth decomposition π iS for a surface S , one could expect theChow–K¨unneth decomposition (4) of S [ n ] to be multiplicative. This was answered affirmatively when n = 2 for any smooth projective variety X with a self-dual Chow–K¨unneth decomposition (undersome additional assumptions on the Chern classes of X ) in [17], and a similar result when n = 3 willappear in [18]. (For n > X [ n ] is no longer smooth if X is smooth of dimension > S is a K3 surface or an abelian surface and will prove Theorem 1. By theisomorphism (2) of de Cataldo and Migliorini, it is enough to check that(ˆΓ µ ⊗ ˆΓ µ ⊗ ˆΓ µ ) ∗ ( π iS [ n ] ⊗ π jS [ n ] ⊗ π kS [ n ] ) ∗ ∆ = 0for all i + j + k = 8 n and for all partitions µ , µ and µ of { , . . . , n } , or equivalently for all i, j, k such that ( π iS [ n ] ⊗ π jS [ n ] ⊗ π kS [ n ] ) ∗ [∆ ] = 0 in H ∗ ( S [ n ] × S [ n ] × S [ n ] , Q ) and all partitions µ , µ and µ .By (4), it is even enough to show that(7) (cid:16) ˆΓ µ ⊗ ˆΓ µ ⊗ ˆΓ µ (cid:17) ∗ (cid:16)(cid:16) ˆΓ ν ◦ π iS ( ν ◦ t ˆΓ ν (cid:17) ⊗ (cid:16) ˆΓ ν ◦ π jS ( ν ◦ t ˆΓ ν (cid:17) ⊗ (cid:16) ˆΓ ν ◦ π kS ( ν ◦ t ˆΓ ν (cid:17)(cid:17) ∗ ∆ is zero in CH ∗ ( S ( µ ) × S ( µ ) × S ( µ ) ) for all partitions µ , µ , µ , and all partitions ν , ν , ν and all i, j, k such that (cid:16)(cid:16) ˆΓ ν ◦ π iS ( ν ◦ t ˆΓ ν (cid:17) ⊗ (cid:16) ˆΓ ν ◦ π jS ( ν ◦ t ˆΓ ν (cid:17) ⊗ (cid:16) ˆΓ ν ◦ π kS ( ν ◦ t ˆΓ ν (cid:17)(cid:17) ∗ [∆ ] = 0 in H ∗ (( S [ n ] ) , Q ) . Note that the expression (7) is equal to h(cid:16) t ˆΓ µ ⊗ t ˆΓ µ ⊗ t ˆΓ µ (cid:17) ◦ (cid:16) ˆΓ ν ⊗ ˆΓ ν ⊗ ˆΓ ν (cid:17) ◦ (cid:16) π iS ( ν ⊗ π jS ( ν ⊗ π kS ( ν (cid:17) ◦ (cid:16) t ˆΓ ν ⊗ t ˆΓ ν ⊗ t ˆΓ ν (cid:17)i ∗ ∆ . But it is clear from Theorem 1.1 that (cid:16) t ˆΓ µ ⊗ t ˆΓ µ ⊗ t ˆΓ µ (cid:17) ◦ (cid:16) ˆΓ ν ⊗ ˆΓ ν ⊗ ˆΓ ν (cid:17) = (cid:26) ν , ν , ν ) = ( µ , µ , µ ) m µ m µ m µ ∆ S ( µ × S ( µ × S ( µ if ( ν , ν , ν ) = ( µ , µ , µ ) . CHARLES VIAL
Thus we have proved the following criterion for the Chow–K¨unneth decomposition (4) to be multi-plicative :
Proposition 2.1.
The Chow–K¨unneth decomposition (4) is multiplicative (equivalently, the motiveof S [ n ] splits as in Theorem 1) if for all partitions µ , µ and µ of the set { , . . . , n } (cid:16) π iS µ ⊗ π jS µ ⊗ π kS µ (cid:17) ∗ (Γ µ ⊗ Γ µ ⊗ Γ µ ) ∗ ∆ = 0 in CH ∗ ( S µ × S µ × S µ ) as soon as (cid:16) π iS µ ⊗ π jS µ ⊗ π kS µ (cid:17) ∗ (Γ µ ⊗ Γ µ ⊗ Γ µ ) ∗ [∆ ] = 0 in H ∗ ( S µ × S µ × S µ , Q ) . (cid:3) Proof of Theorem 1 and Theorem 2
The proof is inspired by the proof of Claire Voisin’s [23, Theorem 5.1]. In fact, because of [17,Proposition 8.12], Theorem 1 for K3 surfaces implies [23, Theorem 5.1]. The first step towards theproof of Theorem 1 is to understand the cycle (Γ µ ⊗ Γ µ ⊗ Γ µ ) ∗ ∆ . The following proposition, dueto Voisin [23] (see also [20]), builds on the work of Ellingsrud, G¨ottsche and Lehn [8]. Here, S is asmooth projective surface and ∆ k is the class of the small diagonal inside S k in CH ( S k ). Proposition 3.1 (Voisin, Proposition 5.6 in [23]) . For any set of partitions µ := { µ , . . . , µ k } of { , . . . , n } , there exists a universal (i.e., independent of S ) polynomial P µ with the following property : (Γ µ ⊗ . . . ⊗ Γ µ k ) ∗ ∆ k = P µ ( pr ∗ r c ( S ) , pr ∗ r ′ K S , pr ∗ s,t ∆ S ) in CH ∗ ( S µ ) , where the pr r ’s are the projections from S µ := Q i S µ i ≃ S N to its factors, and the pr s,t ’s are theprojections from S µ to the products of two of its factors. (cid:3) In fact, Proposition 3.1 is a particular instance of [23, Theorem 5.12]. Another consequence of [23,Theorem 5.12], which will be used to prove Theorem 2, is
Proposition 3.2 (Voisin) . For any partition µ of { , . . . , n } and any polynomial P in the Chernclasses of S [ n ] , the cycle (Γ µ ) ∗ P of S µ is a universal (i.e., independent of S ) polynomial in thevariables pr ∗ r c ( S ) , pr ∗ r ′ K S , pr ∗ s,t ∆ S , where the pr r ’s are the projections from S µ ≃ S N to its factors,and the pr s,t ’s are the projections from S µ to the products of two of its factors. (cid:3) We first prove Theorems 1 & 2 for S a K3 surface and then for S an abelian surface. Note thatclearly a multiplicative Chow–K¨unneth decomposition { π iS [ n ] , ≤ i ≤ n } induces a multiplicativebigrading on the Chow ring CH ∗ ( S [ n ] ) :CH ∗ ( S [ n ] ) = M i,s CH i ( S [ n ] ) s , where CH i ( S [ n ] ) s = ( π i − sS [ n ] ) ∗ CH i ( S [ n ] ) . Thus once Theorem 1 is established it only remains to show that the Chern classes of S [ n ] sit inCH ∗ ( S [ n ] ) in order to conclude.3.1. The Hilbert scheme of points on a K3 surface.
Let S be a smooth projective surface andlet be a zero-cycle of degree 1 on S . Let m be a positive integer and consider the m -fold product S m of S . Let us define the idempotent correspondences(8) π S := pr ∗ = × S, π S = pr ∗ = S × , and π S := ∆ S − π S − π S . (Note that π S is not quite a Chow–K¨unneth projector, it projects onto H ( S, Q ) ⊕ H ( S, Q ) ⊕ H ( S, Q ).)In this case, the idempotents π iS m := P i + ... + i n = i π i S ⊗ · · · ⊗ π i n S given in (3) are clearly sums ofmonomials of degree 2 m in pr ∗ r and pr ∗ s,t ∆ S . By Proposition 3.1, it follows that for any smoothprojective surface S and any zero-cycle of degree 1 on S (cid:16) π iS µ ⊗ π jS µ ⊗ π kS µ (cid:17) ∗ (Γ µ ⊗ Γ µ ⊗ Γ µ ) ∗ ∆ is a polynomial Q µ ,i,j,k in the variables pr ∗ r c ( S ) , pr ∗ r ′ K S , pr ∗ r ′′ and pr ∗ s,t ∆ S .We now have the following key result which is due to Claire Voisin [23, Corollary 5.9] and whichrelies in an essential way on a theorem due to Qizheng Yin [24] that describes the cohomologicalrelations among the cycles pr ∗ r,s π S . N THE MOTIVE OF SOME HYPERK¨AHLER VARIETIES 7
Proposition 3.3 (Voisin [23]) . For all smooth projective surfaces S and any degree- zero-cycle on S , let P be a polynomial (independent of S ) in the variables pr ∗ r [ c ( S )] , pr ∗ r ′ [ K S ] , pr ∗ r ′′ [ ] and pr ∗ s,t [∆ S ] with value an algebraic cycle of S n . If P vanishes for all smooth projective surfaces with b ( S ) = 0 ,then the polynomial P belongs to the ideal generated by the relations :(a) [ c ( S )] = χ top ( S )[ ] ;(b) [ K S ] = deg( K S )[ ] ;(c) [∆ S ] · pr ∗ [ K S ] = pr ∗ [ K S ] · pr ∗ [ ] + pr ∗ [ ] · pr ∗ [ K S ] ;(d) [∆ ] = pr ∗ , [∆ S ] · pr ∗ [ ] + pr ∗ , [∆ S ] · pr ∗ [ ] + pr ∗ , [∆ S ] · pr ∗ [ ] − pr ∗ [ ] · pr ∗ [ ] − pr ∗ [ ] · pr ∗ [ ] − pr ∗ [ ] · pr ∗ [ ] ;(e) [∆ S ] = χ top ( S ) pr ∗ [ ] · pr ∗ [ ] ;(f ) [∆ S ] · pr ∗ [ ] = pr ∗ [ ] · pr ∗ [ ] . (cid:3) We may then specialize to the case where S is a K3 surface. Consider then a K3 surface S andlet be the class of a point lying on a rational curve of S . Note that by definition of a K3 surface K S = 0. The following theorem of Beauville and Voisin shows that the relations (a)–(f) listed aboveactually hold modulo rational equivalence. Theorem 3.4 (Beauville–Voisin [4]) . Let S be a K3 surface and let be a rational point lying on arational curve on S . The following relations hold :(i) c ( S ) = χ top ( S ) (= 24 ) in CH ( S ) ;(ii) ∆ = pr ∗ , ∆ S · pr ∗ + pr ∗ , ∆ S · pr ∗ + pr ∗ , ∆ S · pr ∗ − pr ∗ · pr ∗ − pr ∗ · pr ∗ − pr ∗ · pr ∗ in CH ( S × S × S ) . The proof of Theorem 1 in the case when S is a K3 surface is then immediate : by the discussionabove if the cycle δ := (cid:16) π iS µ ⊗ π jS µ ⊗ π kS µ (cid:17) ∗ (Γ µ ⊗ Γ µ ⊗ Γ µ ) ∗ ∆ is zero in H ∗ ( S µ × S µ × S µ , Q ),then by Proposition 3.3 it belongs to the ideal generated by the relations (a)–(f). By Theorem 3.4,the relations (a)–(f) actually hold modulo rational equivalence. Therefore, the cycle δ is zero inCH ∗ ( S µ × S µ × S µ ). We may then conclude by invoking Proposition 2.1.It remains to prove that the Chern classes c i ( S [ n ] ) sit in CH i ( S [ n ] ) . It suffices to show that( π jS [ n ] ) ∗ c i ( S [ n ] ) = 0 in CH i ( S [ n ] ) as soon as ( π jS [ n ] ) ∗ [ c i ( S [ n ] )] = 0 in H i ( S [ n ] , Q ) (equivalently assoon as j = 2 i ). By de Cataldo and Migliorini’s theorem, it is enough to show for all partitions µ of { , . . . , n } that (Γ µ ) ∗ ( π jS [ n ] ) ∗ c i ( S [ n ] ) = 0 in CH ∗ ( S µ ) as soon as (Γ µ ) ∗ ( π jS [ n ] ) ∗ [ c i ( S [ n ] )] = 0 inH ∗ ( S µ , Q ). Proceeding as in section 2, it is even enough to show that, for all partitions µ of { , . . . , n } ,( π jS µ ) ∗ (Γ µ ) ∗ c i ( S [ n ] ) = 0 in CH ∗ ( S µ ) as soon as ( π jS µ ) ∗ (Γ µ ) ∗ [ c i ( S [ n ] )] = 0 in H ∗ ( S µ , Q ). By Proposi-tion 3.2, (Γ µ ) ∗ c i ( S [ n ] ) is a universal polynomial in the variables pr ∗ r c ( S ) , pr ∗ r ′ K S , pr ∗ s,t ∆ S . It followsthat ( π jS µ ) ∗ (Γ µ ) ∗ c i ( S [ n ] ) is also a universal polynomial in the variables pr ∗ r c ( S ) , pr ∗ r ′ K S , pr ∗ s,t ∆ S . Wecan then conclude thanks to Proposition 3.3 and Theorem 3.4. (cid:3) The Hilbert scheme of points on an abelian surface.
Let A be an abelian surface. Inthat case, the Chow–K¨unneth projectors { π iA } given by the theorem of Deninger and Murre are symmetrically distinguished in the Chow ring CH ∗ ( A × A ) in the sense of O’Sullivan [16]. (We referto [17, Section 7] for a summary of O’Sullivan’s theory of symmetrically distinguished cycles on abelianvarieties.) Let us mention that the identity element O A of A plays the role of the Beauville–Voisincycle in the case of K3 surfaces, e.g. π A = O A × A . By O’Sullivan’s theorem, the Chow–K¨unnethprojectors π iA m given in (3) are symmetrically distinguished for all positive integers m . By Proposition3.1, the cycle (Γ µ ⊗ Γ µ ⊗ Γ µ ) ∗ ∆ is a polynomial in the variables pr ∗ r c ( A ), pr ∗ r ′ K A and pr ∗ s,t ∆ A .Since c ( A ) = 0 and K A = 0, this cycle is in fact symmetrically distinguished. It immediately followsthat (cid:16) π iA µ ⊗ π jA µ ⊗ π kA µ (cid:17) ∗ (Γ µ ⊗ Γ µ ⊗ Γ µ ) ∗ ∆ is symmetrically distinguished. Thus by O’Sullivan’s theorem [16], this cycle is rationally trivial ifand only if it is numerically trivial. By Proposition 2.1, we conclude that A [ n ] has a multiplicativeChow–K¨unneth decomposition. The proof of Theorem 1 is now complete.It remains to prove that the Chern classes c i ( A [ n ] ) sit in CH i ( A [ n ] ) . As in the case of K3 surfaces,it suffices to show that, for all partitions µ of { , . . . , n } , ( π jA µ ) ∗ (Γ µ ) ∗ c i ( A [ n ] ) = 0 in CH ∗ ( A µ ) as CHARLES VIAL soon as ( π jA µ ) ∗ (Γ µ ) ∗ [ c i ( A [ n ] )] = 0 in H ∗ ( A µ , Q ). By Proposition 3.2, (Γ µ ) ∗ c i ( A [ n ] ) is a polynomialin the variables pr ∗ r c ( A ) = 0 , pr ∗ r ′ K A = 0 , pr ∗ s,t ∆ A . It follows that the cycle ( π jA µ ) ∗ (Γ µ ) ∗ c i ( A [ n ] ) issymmetrically distinguished. We can then conclude thanks to O’Sullivan’s theorem. (cid:3) Decomposition theorems for the relative Hilbert scheme of abelian surfaceschemes and of families of K3 surfaces
In this section, we generalize Voisin’s decomposition theorem [21, Theorem 0.7] for families of K3surfaces to families of Hilbert schemes of points on K3 surfaces or abelian surfaces.Let π : X → B be a smooth projective morphism. Deligne’s decomposition theorem states thefollowing : Theorem 4.1 (Deligne [6]) . In the derived category of sheaves of Q -vector spaces on B , there is adecomposition (which is non-canonical in general) (9) Rπ ∗ Q ∼ = M i R i π ∗ Q [ − i ] . Both sides of (9) carry a cup-product : on the right-hand side the cup-product is the direct sum ofthe usual cup-products R i π ∗ Q ⊗ R j π ∗ Q → R i + j π ∗ Q defined on local systems, while on the left-handside the derived cup-product Rπ ∗ Q ⊗ Rπ ∗ Q → Rπ ∗ Q is such that it induces the usual cup-product incohomology. As explained in [21], the isomorphism (9) does not respect the cup-product in general.Given a family of smooth projective varieties π : X → B , Voisin [21, Question 0.2] asked if thereexists a decomposition as in (9) which is multiplicative, i.e. , which is compatible with cup-product.By Deninger–Murre [7], there does exist such a decomposition for an abelian scheme π : A → B . Themain result of [21] is : Theorem 4.2 (Voisin [21]) . For any smooth projective family π : X → B of K3 surfaces, there exista decomposition isomorphism as in (9) and a nonempty Zariski open subset U of B , such that thisdecomposition becomes multiplicative for the restricted family π | U : X | U → U . Our main result in this section is the following extension of Theorem 4.2 :
Theorem 4.3.
Let π : X → B be either an abelian surface over B or a smooth projective family ofK3 surfaces. Consider π [ n ] : X [ n ] → B the relative Hilbert scheme of length- n subschemes on X → B .Then there exist a decomposition isomorphism for π [ n ] : X [ n ] → B as in (9) and a nonempty Zariskiopen subset U of B , such that this decomposition becomes multiplicative for the restricted family π [ n ] | U : X [ n ] | U → U .Proof. The proof follows the original approach of Voisin [21] (after reinterpreting, as in [17, Proposi-tion 8.14], the vanishing of the modified diagonal cycle of Beauville–Voisin [4] as the multiplicativityof the Beauville–Voisin Chow–K¨unneth decomposition).First, we note that there exist a nonempty Zariski open subset U of B and relative Chow–K¨unnethprojectors Π i := Π i X [ n ] | U /U ∈ CH n ( X [ n ] | U × U X [ n ] | U ), which means that ∆ X | U /U = P i Π i , Π i ◦ Π i =Π i , Π i ◦ Π j = 0 for i = j , and Π i acts as the identity on R i ( π [ n ] | U ) ∗ Q and as zero on R j ( π [ n ] | U ) ∗ Q for j = i . Indeed, Let X be the generic fiber of π : X → B . If X is a K3 surface, then we considerthe degree 1 zero-cycle := c ( X ) ∈ CH ( X ). We then have a Chow–K¨unneth decompositionfor X given by π X := pr ∗ , π X := pr ∗ and π X := ∆ X − π X − π X . If X is an abelian surface, wemay consider the Chow–K¨unneth decomposition of Deninger–Murre [7]. In both cases, these Chow–K¨unneth decompositions induce as in (4) a Chow–K¨unneth decomposition ∆ X [ n ] = P i π iX [ n ] of theHilbert scheme of points X [ n ] . By spreading out, we obtain the existence of a sufficiently small butnonempty open subset U of B such that this Chow–K¨unneth decomposition spreads out to a relativeChow–K¨unneth decomposition ∆ X | U /U = P i Π i .By [21, Lemma 2.1], the relative idempotents Π i induce a decomposition in the derived category Rπ ∗ Q ∼ = L ni =0 H i ( Rπ ∗ Q )[ − i ] = L ni =0 R i π ∗ Q [ − i ] with the property that Π i acts as the identity onthe summand H i ( Rπ ∗ Q )[ − i ] and acts as zero on the summands H j ( Rπ ∗ Q )[ − j ] for j = i . Thus, inorder to show the existence of a decomposition as in (9) that is multiplicative, it is enough to show, N THE MOTIVE OF SOME HYPERK¨AHLER VARIETIES 9 up to further shrinking U if necessary, that the relative Chow–K¨unneth decomposition { Π i } abovesatisfies(10) Π k ◦ ∆ ◦ (Π i ⊗ Π j ) = 0 in CH n (( X [ n ] × B X [ n ] × B X [ n ] ) | U ) for all k = i + j. Here, ∆ is the class of the relative small diagonal inside CH n ( X [ n ] × B X [ n ] × B X [ n ] ). But then,by Theorem 1, the relation (10) holds generically. Therefore, by spreading out, (10) holds over anonempty open subset of B . This concludes the proof of the theorem. (cid:3) References [1] A. Beauville,
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