On the motive of some hyperKähler varieties

Introduction

In this work, we fix a field k and all varieties are defined over this field k. Chow groups are always meant with rational coefficients and H*⁢(-,ℚ) is Betti cohomology with rational coefficients. Up to replacing 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 language of (Chow) motives as is described in [11].

Work of Shermenev [18], Beauville [2], and Deninger and Murre [7] unravelled the structure of the motives of abelian varieties:

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 [13]. Such a decomposition is called a Chow–Künneth decomposition. However, in general, there is no analogue of the multiplication by n morphisms, and the existence of a Chow–Künneth decomposition of the motive of X satisfying (ii) (in that case, the Chow–Künneth decomposition is said to be multiplicative) is very restrictive. We refer to [16, Section 8] for some discussion on the existence of such a multiplicative decomposition.

Nonetheless, inspired by the seminal work of Beauville and Voisin [4, 3, 19], we were led to ask in [16] whether the motives of hyperKähler varieties admit a multiplicative decomposition similar to that of the motive of abelian varieties as in the theorem of Beauville, Deninger and Murre, and Shermenev. Here, by hyperKähler variety we mean a simply connected smooth projective variety X whose space of global 2-forms H0⁢(X,ΩX2) is spanned by a nowhere degenerate 2-form. When k=ℂ, a hyperKähler variety is nothing but a projective irreducible holomorphic symplectic manifold [1].

An important class of hyperKähler 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 motive of S[n] for S a K3 surface admits a decomposition with properties (i) and (ii) and thus answers affirmatively the question raised in Conjecture 1 in that case.

Theorem 6 of [16] can then be improved by including the Hilbert schemes of length-n subschemes on K3 surfaces. Theorem 1 is due for S a K3 surface and n=1 to Beauville and Voisin [4] (see [16, Proposition 8.14] for the link between the original statement of [4] (recalled in Theorem 3.4) and the statement given here), and was established in [16] for n=2. Its proof in full generality is given in Section 3. Note that, as explained in Section 1, the existence of a Chow–Künneth decomposition for the 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 [12]). Our main contribution is the claim that by choosing the Beauville–Voisin decomposition of K3 surfaces [4], the induced Chow–Künneth 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≥0 and all s∈ℤ

We have the following corollary to Theorem 1:

Theorem 2 answers partially a question raised by Beauville in [3]: the filtration F∙ defined by Fl⁢CHi⁢(X):=⊕s≥lCHi⁢(X)s is a filtration on the Chow ring CH*⁢(S[n]) that is split. Moreover, this filtration is expected to be the one predicted by Bloch and Beilinson (because it is induced by a Chow–Künneth 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 CHi⁢(S[n])s=0 for s<0 and that ⊕s>0CHi⁢(S[n])s is exactly the kernel of the cycle class map CHi⁢(S[n])→H2⁢i⁢(S[n],ℚ). Note that for i=0,1,2⁢n-1 or 2⁢n, it is indeed the case that CHi⁢(S[n])s=0 for s<0 and that

Therefore, we have

Let us mention that Theorem 1 and Theorem 2 (and a fortiori Corollary 1) are also valid for hyperKähler varieties that are birational to S[n], for some K3 surface S. Indeed, Riess [15] showed that birational hyperKähler varieties have isomorphic Chow rings and isomorphic Chow motives (as algebras in the category of Chow motives); see also [16, Section 6]. As for more evidence as why Conjecture 1 should be true, Mingmin Shen and I showed [16] that the variety of lines on a very general cubic fourfold satisfies the conclusions of Theorem 2.

Finally, we use the notion of multiplicative Chow–Künneth decomposition to obtain new decomposition results in the spirit of [20]; see Theorem 4.3.

1 Chow–Künneth decompositions

A Chow motive M is said to have a Chow–Künneth decomposition if it splits as

with H*⁢(Mi,ℚ)=Hi⁢(M,ℚ). In other words, M admits a Künneth decomposition that lifts to rational equivalence. Concretely, if M=(X,p,n) with X a smooth projective variety of pure dimension d and p∈CHd⁢(X×X) an idempotent and n an integer, then M has a Chow–Künneth decomposition if there exist finitely many correspondences pi∈CHd⁢(X×X), i∈ℤ, such that p=∑ipi, pi∘pi=pi, p∘pi=pi∘p=pi, pi∘pj=0 for all i∈ℤ and all j≠i and such that p*i⁢H*⁢(X,ℚ)=p*⁢Hi+2⁢n⁢(X,ℚ).

A smooth projective variety X of dimension d has a Chow–Künneth decomposition if its Chow motive 𝔥⁢(X) has a Chow–Künneth decomposition, that is, there exist correspondences πi∈CHd⁢(X×X) such that ΔX=∑i=02⁢dπi, with πi∘πi=πi, πi∘πj=0 for i≠j and π*i⁢H*⁢(X,ℚ)=Hi⁢(X,ℚ). A Chow–Künneth decomposition {πi:0≤i≤2⁢d} of X is said to be self-dual if π2⁢d-i=πit for all i.

If 𝔬 is the class of a rational point on X (or more generally a zero-cycle of degree 1 on X), then π0:=pr1*⁢𝔬=𝔬×X and π2⁢d=pr2*⁢𝔬=X×𝔬 define mutually orthogonal idempotents such that π*0⁢H*⁢(X,ℚ)=H0⁢(X,ℚ) and π*2⁢d⁢H*⁢(X,ℚ)=H2⁢d⁢(X,ℚ). Note that pairs of idempotents with the property 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 CHd⁢(X×X). From the above, one sees that every curve C admits a Chow–Künneth decomposition: one defines π0 and π2 as above and then π1 is simply given by ΔC-π0-π2. It is a theorem of Murre [12] that every smooth projective surface S admits a Chow–Künneth decomposition ΔS=πS0+πS1+πS2+πS3+πS4.

The notion of Chow–Künneth decomposition is significant because when it exists it induces a filtration

on the Chow group CH*⁢(X) which should not depend on the choice of the Chow–Künneth decomposition ΔX=∑i=02⁢dπi and which should be of Bloch–Beilinson type; cf. [10, 13].

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 a morphism S[n]→S(n) to the n-th symmetric product of S, called the Hilbert–Chow morphism. De Cataldo and Migliorini [5] have given an explicit description of the motive of S[n]. Let us introduce some notations related to this description. Let μ={A1,…,Al} be a partition of the set {1,…,n}, where all the Ai are nonempty. The integer l, also denoted l⁢(μ), is the length of the partition μ. Let Sμ≃Sl⊆Sn be the set

and let

where the subscript “red” means the underlying reduced scheme. It is known that Γμ is irreducible of dimension n+l⁢(μ). The subgroup 𝔖μ of 𝔖n that acts on {1,…,n} by permuting the Ai with same cardinality acts on the first factor of the product Sμ×S[n], and the correspondence Γμ is invariant under this action. We can therefore define

where S(μ):=Sμ/𝔖μ. Since for a variety X endowed with the action of a finite group G we have CH*⁢(X/G)=CH*⁢(X)G (with rational coefficients), the calculus of correspondences and the theory of motives in the setting of smooth projective varieties endowed with the action of a finite group is similar in every way to the usual case of smooth projective varieties. We will therefore freely consider actions of correspondences and motives of quotient varieties by the action of a finite group.

The symmetric groups 𝔖n acts naturally on the set of partitions of {1,…,n}. By choosing one element in each orbit for the above action, we may define a subset 𝔅⁢(n) of the set of partitions of {1,…,n}. This set is isomorphic to the set of partitions of the integer n.

Let now ΔS=πS0+πS1+πS2+πS3+πS4 be a Chow–Künneth decomposition of S. For all nonnegative integers m, the correspondences

define a Chow–Künneth decomposition of Sm that is clearly 𝔖m-equivariant. Therefore, these correspondences can be seen as correspondences of CH2⁢m⁢(S(m)×S(m)) and they do define a Chow–Künneth decomposition of the m-th symmetric product S(m). Let us denote this decomposition

Since Sl is endowed with a 𝔖l-equivariant Chow–Künneth decomposition as above and since 𝔖μ is a subgroup of 𝔖l, Sμ≃Sl is endowed with a 𝔖μ-equivariant Chow–Künneth decomposition. Therefore S(μ) is endowed with a natural Chow–Künneth decomposition

coming from that of S. In particular, the isomorphism of de Cataldo and Migliorini gives a natural Chow–Künneth decomposition for the Hilbert scheme S[n] coming from that of S. Precisely, this Chow–Künneth decomposition is given by

Note that if the Chow–Künneth decomposition {πSi} of S is self-dual, then the Chow–Künneth decomposition {πS[n]i} of S[n] is also self-dual.

We will show that when S is either a K3 surface or an abelian surface the Chow–Künneth decomposition above induces a decomposition of the motive 𝔥⁢(S[n]) that satisfies the conclusions of Theorem 1 for an appropriate choice of Chow–Künneth decomposition for S.

2 Multiplicative Chow–Künneth decompositions

Let X be a smooth projective variety of dimension d and let Δ3∈CHd⁢(X×X×X) be the small diagonal, that is, the class of the subvariety

If we view Δ3 as a correspondence from X×X to X, then Δ3 induces the multiplication morphism

Note that if α and β are cycles in CH*⁢(X), then (Δ3)*⁢(α×β)=α⋅β in CH*⁢(X).

If X admits a Chow–Künneth decomposition

then this decomposition is said to be multiplicative if the multiplication morphism

factors through the direct summand 𝔥i+j⁢(X) for all i and j. For a variety to be endowed with a multiplicative Chow–Künneth decomposition is very restrictive; we refer to [16], where this notion was introduced, for some discussions. For instance, a very general curve of genus ≥3 does not admit such a decomposition. Examples of varieties admitting a multiplicative Chow–Künneth decomposition are provided by [16, Theorem 6] and include hyperelliptic curves, K3 surfaces, abelian varieties, and their Hilbert squares.

If one writes ΔX=πX0+…+πX2⁢d for the Chow–Künneth decomposition (6) of X, then by definition this decomposition is multiplicative if

or equivalently if

If the Chow–Künneth decomposition {πXi} is self-dual, then it is multiplicative if

Note that the above three relations always hold modulo homological equivalence.

Given a multiplicative Chow–Künneth decomposition πSi for a surface S, one could expect the Chow–Künneth 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–Künneth decomposition (under some additional assumptions on the Chern classes of X) in [16], and a similar result when n=3 can be found in [17]. (For n>3, X[n] is no longer smooth if X is smooth of dimension >2.) Here we deal with the case when S is a K3 surface or an abelian surface and will prove Theorem 1. By the isomorphism (2) of de Cataldo and Migliorini, it is enough to check that

for all i+j+k≠8⁢n and for all partitions μ1,μ2 and μ3 of {1,…,n}, or equivalently for all i,j,k such that (πS[n]i⊗πS[n]j⊗πS[n]k)*⁢[Δ3]=0 in H*⁢(S[n]×S[n]×S[n],ℚ) and all partitions μ1,μ2 and μ3. By (4), it is even enough to show that

is zero in CH*⁢(S(μ1)×S(μ2)×S(μ3)) for all partitions μ1,μ2,μ3, and all partitions ν1,ν2,ν3 and all i,j,k such that

in H*⁢((S[n])3,ℚ). Note that the expression (7) is equal to

But it is clear from Theorem 1.1 that

Thus we have proved the following criterion for the Chow–Künneth decomposition (4) to be multiplicative.

3 Proof of Theorem 1 and Theorem 2

The proof is inspired by the proof of Claire Voisin’s [21, Theorem 5.1]. In fact, because of [16, Proposition 8.12], Theorem 1 for K3 surfaces implies [21, Theorem 5.1]. The first step towards the proof of Theorem 1 is to understand the cycle (Γμ1⊗Γμ2⊗Γμ3)*⁢Δ3. The following proposition, due to Voisin [21] (see also [19]), builds on the work of Ellingsrud, Göttsche and Lehn [8]. Here, S is a smooth projective surface and Δk is the class of the small diagonal inside Sk in CH2⁢(Sk).

In fact, Proposition 3.1 is a particular instance of [21, Theorem 5.12]. Another consequence of [21, Theorem 5.12], which will be used to prove Theorem 2, is

We first prove Theorems 1–2 for S a K3 surface and then for S an abelian surface. Note that clearly a multiplicative Chow–Künneth decomposition {πS[n]i:0≤i≤4⁢n} induces a multiplicative bigrading on the Chow ring CH*⁢(S[n]) :

Thus once Theorem 1 is established it only remains to show that the Chern classes of S[n] sit in CH*⁢(S[n])0 in order to conclude.