The Chow ring of hyperk\"ahler varieties of $K3^{[2]}$-type via Lefschetz actions

We propose an explicit conjectural lift of the Neron-Severi Lie algebra of a hyperk\"ahler variety $X$ of $K3^{[2]}$-type to the Chow ring of correspondences ${\rm CH}^\ast(X \times X)$ in terms of a canonical lift of the Beauville-Bogomolov class obtained by Markman. We give evidence for this conjecture in the case of the Hilbert scheme of two points of a $K3$ surface and in the case of the Fano variety of lines of a very general cubic fourfold. Moreover, we show that the Fourier decomposition of the Chow ring of $X$ of Shen and Vial agrees with the eigenspace decomposition of a canonical lift of the grading operator.


Introduction and results
The main purpose of this note is to hint at a connection between two lines of reasoning which are both relevant for the study of the Chow ring of hyperkähler varieties of K3 [2]type. One focuses on Hilbert schemes of points of K3 surfaces [Obe19, NOY20] using Nakajima operators, and the other investigates an analog of the Fourier transform and injectivity results for the cycle class map on subrings of universal classes [SV16,FLVS19]. The present text suggests a connection in two directions. The first is provided by Conjecture 1.2, asking for an extension of the main theorem of [Obe19] to the K3 [2] -case. The second comes in the form of Theorem 1.7, stating that the Fourier decomposition of the Chow ring from [SV16, Theorem 2] agrees with the eigenspace decomposition of a canonical lift of the cohomological grading operator h.
1.1 Conventions. We denote the Chow ring of a smooth projective variety X over C by CH * (X), and it will always be with coefficients in Q. Similarly, we often abbreviate the cohomology ring H * (X, Q) by H * (X). For a cycle class Z ∈ CH * (X) or a cohomology class β ∈ H * (X) we denote the pullbacks to X × X via the two projections by Z 1 , Z 2 and β 1 , β 2 , and similar conventions will be followed throughout.
Conjecture 1.2. Let g NS (X) be the Neron-Severi Lie algebra of X and cl the cycle class map. The linear map ϕ in the commutative diagram g NS (X) CH * (X × X) End Q (H * (X, Q)), ϕ cl given by ϕ(e a ) = ∆ * (a), ϕ(f a ) = F a and ϕ(h) = H is a well-defined Lie algebra homomorphism.
In the case X = S [2] for a projective K3 surface S we use the explicit description from Theorem 2.4 of Markman's lift L in order to prove in Section 3.3 that our formulas for the lifts F a and H agree with the canonical lifts provided in [Obe19] in terms of Nakajima operators. This is the content of Proposition 3.21. Hence, the main theorem of loc. cit.
shows that Conjecture 1.2 is true if X = S [2] . In the case of the Fano variety of lines we use a result of Fu, Laterveer, Vial and Shen [FLVS19] and the explicit description of Markman's lift from Theorem 2.7 to obtain the following partial confirmation of Conjecture 1.2.
Theorem 1.3. Let F = F (Y ) be the Fano variety of lines of a smooth cubic fourfold Y and g ∈ CH 1 (F ) the Plücker polarization class. Let L ∈ CH 2 (F × F ) be Markman's lift. Then there is a Lie algebra homomorphism lifting the sl 2 (Q)-action on cohomology given by the Lefschetz triple (e g , f g , h).
Whenever F has Picard rank 1, i.e., CH 1 (F ) = g , which is true for very general cubic fourfolds Y , then Conjecture 1.2 reduces to Theorem 1.3. We also propose two new relations in Conjecture 3.13 which would yield a generalization of Theorem 1.3 to the case of divisor classes different from g, see Proposition 3.16. More evidence for Conjecture 1.2 is provided by Remark 3.12, where we note that [F a , F b ] = 0 in full generality. The obstacle to proving the conjecture in the case of Picard rank > 1 is that we neither have at our disposal an analog of the machinery of Nakajima operators nor sufficiently strong injectivity results for the cycle class map yet, involving non-tautological divisor classes and hence extending results such as Theorem 2.9, which is the main geometrical input for Theorem 1.3.
1.3 Eigenspace decomposition of H. If the lift H of h is diagonalizable, it could be expected to be multiplicative with respect to the intersection product. This is because the eigenspace decomposition of H can be viewed as an analog of the Beauville decomposition in the abelian variety case, as discussed in the introduction of [NOY20].
Theorem 1.4. Let X be a hyperkähler variety of K3 [2] -type endowed with a lift L ∈ CH 2 (X × X) of B satisfying the relations (2.8)-(2.10). Let Λ i λ ⊆ CH i (X) be the eigenspace for the eigenvalue λ of H * and denote CH i (X) s : (1.1) All direct summands outside the leftmost column belong to the homologically trivial cycle classes CH * (X) hom . We have ), so that the cycle class map is injective on CH i (X) 0 except maybe for i = 2. Moreover, all elements of CH 4 (X) 0 ⊕ CH 4 (X) 2 are multiples of l, and multiplication by l gives an injective map CH 1 (X) → CH 3 (X) 0 . Furthermore, L * (CH 4 (X) 4 ) = 0.
The number s should be seen as a sort of defect, e.g., the terms with s = 0 give the expected eigenvalue 2i − 4 of H * . The eigenspace decomposition only shows terms with s ≥ 0. In fact, there is a conjecture by Beauville in the abelian variety case which predicts this behavior, see [Bea07]. It is also expected that the cycle class map is injective on CH * (X) 0 , and Theorem 1.4 confirms this in all codimensions except 2.
Conjecture 1.5. Let X be as in Theorem 1.4 such that L additionally satisfies (2.11). For all occurring s, t ∈ Z we conjecture that the intersection product gives a well-defined map This can be rewritten in several ways. For this and a generalization in the case of Hilbert schemes of points of K3 surfaces see [NOY20,eq. (6) and (44)]. Evidence for the conjecture is provided by Corollary 1.8 below.
We define the Fourier decomposition groups by They do not depend on the precise values of the coefficients of the powers L i as long as they are non-zero.
and the first one was dealt with for Y not necessarily very general in [FLVS19, Proposition A.7]. The second inclusion, however, still remains open for arbitrary smooth cubic fourfolds Y . In the very general case these inclusions are even equalities.

2.1
The Neron-Severi Lie algebra. Let X be a smooth projective variety over C of complex dimension n. An element a ∈ H 2 (X, Q) is called Lefschetz if the conclusion of the hard Lefschetz theorem holds, i.e., for the operator e a of cup product by a the k-fold iteration e k a : H n−k (X) → H n+k (X) is an isomorphism for all 1 ≤ k ≤ n. The operator e a is called the Lefschetz operator. If a is Lefschetz, there exists a unique operator f a , the Lefschetz dual, such that the commutator [e a , f a ] = h is the grading operator, given by multiplication with the integer k − n in degree k. The triple (e a , f a , h) satisfies the sl 2 -commutation relations and is called a Lefschetz triple. Looijenga and Lunts [LL97] and Verbitsky [Ver96] introduced the total Lie algebra g(X) ⊆ End Q (H * (X)) of X which is generated by all Lefschetz triples (e a , f a , h). The Neron-Severi Lie algebra of X is the Lie subalgebra g NS (X) ⊆ g(X) generated by only those Lefschetz triples where a is algebraic, i.e., a ∈ H 1,1 (X, Q). In view of the Grothendieck standard conjecture of Lefschetz type, a natural question to ask, then, is whether the Neron-Severi Lie algebra action on the cohomology ring can be lifted to an action on the Chow ring or, slightly stronger, whether there is a Lie algebra homomorphism g NS (X) → CH * (X × X) to the ring of correspondences CH * (X × X) such that End Q (H * (X, Q)) cl commutes, where cl denotes the cycle class map. By the main theorem of [Obe19], this is the case if X is the Hilbert scheme of points of a projective K3 surface. We investigate this question for the Fano variety of lines of a smooth cubic fourfold, obtaining Theorem 1.3 as a partial analog. We view the Fano variety of lines as a test case for Conjecture 1.2.
2.2 Markman's lift of the Beauville-Bogomolov class. Let X be a hyperkähler variety and (−, −) : H 2 (X, Q) × H 2 (X, Q) → Q the non-degenerate symmetric bilinear form associated to the Beauville-Bogomolov quadratic form on X [Huy97]. Via the Künneth isomorphism we obtain from this the Beauville-Bogomolov cohomology class B ∈ H 4 (X × X, Q). Extending the coefficients to C and choosing an orthonormal basis (e i ) of H 2 (X, C) with respect to (−, −), we can write B = r i=1 e i ⊗ e i , where r = b 2 (X) is the second Betti number. We denote by b := ∆ * (B) ∈ H 4 (X, Q) the pullback along the diagonal embedding ∆ : X ֒→ X × X. Over C we can write b = r i=1 e 2 i . Theorem 2.1 ( [Mar20]). Let X be a hyperkähler variety deformation equivalent to the Hilbert scheme of n ≥ 2 points of a K3 surface. Then there exists a lift L ∈ CH 2 (X × X) of the Beauville-Bogomolov class B.
A summary of Markman's construction of L in [Mar20] is given in [SV16, Section 1.9], particularly Theorem 9.15 where Shen and Vial use the κ-class κ 2 (M ) ∈ CH 2 (X × X) of Markman's sheaf M , a twisted sheaf in the sense of [Mar20, Definition 2.1], in order to define L. The twisted sheaf M is constructed as in the theorem below, following closely [SV16, Theorem 9.12].
Theorem 2.2 ([Mar20]). Let X be a hyperkähler manifold of K3 [n] -type. There exists a K3 surface S and a suitable Mukai vector v with a v-generic ample divisor H such that there is a proper flat family π : X → C having the following properties: (1) The curve C is connected but possibly reducible of arithmetic genus 0.
(2) There exist t 1 , t 2 ∈ C such that X t 1 = X and X t 2 = M H (v), the latter denoting the moduli space of stable sheaves on S with Mukai vector v.
(4) There is a torsion-free reflexive coherent twisted sheaf G on X × C X , flat over C, . Then set Markman's sheaf to be the twisted sheaf M := G t 1 on X × X. A priori, M might depend on the chosen deformation, but as it turns out, κ 2 (M ) does not. As in [MMV19] we denote by M Λ the moduli space of isomorphism classes of marked hyperkähler manifolds.
where p i : X ×X → X are the projections and c i (X) are the Chern classes of the tangent bundle of X. This is indeed a lift of b by [SV16, Lemma 9.14]. They then set which is indeed a lift of B by the same lemma. By Proposition 2.3, L and l are, in fact, canonical lifts of their cohomology classes, and we are justified in calling L Markman's lift of B without ambiguity. The equation ∆ * (L) = l holds if and only if l is a rational multiple of c 2 (X) in CH 2 (X).

Markman's lift in terms of tautological classes for the Hilbert scheme.
In the case of the Hilbert scheme S [2] of two points of a projective K3 surface S we have an explicit description of Markman's lift of B, see [SV16] for details. Denote F = S [2] and let Z ⊆ F × S be the universal family. Its set of closed points consists of the pairs (η, x) where x ∈ supp(η). This is a codimension 2 closed subscheme of the product, and we denote by the projections. Let c ∈ CH 0 (S) be the canonical 0-cycle, represented by any point on a rational curve in S [BV01, Theorem 1]. We let Moreover, denote by ∆ Hilb ∈ CH 1 (F ) the divisor class on F parametrizing the nonreduced length 2 subschemes of S and set This agrees with the convention of [SV16] which differs in the sign from [Obe19]. Finally, let I ⊆ F × F be the subset of pairs of length 2 subschemes which share a common support point. This is closed and irreducible (see the proof of [SV16, Lemma 11.2]), and endowed with the reduced induced subscheme structure it gives a closed subvariety of codimension 2, called the incidence subscheme. Its cycle class in CH 2 (F × F ) is also denoted I and called the incidence correspondence. By loc. cit. we have I = t Z • Z.
Moreover, by [SV16,Proposition 16.1], L agrees with Markman's lift. For its pullback along the diagonal embedding i ∆ : In fact, l = 5 6 c 2 (T F ) where c 2 (T F ) is the second Chern class of the tangent bundle.

Markman's lift in terms of tautological classes for the Fano variety.
For the Fano variety of lines F := F (Y ) ⊆ Gr(2, 6) of a smooth cubic fourfold Y we define several tautological cycle classes as follows.
Definition 2.5. Let E be the tautological bundle on Gr(2, 6). Let g ∈ CH 1 (F (Y )) be the first Chern class c 1 (E| ∨ F (Y ) ). Then g is called the Plücker polarization class. Moreover, we denote by c ∈ CH 2 (F (Y )) the second Chern class c 2 (E| ∨ F (Y ) ). Definition 2.6. The incidence subscheme is the closed subset I ⊆ F × F with the reduced subscheme structure, given by the set of pairs of intersecting lines inside Y .
Its cycle class in CH 2 (F × F ), also denoted I, is called the incidence correspondence.
. Let Y be a smooth cubic fourfold and F = F (Y ) its Fano variety of lines. An explicit lift L of the Beauville-Bogomolov class B is given by Moreover, we have l := ∆ * (L) = 5 6 c 2 (T F ), where c 2 (T F ) is the second Chern class of the tangent bundle, and c 2 (T F ) = 5g 2 − 8c. Hence, the tautological subring R * (F × F ) contains L, l 1 , l 2 . By the quadratic equation (2.8) below for L, R * (F × F ) can in fact be generated by L, l 1 , l 2 , g 1 , g 2 .
In fact, Shen and Vial in [SV16] prove the following relations for the explicit lifts L of Theorems 2.4 and 2.7. The first is referred to as the quadratic equation for L: Additionally, we consider the following three relations for all σ ∈ CH 4 (X) and all τ ∈ CH 2 (X): These are the relations which Shen and Vial in [SV16, Theorem 2] established to be the core relations necessary to obtain a Fourier decomposition. The lifts from Theorems 2.4 and 2.7 indeed satisfy all of these, see Theorem 1 and the paragraphs after Theorem 2 of loc. cit. In the case of the Fano variety of lines they all follow more directly from cohomology, Theorem 2.9 and the fact that the tautological subring R * (F × F ) is closed under the composition of correspondences. The latter is a consequence of [FLVS19, Proposition 6.3].
3.1 Hyperkähler fourfolds. Let X be a hyperkähler variety of complex dimension 4 without odd cohomology over Q or, equivalently, H 3 (X) = 0. Then the cup product is commutative and we always write αβ instead of α ∪ β. By (e i ) we denote an arbitrary orthonormal basis of H 2 (X, C) with respect to the Beauville-Bogomolov form and we denote the second Betti number by r := b 2 (X). The (modified) Fujiki constant c X equals 1 for hyperkähler varieties of K3 [n] -type. Finally, we write 1 ∈ H 8 (X) for the generator of H 8 (X) with integral 1.
If (a, a) = 0, we set f a := 1 (a,a) f a to obtain the usual sl 2 (Q)-commutation relations. By Lemma 3.5 below, (a, a) = 0 is equivalent to a being Lefschetz in the sense of Subsection 2.1.
Remark 3.2. If (a, a) = 0, then f a is uniquely determined by the commutation relations while this fails if (a, a) = 0 in which case the zero map also satisfies them. It should be emphasized also that f a is linear in a ∈ H 2 (X) because for one, f a is not, and neither is it clear from the abstract description of f a that it would suffice to multiply it by some quadratic form (a, a) in order to make it linear in a.
Before proving Proposition 3.1, we need some lemmas. They are simple computations in cohomology using the definition of the Beauville-Bogomolov form, so we omit the proofs.
Proposition 3.11. Let X be a hyperkähler variety of complex dimension 4 with vanishing H 3 (X). Whenever L ∈ CH 2 (X × X) is a lift of B and l ∈ CH 2 (X) a lift of b then is a lift of the grading operator h.
Finally, let Z ∈ e L CH 4 (X) 4 which is equivalent to the vanishing of both L * (Z) and i.e., Z ∈ Λ 4 0 = CH 4 (X) 4 . Remark 3.12. Let X be a hyperkähler variety of complex dimension 4 with H 3 (X) = 0 and L ∈ CH 2 (X ×X) any lift of B as well as l ∈ CH 2 (X) any lift of b. Then for all divisor classes a, b ∈ CH 1 (X) we have [ F a , F b ] = 0. We omit the proof as it is a straightforward formal computation, although a little lengthy if spelled out in detail. Just use that all terms symmetric in a and b cancel out and that (p 13 ) * (L 12 · L 23 · a 2 ) is a divisor on X × X vanishing in cohomology.

Fano variety of lines.
Proof of Theorem 1.3. By Corollary 3.6, F g lifts f g , and by Proposition 3.11, H lifts h. Moreover, ∆ * (g) lifts e g . By the explicit formulas for ∆ * (g) = ∆g 1 , F g and H we know that all of them lie in the tautological subring R * (F × F ) of Theorem 2.9. As a consequence of [FLVS19, Proposition 6.3], so do their compositions. Hence all the commutators lie in R * (F ×F ), and as the commutation relations are true in cohomology, they are true in the Chow ring as well.

(3.3)
Remark 3.14. The second relation follows from the first one by [Voi06,Theorem 1.4]. Moreover, the first relation and its transpose would imply the important equality H = H a by the proof of Proposition 3.11 carried out in the Chow ring, replacing B by L and b by l. This equality is necessary for establishing Conjecture 1.2. Note also that we indirectly show H = H a in the Hilbert scheme setting in Section 3.3, see Remark 3.22. In the Hilbert scheme setting, the above relations can be checked by computations similar to those in Section 3.4, and we expect them to hold true.
Proposition 3.15. Conjecture 3.13 is true in the case of the Fano variety if a is a multiple of the Plücker polarization class g. In particular, H = H g .
Proof. By the aforegoing Propositions 3.8-3.10 the relations are true in cohomology for all a. If a is a multiple of g, then by Theorem 2.9 they hold in the Chow ring as well because all occurring terms then lie in the tautological subring R * (F × F ). Moreover, (g, g) = 0 by [Ott15, Section 2].
Under Conjecture 3.13, we have the following generalization of Theorem 1.3.
Proposition 3.16. Let X be a hyperkähler variety of K3 [2] -type endowed with a symmetric lift L ∈ CH 2 (X × X) satisfying the relations (2.8) and (2.9). Let a ∈ CH 1 (X) be a divisor class satisfying the relations of Conjecture 3.13 with respect to L. Then the analogous version of Theorem 1.3 holds with respect to L and for g replaced by a. For the last of these we already indicated this in Remark 3.14. For the first one, observe that the left hand side equals (a 1 − a 2 )H = 4 r(r + 2) (l 2 1 a 2 + l 2 2 a 1 ) + 2 r + 2 L(l 1 a 2 + l 2 a 1 ) − 2 r + 2 L(l 1 a 1 + l 2 a 2 ).
For the right hand side use the quadratic equation for L and the second and third relation of the conjecture. Then 2∆ * (a) = 2 r + 2 L(l 1 a 2 + l 2 a 1 ) + 2 r(r + 2) (l 2 1 a 1 + l 2 2 a 2 ).
Hence the difference equals 2 r(r+2) (l 2 1 a 2 + l 2 2 a 1 ) − 2 r+2 L(l 1 a 1 + l 2 a 2 ) = 0 by another application of the second relation of the conjecture and its transpose. For the second commutation relation we show without loss of generality [H, F a ] = −2 F a . Here, consider first the composition H • F a . The only summand of the latter which is not easily computed using the projection formula, L * (l 2 ) = 0 and the second relation of Conjecture 3.13 is − 4 r + 2 (p 13 ) * (p 13 ) * (Ll 2 a 1 ) · (p 23 ) * (L) , where p ij : X × X × X → X × X denote the projections to the factors according to the indices. Here, we use the transpose of the third relation of the conjecture in order to write Ll 2 a 1 = r+2 2 L 2 a 2 − 1 2 l 1 l 2 a 2 . From this we deduce that the above summand equals 2l 1 a 2 − 2(p 13 ) * (p 13 ) * (L 2 ) · (p 23 ) * (La 1 ) .
Next, using the quadratic equation for L 2 we obtain that the latter agrees with −4La 1 . We obtain H • F a = − 16 r+2 l 1 a 1 + 8 r+2 l 2 a 2 − 4La 1 . Now, t H = −H and t F a = F a , so as desired.
3.3 Hilbert scheme of two points of a K3 surface. Let S be a smooth projective surface. Recall the Nakajima operators on the Chow ring of Hilbert schemes of points of S, We refer to [Nak97,Gro95] and to [Obe19, Section 2.3] for a very brief overview. Recall that q i for i ≥ 1 is given as a correspondence by the reduced closed subscheme , where I ξ is the ideal sheaf corresponding to the length n closed subscheme ξ of S. Recall also that q −i is just the transpose of (−1) i q i . Let Z n ⊆ S [n] × S be the universal family. We write Z := Z 2 . where t Z n is the transpose of Z n .
Proof. First recall that Z n−1,n is a closed subvariety of the product. This can be seen using the projection map Z n−1,n → S [n−1,n] onto the nested Hilbert scheme [Leh99, Section 1.2] and the fact that S [n−1,n] is irreducible by [Leh99,Theorem 1.9]. Indeed, the projection is proper, hence closed, and it actually is a bijection on closed points. It follows that Z n−1,n is irreducible, hence a closed subvariety as it is endowed with the reduced subscheme structure. Now, to prove the claim, note that the equation is clearly true set-theoretically. As Z n−1,n is a closed subvariety of the product, it suffices to show that the degree of p S×S [n] | Z n−1,n is 1. Considering a general fiber suffices, and if (x, η) ∈ t Z n with η supported at n distinct points x, x 2 , . . . , x n then the only preimage in Z n−1,n is ([x 2 , . . . , x n ], x, η) with multiplicity 1.
The third equation is a consequence of the first one and the self-intersection formula. Now we show that the canonical lifts F a of the Lefschetz dual f a and H of the grading operator h agree with Oberdieck's canonical lifts in [Obe19]. Recall the notation set up in Subsection 2.3.
Proposition 3.19. Let S be a projective K3 surface and F = S [2] . We have the following expressions for correspondences in terms of Nakajima operators: Of course, as correspondences we interpret mult Z to be (∆ F ) * (Z).
Proof. The first of these equations follows immediately from Corollary A.2 and [F ] = Using the commutation relations for Nakajima operators, the second summand inside the brackets equals −2q 1 q 1 q −1 q −1 (∆ 1234 ). For the first summand, we obtain where in the last step we used ∆ 1234 = ∆ 12 ∆ 23 ∆ 34 and ∆ 2 S = 24c 1 c 2 from (3.4) as well as ∆ S c 1 = ∆ S c 2 = c 1 c 2 , concluding the proof.

A An auxiliary lemma
Lemma A.1. Let X, Y and Z be smooth projective varieties and Γ ∈ CH * (X × Y ), Γ ∈ CH * (X × Z) correspondences. Then as correspondences in CH * (Y × Z) we have Proof. Writing out the definitions of both sides, we see that we only need to show . In here, p Y X and p XZ are the projections from the triple product Y × X × Z to Y × X and X × Z, respectively. The projections p X 1 Y , p X 2 Z and p X 1 X 2 are the projections from the product X × Y × X × Z to the factors indicated by the indices. Moreover, ∆ X denotes both the diagonal embedding X ֒→ X × X and the cycle class of its image. We can now rewrite the argument of the right hand side of (A.1) as . Hence, by the projection formula, the entire right hand side of (A.1) equals This seemingly trivial result has an interesting consequence in the case X = Spec(C) where ∆ X is an isomorphism. The following was pointed out to me by my advisor Georg Oberdieck and uses only the fact that the Nakajima correspondences q i and q −i are the transpose of each other up to sign. acting as the Nakajima operator q i on the first factor and as the identity on the second factor respectively as q j on the second factor and as the identity on the first factor. Obviously, q i commutes with q ′ j . Let Γ ∈ CH * (S k+l ). Then we have the equation of correspondences q i 1 · · · q i k q ′ j 1 · · · q ′ j l (Γ) · 1 S [0] = (−1) i 1 +...+i k q j 1 · · · q j l q −i k · · · q −i 1 (τ * (Γ)), where τ : S k+l → S k+l permutes the factors according to the permutation of the indices.
Proof. This follows from the definition of the Nakajima operators and Lemma A.1 on noting that ∆ S