On the Chow ring of certain Fano fourfolds

We prove that certain Fano fourfolds of K3 type constructed by Fatighenti-Mongardi have a multiplicative Chow-K\"unneth decomposition. We present some consequences for the Chow ring of these fourfolds.


INTRODUCTION
This note is part of a program aimed at understanding the class of varieties admitting a multiplicative Chow-Künneth decomposition, in the sense of [18]. The concept of multiplicative Chow-Künneth decomposition was introduced in order to better understand the (conjectural) behaviour of the Chow ring of hyperkähler varieties, while also providing a systematic explanation of the peculiar behaviour of the Chow ring of K3 surfaces and abelian varieties. In [12], the following conjecture is raised: Conjecture 1.1. Let X be a smooth projective Fano variety of K3 type (i.e. dim X = 2m and the Hodge numbers h p,q (X) are 0 for all p = q except for h m−1,m+1 (X) = h m+1,m−1 (X) = 1). Then X has a multiplicative Chow-Künneth decomposition.
This conjecture is verified in some special cases [11], [10], [12]. The aim of the present note is to provide some more evidence for Conjecture 1.1. We consider two families of Fano fourfolds of K3 type (these are the families labelled B1 and B2 in [5]).
Theorem 4.1 has interesting consequences for the Chow ring A * (X) Q of these fourfolds: Corollary (=Corollary 5.1). Let X and M be as in Theorem 4.1. Let R 3 (X) ⊂ A 3 (X) Q be the subgroup generated by the Chern class c 3 (T X ), the image of the restriction map A 3 (M) Q → A 3 (X) Q , and intersections A 1 (X) Q · A 2 (X) Q of divisors with 2-cycles. The cycle class map induces an injection R 3 (X) ֒→ H 6 (X, Q) .
This is reminiscent of the famous result of Beauville-Voisin describing the Chow ring of a K3 surface [2]. More generally, there is a similar injectivity result for the Chow ring of certain self-products X m (Corollary 5.1).
Another consequence is the existence of a multiplicative decomposition in the derived category for families of Fano fourfolds as in Theorem 4.1 (Corollary 5.4).
Conventions. In this note, the word variety will refer to a reduced irreducible scheme of finite type over C. For a smooth variety X, we will denote by A j (X) the Chow group of codimension j cycles on X with Q-coefficients.
The notation A j hom (X) will be used to indicate the subgroups of homologically trivial cycles. For a morphism between smooth varieties f : X → Y , we will write Γ f ∈ A * (X × Y ) for the graph of f . The contravariant category of Chow motives (i.e., pure motives with respect to rational equivalence as in [17], [15]) will be denoted M rat .
We will write H * (X) := H * (X, Q) for singular cohomology with Q-coefficients.

THE FANO FOURFOLDS
Proposition 2.1. (i) Let X ⊂ P 3 × P 1 × P 1 be a smooth hypersurface of multidegree (2, 1, 1) (following [5], we will say X is "of type B1"). Then X is Fano, and the Hodge numbers of X are (2,4) × P 1 be a smooth hypersurface of multidegree (2, 1) with respect to the Plücker embedding (following [5], we will say X is "of type B2"). Then X is Fano, and the Hodge numbers of X are Proof. An easy way to determine the Hodge numbers is to use the following identification: Lemma 2.2. Let Z be a smooth projective variety of Picard number 1, and X ⊂ Z × P 1 a general hypersurface of bidegree (d, 1). Then X is isomorphic to the blow-up of Z with center S, where S ⊂ Z is a smooth dimensionally transversal intersection of 2 divisors of degree d.
Conversely, given a smooth dimensionally transversal intersection S ⊂ Z of 2 divisors of degree d, the blow-up of Z with center S is isomorphic to a smooth hypersurface X ⊂ Z × P 1 of bidegree (d, 1).
Proof. This is [5,Lemma 2.2]. The gist of the argument is that X determines a pencil of divisors in Z, of which S is the base locus. In terms of equations, if X is defined by y 0 f + y 1 g = 0 (where [y 0 : y 1 ] ∈ P 1 and f, g ∈ H 0 (Z, O Z (d))) then S is defined by f = g = 0. It follows that for X general (in the usual sense of "being parametrized by a Zariski open in the parameter space") the locus S is smooth.
In case (i), Z = P 3 × P 1 and S is a genus 7 K3 surface. In case (ii), Z = Gr(2, 4) (which is a quadric in P 5 ), and S is a genus 5 K3 surface. This readily gives the Hodge numbers.

MULTIPLICATIVE CHOW-KÜNNETH DECOMPOSITION
Definition 3.1 (Murre [14]). Let X be a smooth projective variety of dimension n. We say that X has a CK decomposition if there exists a decomposition of the diagonal such that the π i X are mutually orthogonal idempotents and the action of π i X on H j (X) is the identity for i = j and zero for i = j. Given a CK decomposition for X, we set (Here t π denotes the transpose of a cycle π.) (NB: "CK decomposition" is short-hand for "Chow-Künneth decomposition".) Remark 3.2. The existence of a Chow-Künneth decomposition for any smooth projective variety is part of Murre's conjectures [14], [15]. It is expected that for any X with a CK decomposition, one has A i (X) (j) ??
= 0. These are Murre's conjectures B and D, respectively.  [18]). Let X be a smooth projective variety of dimension n. Let ∆ sm X ∈ A 2n (X × X × X) be the class of the small diagonal In that case, A i (X) (j) := (π 2i−j X ) * A i (X) defines a bigraded ring structure on the Chow ring ; that is, the intersection product has the property that (For brevity, we will write MCK decomposition for "multiplicative Chow-Künneth decomposition".) Remark 3.4. The property of having an MCK decomposition is severely restrictive, and is closely related to Beauville's "(weak) splitting property" [1]. For more ample discussion, and examples of varieties admitting a MCK decomposition, we refer to [18,Chapter 8], as well as [20], [19], [7], [13]. There are the following useful general results: Proof. This is [18,Theorem 8.6], which shows more precisely that the product CK decomposition Proof. This is [19,Proposition 2.4]. (NB: in loc. cit., M and N are required to have a self-dual MCK decomposition; however, the self-duality is actually a redundant hypothesis, cf. Remark 3.5.) In a nutshell, the construction of loc. cit. is as follows. Given MCK decompositions π * M and π * N (of M resp. N), one defines where r + 1 is the codimension of N in M, and Ψ, Ψ −1 are certain explicit correspondences (this is [19,Equation (13)]). Then one checks that the π * M form an MCK decomposition.

MAIN RESULT
Theorem 4.1. Let X be a smooth fourfold of one of the following types: • a hypersurface of multidegree (2, 1, 1) in P 3 × P 1 × P 1 ; • a hypersurface of multidegree (2,1) in Gr(2, 4) × P 1 (with respect to the Plücker embedding). Then X has an MCK decomposition. Moreover, the Chern classes c j (T X ) are in A * (0) (X). Proof. The argument relies on the alternative description of the general X given by Lemma 2.2.
Step 1: We restrict to X sufficiently general, in the sense that X is a blow-up as in Lemma 2.2 with smooth center S.
To construct an MCK decomposition for X, we apply the general Proposition 3.7, with M being either P 3 × P 1 or Gr (2,4), and N being the K3 surface S ⊂ M determined by Lemma 2.2. All we need to do is to check that the assumptions of Proposition 3.7 are met with.
Assumption (1) . This is an MCK decomposition for S [18,Example 8.17]. Assumption (4) is trivially satisfied: one has A * hom (M) = 0 and so (because π j M acts as zero To check assumptions (2) and (3), we consider things family-wise. That is, we writē 4)), and we consider the universal complete intersection S →B .
We write B 0 ⊂B for the Zariski open parametrizing smooth dimensionally transversal intersections, and S → B 0 for the base change (so the fibres S b of S → B 0 are exactly the K3 surfaces that are the centers of the blow-up occurring in Lemma 2.2). We now make the following claim:

Then also
We argue that the claim implies that assumptions (2) and (3) of Proposition 3.7 are met with (and thus Proposition 3.7 can be applied to prove Theorem 4.1). Indeed, let p j : S × B 0 S → S, j = 1, 2, denote the two projections. We observe that defines a "relative MCK decomposition", in the sense that for any b ∈ B 0 , the restriction π i S | S b ×S b defines an MCK decomposition for S b which agrees with (2).
Let us now check that assumption (2) is satisfied. Since A 1 (S b ) = A 1 (0) (S b ), we only need to consider c 2 of the normal bundle. That is, we need to check that for any b ∈ B 0 there is vanishing But we can write (for the formalism of relative correspondences, cf. [15,Chapter 8]), and besides we know that ). Thus, Claim 4.2 implies the necessary vanishing (3).
Assumption (3) is checked similarly. Let ι b : S b → M and ι : S → M ×B denote the inclusion morphisms. To check assumption (3), we need to convince ourselves of the vanishing is homologically trivial for any ℓ = 8. Furthermore, we can write the cycle we are interested in as the restriction of a universal cycle: For any b ∈ B 0 , there is a commutative diagram where horizontal arrows are restriction to a fibre, and where vertical arrows are isomorphisms because M has trivial Chow groups. Claim 4.2 applied to the lower horizontal arrow shows the vanishing (4), and so assumption (3) holds. It is only left to prove the claim. Since A i hom (S b ) = 0 for i ≤ 1, the only non-trivial case is i = 2. Given Γ ∈ A 2 (S) as in the claim, letΓ ∈ A 2 (S) be a cycle restricting to Γ. We consider the two projectionsS Since any point of M imposes exactly one condition onB, the morphism π has the structure of a projective bundle. As such, anyΓ ∈ A 2 (S) can be written where a ℓ ∈ A 2−ℓ (M) and ξ ∈ A 1 (S) is the relative hyperplane class.
Let h := c 1 (OB(1)) ∈ A 1 (B). There is a relation where α ∈ Q and h 1 ∈ A 1 (M). As in [16, Proof of Lemma 1.1], one checks that α = 0 (if α were 0, we would have φ * (h dimB ) = π * (h dimB 1 ), which is absurd since dimB > 4 and so the right-hand side must be 0). Hence, there is a relation For any b ∈ B 0 , the restriction of φ * (h) to the fibre S b vanishes, and so it follows that is generated by intersections of divisors in case M = P 3 ×P 1 , and A 2 (M) is generated by divisors and c 2 of the tautological bundle in case M = Gr (2,4). In both cases, it follows thatΓ , (in the second case, this is proven as in [16, Proposition 2.1]). Given Γ ∈ A 2 (S) a cycle such that the fibrewise restriction has degree zero, this shows that the fibrewise restriction is zero in A 2 (S b ). Claim 4.2 is proven.
Step 2: It remains to extend to all smooth hypersurfaces as in the theorem. That is, let B ⊂B be the open such that the Fano fourfold X b (which is the blow-up of M with center S b ) is smooth. One has B ⊃ B 0 . Let X → B and X 0 → B 0 denote the universal families of Fano fourfolds over B resp. B 0 .
From step 1, one knows that X b has an MCK decomposition for any b ∈ B 0 . A closer look at the proof reveals more: the family X 0 → B 0 has a "universal MCK decomposition", in the sense that there exist relative correspondences π * X 0 ∈ A 4 (X 0 × B 0 X 0 ) such that for each b ∈ B 0 the restriction π * forms an MCK decomposition for X b . (To see this, one observes that Proposition 2.2 is "universal": given families M → B, N → B and universal MCK decompositions π * M , π * N , the result of (1) is a universal MCK decomposition for M → B.) A standard argument now allows to spread out the MCK property from B 0 to the larger base B. That is, we define π j X :=π j X 0 ∈ A 4 (X × B X ) (whereπ refers to the closure of a representative of π). The "spread lemma" [22,Lemma 3.2] (applied to X × B X ) gives that the π * X are a fibrewise CK decomposition, and the same spread lemma (applied to X × B X × B X ) gives that the π * X are a fibrewise MCK decomposition. This ends step 2. Remark 4.3. Claim 4.2 states that the families S → B 0 verify the "Franchetta property" as studied in [6]. It is worth mentioning that the Franchetta property for the universal K3 surface of genus g ≤ 10 (and for some other values of g) was already proven in [16]; the families considered in Claim 4.2 are different, however, so Claim 4.2 is not covered by [16] (e.g., in case M = P 3 × P 1 the K3 surfaces of Claim 4.2 have Picard number at least 2, so they correspond to a Noether-Lefschetz divisor in F 7 ).
As a corollary of Claim 4.2, the universal families X → B of Fano fourfolds of type B1 or B2 also verify the Franchetta property. (Indeed, in view of [22,Lemma 3.2] it suffices to prove this for X 0 → B 0 . In view of Lemma 2.2, X 0 can be constructed as the blow-up of M × B 0 with center S. This blow-up yields a relative correspondence from X 0 to S, inducing a commutative diagram where horizontal arrows are injective (by the blow-up formula). The Franchetta property for S → B 0 thus implies the Franchetta property for X 0 → B 0 .) (2) (X) = A 3 hom (X)).

An injectivity result.
Corollary 5.1. Let X and M be as in Theorem 4.1, and let m ∈ N. Let R * (X m ) ⊂ A * (X m ) be the Q-subalgebra (Here p i : X m → X and p ij : X m → X 2 denote projection to the ith factor, resp. to the ith and jth factor.) The cycle class map induces injections R j (X m ) ֒→ H 2j (X m ) in the following cases: (1) m = 1 and j arbitrary; (2) m = 2 and j ≥ 5; (3) m = 3 and j ≥ 9.
Proof. Theorem 4.1, in combination with Proposition 3.6, ensures that X m has an MCK decomposition, and so A * (X m ) has the structure of a bigraded ring under the intersection product. The corollary is now implied by the conjunction of the two following claims:

Claim 5.3. The cycle class map induces injections
provided m = 1, or m = 2 and j ≥ 5, or m = 3 and j ≥ 9.
To prove Claim 5.2, we note that A k hom (X) = 0 for k = 3, which readily implies the equality is a general fact for any X with a (necessarily self-dual) MCK decomposition [19,Lemma 1.4]. It remains to prove that codimension three cycles coming from the ambient space M are in A 3 (0) (X). To this end, we observe that such cycles are universally defined, i.e.
where X → B is the universal family as before, and X = X b 0 for some b 0 ∈ B. Given a ∈ A 3 (X ), applying the Franchetta property (Remark 4.3) to one finds that the restriction a| X ∈ A 3 (X) lives in A 3 (0) (X). In particular, it follows that as desired. Since the projections p i and p ij are pure of grade 0 [19, Corollary 1.6], and A * (0) (X m ) is a ring under the intersection product, this proves Claim 5.2.
To prove Claim 5.3, we observe that Manin's blow-up formula [17,Theorem 2.8] gives an isomorphism of motives Moreover, in view of Proposition 3.7 (cf. also [19,Proposition 2.4]), the correspondence inducing this isomorphism is of pure grade 0.
In particular, for any m ∈ N we have isomorphisms of Chow groups and this isomorphism respects the A * (0) () parts. Claim 5.3 now follows from the fact that for any surface S with an MCK decomposition, and any m ∈ N, the cycle class map induces injections (this is noted in [20,Introduction], cf. also [9, Proof of Lemma 2.20]).

5.2.
Decomposition in the derived category. Given a smooth projective morphism π : X → B, Deligne [3] has proven a decomposition in the derived category of sheaves of Q-vector spaces on B: As explained in [21], for both sides of this isomorphism there is a cup-product: on the right-hand side, this is the direct sum of the usual cup-products of local systems, while on the left-hand side, this is the derived cup-product (inducing the usual cup-product in cohomology). In general, the isomorphism (5) is not compatible with these cup-products, even after shrinking the base B (cf. [21]). In some rare cases, however, there is such a compatibility (after shrinking): this is the case for families of abelian varieties [4], and for families of K3 surfaces [21], [ Given a family X → B and m ∈ N, let us write X m/B for the m-fold fibre product Proof. The argument is the same as [21, Proposition 0.9]. First, one observes that divisors d i and codimension 2 cycles e j on X admit a cohomological decomposition (with respect to the Leray spectral sequence) in H 0 (B, R 2 π * Q) ⊕ π * H 2 (B, Q) ∼ = H 2 (X , Q) , e j = e j0 + π * (e j2 ) + π * (e j4 ) in H 0 (B, R 4 π * Q) ⊕ π * H 2 (B) ⊕2 ⊕ π * H 4 (B) ∼ = H 4 (X , Q) .
We claim that the cohomology classes d ik and e jk are algebraic. This claim implies the corollary: indeed, given a polynomial z = p(d i , e j ), one may take B ′ to be the complement of the support of the cycles d i2 , e j2 and e j4 . Then over the restricted base one has equality z := p(d i , e j ) = p(d i0 , e j0 ) in H 2r (X ′ ) m/B ′ , Q .