Voisin's Conjecture for Zero--cycles on Calabi--Yau Varieties and their Mirrors
aa r X i v : . [ m a t h . AG ] J un VOISIN’S CONJECTURE FOR ZERO–CYCLES ON CALABI–YAU VARIETIESAND THEIR MIRRORS
GILBERTO BINI, ROBERT LATERVEER, AND GIANLUCA PACIENZAA
BSTRACT . We study a conjecture, due to Voisin, on 0-cycles on varieties with p g = 1 . UsingKimura’s finite dimensional motives and recent results of Vial’s on the refined (Chow-)K¨unnethdecomposition, we provide a general criterion for Calabi-Yau manifolds of dimension at most toverify Voisin’s conjecture. We then check, using in most cases some cohomological computationson the mirror partners, that the criterion can be successfully applied to various examples in eachdimension up to .
1. I
NTRODUCTION
For X a smooth projective variety over C , let A j ( X ) denote the Chow groups of codimension j algebraic cycles on X modulo rational equivalence. Chow groups of cycles of codimension > are still mysterious. As an example, we recall the famous Bloch Conjecture, namely: Conjecture 1.1 (Bloch, [Blo80]) . Let X be a smooth projective complex variety of dimension n .The following are equivalent:(i) A n ( X ) ∼ = Q ;(ii) the Hodge numbers h j, ( X ) are for all j > . The implication from (i) to (ii) is actually a theorem [BS83]. The conjectural part is theimplication from (ii) to (i), which has been verified for surfaces not of general type in [BKS76],but it is wide open for surfaces of general type despite several significant cases have been dealtwith over the years. (see e.g. [Bar85, Voi93, BCGP12, Voi14a, PW15]).A natural next step is to consider varieties X with geometric genus p g = 1 . Here, the kernel A nAJ ( X ) of the Albanese map is huge; in a sense it is “infinite–dimensional” [Mum68] and[Voi02]. Yet, this huge group should have controlled behaviour on the self–product X × X ,according to a conjecture due to Voisin, which is motivated by the Bloch–Beilinson conjectures(see [Voi14b, Section 4.3.5.2] for a detailed discussion). Conjecture 1.2 ([Voi94], see [Voi14b] Conjecture 4.37 for this precise form) . Let X be a smoothprojective complex variety of dimension n with h j, ( X ) = 0 for < j < n . The following areequivalent:(i) For any zero–cycles a, a ′ ∈ A n ( X ) of degree zero, we have a × a ′ = ( − n a ′ × a in A n ( X × X ) . Mathematics Subject Classification.
Primary 14C15, 14C25, 14C30.
Key words and phrases.
Algebraic Cycles, Chow Groups, Motives, Finite–dimensional Motives, Calabi–YauVarieties. (Here a × a ′ is a short–hand for the cycle class ( p ) ∗ ( a ) · ( p ) ∗ ( a ′ ) ∈ A n ( X × X ) , where p , p denote projection on the first, resp. second factor.)(ii) the geometric genus p g ( X ) is ≤ . Again, the implication from (i) to (ii) is actually a theorem (this can be proven `a la Bloch–Srinivas [BS83], see Lemma 2.1 below). The conjectural part is the implication from (ii) to (i),which is still wide open for a general K surface (cf. [Voi94], [Lat16a], [Lat16b], [Lat16c],[Lat17] for some cases where this conjecture is verified).In the present article we present a general criterion to check Voisin’s conjecture (or a weakvariant of it, cf. Theorem 4.12) for specific varieties (see section 1 for all the relevant definitionsand explanations). Theorem (=theorem 4.1) . Let X be a smooth projective variety of dimension n ≤ with h i, ( X ) = 0 , < i < n and p g ( X ) = 1 . Assume moreover that:(i) X is rationally dominated by a variety X ′ of dimension n , and X ′ has finite–dimensionalmotive and B ( X ′ ) is true;(ii) X is e N -maximal.(iii) e N H i ( X ) = H i ( X ) , for < i < n .Then conjecture 1.2 is true for X , i.e. any a, a ′ ∈ A nhom ( X ) verify a × a ′ = ( − n a ′ × a in A n ( X × X ) . The proof of Theorem 4.1 relies, among other things, on results by Vial on the refined Chow-K¨unneth decomposition [Via13c], from which the hypotheses on X ′ are thus inherited.Our criterion can be effectively used to provide explicit examples in any dimension ≤ . Mostof them are given by hypersurfaces of Fermat type in a (weighted) projective space (see Section5 for all the examples).The first and third hypotheses of our criterion hold for any Fermat hypersurface. As for thesecond, it seems the most delicate to verify in pratice. In certain cases it is possible to check thesecond hypothesis by direct computation—e.g. for the Fermat sextic X in P , using results byBeauville, Movasati and the classical inductive structure of Fermat hypersurfaces (proposition5.10). Hence, we obtain the following explicit example: Corollary (=proposition 5.10) . Let X ⊂ P ( C ) be the sextic fourfold defined as x + · · · + x = 0 . Then conjecture 1.2 is true for X , i.e. any a, a ′ ∈ A hom ( X ) verify a × a ′ = a ′ × a in A ( X × X ) . In other cases (for instance for the Fermat quintic 3-fold), despite the fact that the dimension of H n ( X ) is quite large, it is possible to control the dimension of H ntr ( X ) by passing to the mirrorpartner of X , which can be explicitly described in the Fermat case. Among other examples, weobtain in this way the e N -maximality and therefore Voisin’s conjecture in the following case: Corollary (=proposition 5.8) . Let X ⊂ P (1 , be the Calabi–Yau threefold defined as x + x + x + x + x = 0 . OISIN’S CONJECTURE FOR ZERO–CYCLES ON CALABI–YAU VARIETIES AND THEIR MIRRORS 3
Then conjecture 1.2 is true for X , i.e. any a, a ′ ∈ A hom ( X ) verify a × a ′ = − a ′ × a in A ( X × X ) . Conventions.
In this note, the word variety will refer to a reduced irreducible scheme of finitetype over C . All Chow groups will be with rational coefficients:
For a variety X , we will write A j ( X ) forthe 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 ) and A jAJ ( X ) will be used to indicate the subgroups of homologi-cally, resp. Abel–Jacobi trivial cycles. The (contravariant) category of Chow motives (i.e., puremotives with respect to rational equivalence as in [Scho94], [MNP13]) will be denoted M rat .We will write H j ( X ) for singular cohomology H j ( X, Q ) .2. P RELIMINARIES
Warm-up.
We begin with the following result for which we could not find a reference inthe literature, although it may be well-known to experts.
Lemma 2.1.
Let X be a smooth projective complex variety of dimension n with h j, ( X ) = 0 for < j < n . Consider the following conditions:(i) For any zero–cycles a, a ′ ∈ A n ( X ) of degree zero, we have a × a ′ = ( − n a ′ × a in A n ( X × X ) . (Here a × a ′ is a short–hand for the cycle class ( p ) ∗ ( a ) · ( p ) ∗ ( a ′ ) ∈ A n ( X × X ) , where p , p denote projection on the first, resp. second factor.)(ii) the geometric genus p g ( X ) is ≤ .Then (i) implies (ii).Proof. This is a “decomposition of the diagonal” argument `a la Bloch-Srinivas: Let us define acorrespondence π := ∆ X − x × X − X × x ∈ A n ( X × X ) , where ∆ X denotes the diagonal and x ∈ X . Next, we consider the correspondence p := (∆ X − ( − n Γ ι ) ◦ ( π × π ) ∈ A n (cid:0) ( X × X ) × ( X × X ) (cid:1) , where ι is the involution on X × X switching the two factors.Hypothesis (i) implies that p acts trivially on –cycles of X × X , i.e. p ∗ A n ( X × X ) = 0 . The Bloch–Srinivas argument [BS83] then implies there exists a rational equivalence p = γ in A n (cid:0) ( X × X ) × ( X × X ) (cid:1) , where γ is a cycle supported on X × X × D , for some divisor D ⊂ X × X . It follows that ∧ H n ( X ) = p ∗ (cid:0) H n ( X ) ⊗ H n ( X ) (cid:1) ⊂ H n ( X × X ) GILBERTO BINI, ROBERT LATERVEER, AND GIANLUCA PACIENZA is supported on the divisor D . In particular, we see that ∧ H n, ( X, C ) ⊂ H n, ( X × X, C ) is (supported on a divisor and hence) zero. This proves (ii). (cid:3) Remark 2.2.
We have actually proven more than the implication from (i) to (ii). We have provena special instance of the generalized Hodge conjecture: for any variety X satisfying Lemma 2.1,the sub Hodge structure ∧ H n ( X ) ⊂ H n ( X × X ) is supported on a divisor. This implication was already observed by Voisin [Voi14b, Corollary3.5.1] . Finite–dimensional motives.
We refer to [Kim05], [And04], [Ivo11], [Jan07], [MNP13]for the definition of finite–dimensional motive. An essential property of varieties with finite–dimensional motive is embodied by the nilpotence theorem.
Theorem 2.3 (Kimura, Proposition 7.2, (ii), [Kim05]) . Let X be a smooth projective variety ofdimension n with finite–dimensional motive. Let Γ ∈ A n ( X × X ) Q be a correspondence whichis numerically trivial. Then there exists N ∈ N such that Γ ◦ N = 0 ∈ A n ( X × X ) Q . 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 [Jan07, Corollary 3.9]. Conjec-turally, any variety has finite–dimensional motive [Kim05]. We are still far from knowing this,but at least there are quite a few non–trivial examples. Remark 2.4.
The following varieties have finite–dimensional motive: varieties dominated byproducts of curves (which is the case of the Fermat hypersurfaces) and abelian varieties [Kim05], K surfaces with Picard number or [Ped12], surfaces not of general type with vanishinggeometric genus [GuP02, Theorem 2.11], Godeaux surfaces [GuP02], certain surfaces of generaltype with p g = 0 [Voi14a], [BF15],[PW15], Hilbert schemes of surfaces known to have finite–dimensional motive [dCM02], generalized Kummer varieties [Xu15, Remark 2.9(ii)], 3–foldswith nef tangent bundle [Iye08] (an alternative proof is given in [Via11, Example 3.16]), 4–folds with nef tangent bundle [Iye11], log–homogeneous varieties in the sense of [Bri07] (thisfollows from [Iye11, Theorem 4.4]), certain 3–folds of general type [Via15, Section 8], varietiesof dimension ≤ rationally dominated by products of curves [Via11, Example 3.15], varieties X with A iAJ ( X ) = 0 for all i [Via13b, Theorem 4], products of varieties with finite–dimensionalmotive [Kim05]. (cid:3) Remark 2.5.
It is a (somewhat embarrassing) fact that all examples known so far of finite-dimensional motives happen to be in the tensor subcategory generated by Chow motives of curves(i.e., they are “motives of abelian type” in the sense of [Via11]). That is, the finite–dimensionalityconjecture is still unknown for any motive not generated by curves (on the other hand, there existmany motives not generated by curves, cf. [Del72, 7.6]). (cid:3)
OISIN’S CONJECTURE FOR ZERO–CYCLES ON CALABI–YAU VARIETIES AND THEIR MIRRORS 5
Lefschetz standard conjecture and (co-)niveau filtration.
Let X be a smooth projectivevariety of dimension n , and h ∈ H ( X, Q ) the class of an ample line bundle. The hard Lefschetztheorem asserts that the map L n − i : H i ( X, Q ) → H n − i ( X, Q ) obtained by cupping with h n − i is an isomorphism, for any i < n . One of the standard conjec-tures, also known as Lefschetz standard conjecture B ( X ) , asserts that the inverse isomorphismis algebraic: Conjecture 2.6.
Given a smooth projective variety X , the class h ∈ H ( X, Q ) of an ample linebundle, and an integer ≤ i < n , the isomorphism ( L n − i ) − : H n − i ( X, Q ) ∼ = −→ H i ( X, Q ) is induced by a correspondence. We recall the following filtration which, via Proposition 3.3, will play a central rˆole in ourcriterion (Theorem 4.1) to check Conjecture 1.2.
Definition 2.7 (Coniveau filtration [BO74]) . Let X be a quasi-projective variety. The coniveaufiltration on cohomology and on homology is defined as N c H i ( X, Q ) = X Im (cid:0) H iY ( X, Q ) → H i ( X, Q ) (cid:1) ; N c H i ( X, Q ) = X Im (cid:0) H i ( Z, Q ) → H i ( X, Q ) (cid:1) , where Y (respectively Z ) runs over codimension ≥ c (resp. dimension ≤ i − c ) subvarieties of X , and H iY ( X, Q ) denotes the cohomology with support along Y . Remark 2.8.
It is known that B ( X ) holds for the following varieties: curves, surfaces, abelianvarieties [Klei68], [Klei94], threefolds not of general type [Tan11], hyperk¨ahler varieties of K [ n ] –type [CM13], n –dimensional varieties X which have A i ( X ) supported on a subvarietyof dimension i + 2 for all i ≤ n − [Via13a, Theorem 7.1], n –dimensional varieties X whichhave H i ( X ) = N x i y H i ( X ) for all i > n [Via13b, Theorem 4.2], products and hyperplane sec-tions of any of these [Klei68], [Klei94] (in particular it holds for projective hypersurfaces, a factthat we will use).For smooth projective varieties over C , the standard conjecture B ( X ) implies the standardconjecture D ( X ) , i.e homological and numerical equivalence coincide on X and X × X [Klei68],[Klei94]. (cid:3) Friedlander, and independently Vial, introduced the following variant of the coniveau filtra-tion:
Definition 2.9 (Niveau filtration [Fried95], [FM94] [Via13c]) . Let X be a smooth projectivevariety. The niveau filtration on homology is defined as e N j H i ( X ) = X Γ ∈ A i − j ( Z × X ) Im (cid:0) H i − j ( Z ) Γ ⋆ −→ H i ( X ) (cid:1) , GILBERTO BINI, ROBERT LATERVEER, AND GIANLUCA PACIENZA 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 e N c H i X := e N c − i + n H n − i X .
Remark 2.10.
In [Fried95], [FM94], the filtration e N ∗ is called the “correspondence filtration”rather than niveau filtration. (cid:3) The relation between the standard conjecture B ( X ) and the niveau and coniveau filtrations ismade clear in the following. Remark 2.11.
The niveau filtration is included in the coniveau filtration: e N j H i ( X ) ⊂ N j H i ( X ) . These two filtrations are expected to coincide; indeed, one can show the two filtrations coincideif and only if the Lefschetz standard conjecture is true for all varieties [Fried95, Proposition 4.2],[Via13c, Proposition 1.1].Using the truth of the Lefschetz standard conjecture in degree ≤ , it can be checked [Via13c,page 415 ”Properties”] that the two filtrations coincide in a certain range: e N j H i ( X ) = N j H i X for all j ≥ i − . In particular e N H ( X ) = N H ( X ) and e N H ( X ) = N H ( X ) . (cid:3) The following “refined K¨unneth decomposition” and “refined Chow–K¨unneth decomposition”are very useful:
Theorem 2.12 (Vial [Via13c]) . Let X be a smooth projective variety of dimension n ≤ . As-sume B ( X ) holds. There exists algebraic cycles π i,j on X × X and a decomposition of thediagonal ∆ X = X i,j π i,j in H n ( X × X ) , where the π i,j ’s are mutually orthogonal idempotents. The correspondence π i,j acts on H ∗ ( X ) as a projector on Gr j e N H i ( X ) . Moreover, π i,j can be chosen to factor over a variety of dimension i − j (i.e., for each π i,j there exists a smooth projective variety Z i,j of dimension i − j , andcorrespondences Γ i,j ∈ A n − j ( Z i,j × X ) , Ψ i,j ∈ A i − j ( X × Z i,j ) such that π i,j = Γ i,j ◦ Ψ i,j in H n ( X × X ) ).Proof. This is a special case of [Via13c, Theorem 1]. Indeed, as mentioned in loc. cit., varieties X of dimension ≤ such that B ( X ) holds verify condition (*) of loc. cit. (cid:3) Under the extra hypothesis of the finite–dimensionality of the motive the conclusion can beproved at the level of Chow groups.
Theorem 2.13 (Vial [Via13c]) . Let X be a smooth projective variety of dimension n ≤ . As-sume X has finite–dimensional motive and B ( X ) holds. There exists a decomposition of the OISIN’S CONJECTURE FOR ZERO–CYCLES ON CALABI–YAU VARIETIES AND THEIR MIRRORS 7 diagonal ∆ X = X i,j Π i,j in A n ( X × X ) , where the Π i,j ’s are mutually orthogonal idempotents lifting the π i,j of Theorem 2.12. Moreover, Π i,j can be chosen to factor over a variety of dimension i − j (i.e., for each Π i,j there existsa smooth projective variety Z i,j of dimension i − j , and correspondences Γ i,j ∈ A n − j ( Z i,j × X ) , Ψ i,j ∈ A i − j ( X × Z i,j ) such that Π i,j = Γ i,j ◦ Ψ i,j in A n ( X × X ) ).Proof. This is a special case of [Via13c, Theorem 2]. Indeed, X as in theorem 2.13 satisfiesconditions (*) and (**) of loc. cit. (cid:3) Remark 2.14.
Let X be as in Theorem 2.12. Notice that Conjecture B ( X ) implies in particularthat the π i,j are algebraic, cf. [Klei94, Theorem 4.1, item (3)]. (cid:3) Remark 2.15.
Let X be as in Theorem 2.13. Then, as in [Lat16b], one can define the “mosttranscendental part” of the motive of X by setting t n ( X ) := ( X, Π n, , ∈ M rat . The fact that t n ( X ) is well–defined up to isomorphism follows from [KMP07, Theorem 7.7.3]and [Via13c, Proposition 1.8]. In case n = 2 , t n ( X ) coincides with the “transcendental part” t ( X ) constructed for any surface in [KMP07]. (cid:3) e N – MAXIMAL VARIETIES
Let X be a smooth projective n -dimensional variety. Then H n ( X ) is a polarized Hodge struc-ture, and the niveau N := N H n ( X ) is a Hodge substructure. If X satisfies conjecture B ( X ) it follows from the Hodge–Riemann bilinear relations (cf. for instance [Voi14b, Theorem 2.22])that the Hodge substructure N of the polarized Hodge structure H n ( X, Q ) induces a splitting H n ( X, Q ) = N ⊕ ( N ) ⊥ (see [Via13c, Proposition 1.4 and Remark 1.5] for the details). Definition 3.1.
The “transcendental cohomology” is the orthogonal complement H ntr ( X ) := ( N ) ⊥ ⊂ H n ( X, Q ) . Remark 3.2.
Note that H ntr ( X ) is isomorphic to the graded piece Gr N • H n ( X ) (which is a priorionly a quotient of H n ( X ) ). (cid:3) One could also characterize H ntr ( X ) by saying it is the smallest Hodge substructure V ⊂ H n ( X, Q ) for which V C contains H n, . Proposition 3.3.
Let X be a smooth projective n-fold. The following are equivalent:(i) dim H ntr ( X ) = 2 p g ( X ) ;(ii) the subspace H n, ⊕ H ,n ⊂ H n ( X, C ) is defined over Q ;(iii) dim N H n ( X ) = P i,j> h i,j ( X ) ;(iv) the subspace ⊕ i,j> H i,j ⊂ H n ( X, C ) is defined over Q . GILBERTO BINI, ROBERT LATERVEER, AND GIANLUCA PACIENZA
Proof.
Obviously, (i) ⇔ (iii). The equivalence (ii) ⇔ (iv) is obtained using the polarization on H n ( X, Q ) . Indeed, suppose V ⊂ H n ( X, Q ) is a subspace such that V C = H n, ⊕ H ,n . Then V ⊂ H n ( X, Q ) is a Hodge substructure. As mentioned above, a Hodge substructure V of thepolarized Hodge structure H n ( X, Q ) induces a splitting H n ( X, Q ) = V ⊕ V ⊥ (cf. for instance [Voi14b, Theorem 2.22]). The subspace V ⊥ has ( V ⊥ ) C = ⊕ i,j H i,j . The rest isclear: (i) ⇒ (ii) because (i) forces (cid:0) H ntr ( X ) (cid:1) C (which always contains H n, ⊕ H ,n ) to be equal to H n, ⊕ H ,n . Similarly, (ii) ⇒ (i): if V ⊂ H n ( X, Q ) is such that V C = H n, ⊕ H ,n , then both V and H ntr ( X ) are the smallest Hodge substructure of H n ( X, Q ) containing H n, ; as such, they areequal. (cid:3) Definition 3.4.
A smooth projective n -dimensional variety verifying the equivalent conditions ofProposition 3.3 will be called N –maximal . Definition 3.5.
A smooth projective n –dimensional variety X will be called e N –maximal if it is N –maximal and there is equality N H n ( X ) = e N H n ( X ) . Remark 3.6.
Proposition 3.3 is inspired by [Beau14, Proposition 1], where a similar result isproven for surfaces. A surface with dim H tr ( S ) = 2 p g ( S ) is called a ρ –maximal surface.In dimension n ≤ , the notions of N –maximality and e N –maximality coincide, in view ofremark 2.11. (cid:3) Remark 3.7.
While looking for examples of N -maximal Calabi-Yau 3folds we realised that thenotion of N -maximality was already considered (under a different name) in [M98, Remarks, p.48, item 3)], via the characterization (ii) of Proposition 3.3.As a consequence of Proposition 3.3 we have the following nice property of N –maximal n -folds X : they verify a strong (i.e., non–amended) version of the generalized Hodge conjecture: H n ( X, Q ) ∩ F = N H n ( X, Q ) where F is the first piece of the Hodge filtration.4. A GENERAL RESULT
The following result gives sufficient conditions ensuring that a Calabi–Yau n-fold verifiesVoisin’s conjecture 1.2:
Theorem 4.1.
Let X be a smooth projective variety of dimension n ≤ with h i, ( X ) = 0 ,
Remark 4.2.
You may notice that all hypotheses are satisfied in dimension 1.Let ι : X × X → X × X denote the involution exchanging the two factors. We consider the correspondence Λ := 12 (∆ X × X + ( − n +1 Γ ι ) ∈ A n ( X ) , where ∆ X × X ⊂ X denotes the diagonal of ( X × X ) × ( X × X ) , and Γ ι denotes the graph ofthe involution ι . Notice that Λ is idempotent. To prove the Theorem 4.1 we must check that Λ ∗ Im (cid:16) A nhom ( X ) ⊗ A nhom ( X ) → A n ( X × X ) (cid:17) = 0 . We need to modify Λ a bit as follows.Let Ψ ∈ A n ( X ′ × X ) denote the closure of the graph of the dominant rational map ψ from X ′ to X . We know that(1) Ψ ∗ Ψ ∗ = d · id : A n ( X ) → A n ( X ) , where d is the degree of Ψ .Set Π n, := d Ψ ◦ Π X ′ n, ◦ t Ψ where Ψ is as above and Π X ′ n, is given by Vial’s result Theorem2.12, thanks to the finite dimensionality of the motive of X ′ plus B ( X ′ ) . Thanks to (1) combinedwith the idempotence of Π X ′ n, , we have(2) (Π n, ) ∗ ◦ (Π n, ) ∗ = (Π n, ) ∗ : A n ( X ) → A n ( X ) . Hence, up to dividing by a constant, we may assume that (Π n, ) acts as an idempotent on -cycleson X . We finally introduce the correspondence Λ tr := Λ ◦ (Π n, × Π n, ) ∈ A n ( X ) , where the Π n, are as above (see [Voi14b, Section 4.3.5.2] for a similar construction). Note that Λ tr depends on the choice of Π n, . The key point is the following: Claim 4.3. Λ tr acts as an idempotent on -cycles, i.e. (Λ tr ◦ Λ tr ) ∗ = (Λ tr ) ∗ : A ( X × X ) → A ( X × X ) . Proof of Claim 4.3.
Notice that Λ is an idempotent. Moreover by equation (2) also Π n, acts asan idempotent on -cycles. Write (Λ tr ◦ Λ tr ) ∗ := 14 [(∆ X × X + ( − n +1 Γ ι ) ◦ (Π n, × Π n, ) ◦ (∆ X × X + ( − n +1 Γ ι ) ◦ (Π n, × Π n, )] ∗ = [(Λ ◦ Λ) ◦ (Π n, × Π n, ) ◦ (Π n, × Π n, )] ∗ = Λ ∗ (Π n, × Π n, ) ∗ = (Λ tr ) ∗ where the second equality follows from the fact that Λ and Π n, commute (a fact that caneither be checked by hand, or deduced fro the commutativity between Γ ι and Π n, , which in turnfollows from [Kim05, Lemma 3.4]), while the third follows from equation (2). (cid:3) We will prove some intermediate results.
Lemma 4.4.
Set–up as in Theorem 4.1. The correspondence Λ tr acts on cohomology as a pro-jector on the subspace ∧ H ntr ( X ) ⊂ H n ( X × X ) . Proof.
First we observe that Π n, × Π n, acts as projector onto H ntr ( X ) ⊗ H ntr ( X ) . Next, for β, β ′ ∈ H ntr ( X ) we have (∆ X × X + Γ ι ) ∗ ( β ⊗ β ′ ) = β ⊗ β ′ + ( − n +1 β ′ ⊗ β ∈ H n ( X × X ) . This shows that an element in (Λ tr ) ∗ H ∗ ( X × X ) can be written as a sum of tensors of type β ⊗ β ′ + ( − n +1 β ′ ⊗ β , with β, β ′ ∈ H ntr ( X ) . Since the cup–product map H n ( X ) ⊗ H n ( X ) → H n ( X ) is ( − n -commutative, tensors of this type correspond exactly to elements (cid:8) b ∈ Im (cid:0) H ntr ( X ) ⊗ H ntr ( X ) → H n ( X × X ) (cid:1) | ι ∗ ( b ) = − b (cid:9) . Thus, (Λ tr ) ∗ H ∗ ( X × X ) ∼ = ∧ H ntr ( X ) ⊂ H n ( X × X ) . (cid:3) Remark 4.5.
Just to fix ideas, let us suppose for a moment that X and X ′ coincide, so that Π n, (and hence Λ tr ) is idempotent. In this case, Λ tr defines the Chow motiveSym t n ( X ) ∈ M rat in the language of [Kim05, Definition 3.5], where t n ( X ) is the “transcendental motive” ( X, Π n, , as in Remark 2.15. (cid:3) The next lemma ensures that Λ and Λ tr have the same action on the –cycles that we areinterested in. This is the only place in the proof where we need the full force of hypothesis (iii). Lemma 4.6.
Set–up as in Theorem 4.1. Let A ( n,n ) := Im (cid:16) A n ( X ) ⊗ A n ( X ) × −→ A n ( X × X ) (cid:17) ⊂ A n ( X × X ) and let A (2 , := Im (cid:16) A AJ ( X ) ⊗ A AJ ( X ) × −→ A ( X × X ) (cid:17) ⊂ A ( X × X ) (where × denotes the map sending a ⊗ a ′ to a × a ′ ).Then for any choice of Π n, as in Theorem 2.12, we have (Λ tr ) ∗| A ( n,n ) = Λ ∗| A ( n,n ) , and (Λ tr ) ∗| A (2 , = Λ ∗| A (2 , . OISIN’S CONJECTURE FOR ZERO–CYCLES ON CALABI–YAU VARIETIES AND THEIR MIRRORS 11
Proof.
The point is that according to Theorem 2.12, there is a decomposition ∆ X = Π n, + X ( i,j ) =( n, Π i,j in A n ( X × X ) . We claim that the components Π i,j with ( i, j ) = ( n, do not act on A n ( X ) : (Π i,j ) ∗ A n ( X ) = 0 for all ( i, j ) = ( n, . Indeed, Π i,j may be chosen to factor over a variety Z of dimension i − j (by Theorem 2.12).Hence, the action of Π i,j on A n ( X ) factors as follows: (Π i,j ) ∗ : A n ( X ) → A i − j ( Z ) → A j ( X ) , Now, our hypotheses imply that any Π i,j different from Π n, has j > . Thus, the group in themiddle is (for dimension reasons), and the claim is proven.We now consider the diagonal ∆ X × X of the self–product X × X . There is a decomposition ∆ X × X = X i,j,i ′ ,j ′ Π i,j × Π i ′ ,j ′ in A n ( X ) . Let a, a ′ ∈ A n ( X ) . Using the claim, we find that (Π i,j × Π i ′ ,j ′ ) ∗ ( a × a ′ ) = (Π i,j ) ∗ ( a ) × (Π i ′ ,j ′ ) ∗ ( a ′ ) = 0 for ( i, j, i ′ , j ′ ) = ( n, , n, . It follows that a × a ′ = (∆ X × X ) ∗ ( a × a ′ ) = (Π n, × Π n, ) ∗ ( a × a ′ ) in A n ( X × X ) , which proves the A ( n,n ) statement.The second statement of lemma 4.6 is proven similarly: we claim that the components Π i,j with ( i, j ) = ( n, do not act on A AJ ( X ) . This claim follows from the factorization (Π i,j ) ∗ : A AJ ( X ) → A i − j − nAJ ( Z ) → A ( X ) , where dim Z = i − j (one readily checks that for j > , the middle group vanishes in allcases). (cid:3) We now use the hypothesis that dim H ntr ( X ) = 2 and verify that the Hodge conjecture holdsfor the one–dimensional subspace ∧ H ntr ( X ) . Lemma 4.7.
Set–up as in Theorem 4.1. (i)
The subspace ( H ntr ( X ) ⊗ H ntr ( X )) ∩ F ⊂ H n ( X × X ) has dimension and is generatedby the cycle π n, ∈ A n ( X × X ) given by Theorem 2.12. (ii) ∧ H ntr ( X ) = Q [Π n, ] in H n ( X × X ) .Proof. Set V := H ntr ( X ) .(i) We first note that, thanks to the hypothesis of e N -maximality and Proposition 3.3, we have V C = H n, ⊕ H ,n . Hence ( V ⊗ V ) C = V C ⊗ V C ⊂ H n, ⊕ H n,n ⊕ H , n . It follows that ( V ⊗ V ) ∩ F = ( V ⊗ V ) ∩ F n . The complex vector space F n ( V C ⊗ V C ) = ( H ,n ( X ) ⊗ H n, ( X )) ⊕ ( H n, ( X ) ⊗ H ,n ( X )) is –dimensional, with generators c, d such that c = ¯ d . Let a ∈ ( V ⊗ V ) ∩ F n , i.e. a is such that the complexification a C ∈ H n ( X × X, C ) can be written a C = λc + µ ¯ c . But the class a C , coming from rational cohomology, is invariant under conjugation, so that λ = µ ,i.e. dim( V ⊗ V ) ∩ F n = 1 . Let π n, be the cycle given by Theorem 2.12. The class of π n, in H n ( X × X ) lies in V ⊗ V because π n, is a projector on V , i.e. V = ( π n, ) ∗ H ∗ ( X ) . As the class π n, ∈ H n ( X × X ) is non–zero (for otherwise H ntr ( X ) = 0 and p g ( X ) = 0 ), π n, generates the one–dimensionalsubspace ( V ⊗ V ) ∩ F n .(ii) Since p g ( X ) = 1 , we have ∧ H ntr ( X ) ⊂ H n ( X × X ) ∩ F . It follows that ∧ H ntr ( X ) = (cid:0) ∧ H ntr ( X ) (cid:1) ∩ F ⊂ (cid:0) H ntr ( X ) ⊗ H ntr ( X ) (cid:1) ∩ F . By item (i) we have that (cid:0) H ntr ( X ) ⊗ H ntr ( X ) (cid:1) ∩ F is one–dimensional with generator π n, andthe conclusion follows. (cid:3) We now have all the ingredients for the:
Proof of Theorem 4.1. (For a related conjecture, the argument that follows was hinted at in [Lat16b,Remark 35].)Consider the correspondence Λ tr ∈ A n ( X ) . By Lemma 4.4 it acts on H ∗ ( X × X ) byprojecting onto ∧ H ntr ( X ) ⊂ H n ( X × X ) . This implies there is a containment Λ tr ∈ (cid:0) ∧ H ntr ( X ) (cid:1) ⊗ (cid:0) ∧ H ntr ( X ) (cid:1) ⊂ H n ( X ) . By Lemma 4.7, the subspace ∧ H ntr ( X ) is one–dimensional and generated by a cycle Π n, ∈ A n ( X × X ) . It follows there is a codimension n subvariety P ⊂ X × X (the support of Π n, )such that Λ tr = γ in H n ( X ) , where γ is a cycle supported on P × P ⊂ X . In other words, we have Λ tr − γ ∈ A nhom ( X ) . Recall that Ψ ∈ A n ( X ′ × X ) denotes the closure of the graph of the dominant rational map ψ from X ′ to X . The correspondence Γ := ( t Ψ × t Ψ) ◦ (Λ tr − γ ) ◦ (Ψ × Ψ) ∈ A n (( X ′ ) ) OISIN’S CONJECTURE FOR ZERO–CYCLES ON CALABI–YAU VARIETIES AND THEIR MIRRORS 13 is homologically trivial (because the factor in the middle is homologically trivial). Using finite–dimensionality and Theorem 2.3, we know there exists N ∈ N such that Γ ◦ N = 0 in A n (( X ′ ) ) . In particular, this implies that (Ψ × Ψ) ◦ Γ ◦ N ◦ ( t Ψ × t Ψ) = 0 in A n ( X ) . Developing this expression, and applying the result to –cycles, and repeatedly using relation(1), we obtain (cid:0) (Λ tr ) ◦ N (cid:1) ∗ = (cid:0) Q + Q + · · · + Q N (cid:1) ∗ : A n ( X × X ) → A n ( X × X ) , where each Q j is a composition of Λ tr and γ in which γ occurs at least once. Since Λ tr is anidempotent, this simplifies to (Λ tr ) ∗ = (cid:0) Q + Q + · · · + Q N (cid:1) ∗ : A n ( X × X ) → A n ( X × X ) . The correspondence γ acts trivially on A n ( X × X ) for dimension reasons, and so the Q j likewiseact trivially on A n ( X × X ) . It follows that (Λ tr ) ∗ = (cid:0) Q + · · · + Q N (cid:1) ∗ = 0 : A n ( X × X ) → A n ( X × X ) . By Lemma 4.6 this ends the proof of Theorem 4.1. (cid:3)
Remark 4.8.
The above proof is somehow indirect as we are able to prove the statement for theauxiliary correspondence Λ tr , and then check that its action on A nhom ( X ) ⊗ A nhom ( X ) coincideswith that of Λ . (cid:3) Remark 4.9.
Hypothesis (i) of theorem 4.1 may be weakened as follows: it suffices that thereexists X ′ of dimension ≤ such that X ′ has finite–dimensional motive and B ( X ′ ) is true, andthere exists a correspondence from X ′ to X inducing a surjection A i ( X ′ ) ։ A ( X ) . The argument is similar. (cid:3)
Remark 4.10.
We have seen (Remark 3.6) that n-dimensional manifolds with dim H ntr ( X ) = 2 are a higher–dimensional analogue of ρ –maximal surfaces. In [Lat16a, Proposition 5], it isshown that surfaces S with finite–dimensional motive and dim H tr ( S ) = 2 (i.e. p g = 1 and S is ρ –maximal) verify Voisin’s conjecture. Theorem 4.1 is a higher–dimensional analogue of thisresult. (cid:3) Remark 4.11.
Following Voisin’s approach [Voi94] one can extend the analysis above to -cycles on higher products of X with itself. In this direction we get the following. Theorem 4.12.
Let X be a smooth projective variety of dimension n less than or equal to .Assume further that h i, ( X ) = 0 for < i < n and p g ( X ) ≤ . Suppose moreover that (1) X is rationally dominated by a variety X ′ , and X ′ has finite dimensional motive and B ( X ′ ) is true; (2) the dimension of H ntr ( X ) is at most ; (3) ˜ N H i ( X ) = H i ( X ) for < i < n . Then any a , a , a , a ∈ A nhom ( X ) verify X σ ∈ S ε ( σ ) σ ∗ ( a × a × a × a ) = 0 in A n ( X × X × X × X ) . Proof.
The proof closely follows that of Theorem 4.1. In that situation, we took into account Λ ( H ntr ( X )) and, after that, described a generator of it via an explicit cycle that is induced bya correspondence. In this situation, it is possible to give a generator of the -dimensional space Λ ( H ntr ( X )) . The rest of the proof is similar to that in Theorem 4.1. (cid:3) Conjecturally, any variety X with h , ( X ) = 0 should have A AJ ( X ) = 0 (this would followfrom the Bloch–Beilinson conjectures, or a strong form of Murre’s conjectures). We cannotprove this for any varieties with p g ( X ) > (such as the Fermat sextic fourfold). However, theabove argument at least gives a weaker statement concerning A AJ ( X ) : Proposition 4.13.
Let X be as in theorem 4.1. Then for any a, a ′ ∈ A AJ ( X ) , we have a × a ′ = − a ′ × a in A ( X × X ) . Proof.
This is really the same argument as theorem 4.1. We have proven there is a rationalequivalence Λ tr = (Λ tr ) ◦ N = Q + Q + · · · + Q N in A ( X ) , where each Q j is a composition of Λ tr and γ in which γ occurs at least once. The correspondence γ does not act on A ( X × X ) for dimension reasons (it factors over A ( P ) where dim P = 3 ),and so the Q j do not act on A ( X × X ) . It follows that (Λ tr ) ∗ = (cid:0) Q + · · · + Q N (cid:1) ∗ = 0 : A ( X × X ) → A ( X × X ) . On the other hand, we know from lemma 4.6 that Λ ∗ = (Λ tr ) ∗ = 0 : Im (cid:16) A AJ ( X ) ⊗ A AJ ( X ) → A ( X × X ) (cid:17) → A ( X × X ) . This means that for any a, a ′ ∈ A AJ ( X ) , we have Λ ∗ ( a × a ′ ) = a × a ′ + a ′ × a = 0 in A ( X × X ) . (cid:3)
5. A
PPLICATIONS
In this section we apply our general result to some Calabi-Yau varieties X of dimension inbetween and . First, we give new examples of ρ -maximal surfaces. Next, we focus on dimen-sion . Here we give examples of different types. In some cases we prove Voisin’s Conjecture asstated in (1.2); in other ones we get the generalization of it on X × X × X × X that appears inTheorem 4.12. Remarkably, one can often study the dimension of the H ntr ( F ) for a Fermat-typehypersurface F in a certain weighted projective spaces by looking at the (topological) mirrorof F . Finally, the conjecture is proved in dimension for the Fermat sextic fourfold and indimension . OISIN’S CONJECTURE FOR ZERO–CYCLES ON CALABI–YAU VARIETIES AND THEIR MIRRORS 15
Examples of Dimension .Remark 5.1. As noted in [Lat16a], examples of general type surfaces verifying the conditions ofTheorem 4.1 are contained in the work of Bonfanti [Bon15]. However, many more examples ofsurfaces verifying the conditions of Theorem 4.1 can be found in [BP16]. Indeed (as explained tous by Roberto Pignatelli), the “duals” (cf. [BP16, Section 9]) of the families in [BP16, Table2] are ρ –maximal surfaces with p g = 1 and q = 0 . Being rationally dominated by a product ofcurves, these surfaces have finite–dimensional motive.5.2. Examples of Dimension of Fermat type: weak version. Let us consider some exam-ples of Calabi-Yau 3folds. One of them is the Fermat quintic F in four dimensional projectivespace, which we work out in full details. We also consider other Fermat type 3folds in weightedprojective spaces (for the basics on weighted projective spaces see e.g. [Dol82]).A different example is taken in [NvG95] and is a small resolution Y ′ of a complete intersection Y of type (2 , , , in seven dimensional projective space. In the Fermat type examples, we aregoing to show that the dimension of H tr is ; in the latter example we do not know whether thedimension of H tr ( Y ′ ) is or . If it were , we could apply our main result and get anotherexample for which Voisin’s conjecture holds. If it is , as in the case of F , we can still deducesomething interesting, namely a weak version of Voisin’s conjecture thanks to Theorem 4.12.We start by collecting a useful fact. Lemma 5.2.
Every Fermat hypersurface { P x di = 0 } ⊂ P n has finite-dimensional motive.Proof. A Fermat hypersurface is rationally dominated by curves by the Katsura–Shioda inductivestructure [Shio79], [KS79, Section 1]. The analysis of the indeterminacy locus allows to show,cf. [GuP02], that this implies that its motive is finite-dimensional. (cid:3)
Consider now the Fermat quintic hypersurface X := { x + . . . + x = 0 } ⊂ P . (Later in the paper we will also denote the Fermat quintic hypersurface by F ). Its Hodge num-bers are h , ( X ) = 101 , h , ( X ) = 1 = h , ( X ) . Its “mirror” ˆ X has been constructed explicitely in [GrP90, CDGP91] as follows. Inside thequotient ( Z / Z ) / diag of ( Z / Z ) under the natural diagonal action, consider the subgroup G defined by the condition ( a , . . . , a ) ∈ G ⇐⇒ X i a i = 0 . The subgroup G , which is abstractly isomorphic to ( Z / Z ) , acts on X and, by [Mar87, Propo-sition 4] and [Roa89, Proposition 2] the quotient X/G possesses a Calabi-Yau resolution ˆ X , inother words we have the following diagram X ↓ p ˆ X f −→ X ′ := X/G .
Notice that the automorphisms σ ∈ G satisfy σ ∗ = id : H , ( X ) → H , ( X ) . The variety ˆ X turns out to be the mirror of X , see e.g. [Mor93, Voi96] for more explanationsand details (the analogous construction and the same result hold for any smooth member of theDwork pencil). In particular its Hodge numbers are h , ( X ) = 101 , h , ( X ) = 1 = h , ( X ) . First of all, as observed in Remark 2.8, X verifies B ( X ) (because it is a projective hypersur-face) and has finite–dimensional motive by Lemma 5.2.We note that X ′ is a quotient variety X/G for a finite group G . As such, there is a well–definedtheory of correspondences with rational coefficients for X ′ (this is because X ′ has A ∗ ( X ′ ) ∼ = A −∗ ( X ′ ) where A ∗ denotes Chow groups and A ∗ denotes operational Chow cohomology [Ful84,Example 17.4.10], [Ful84, Example 16.1.13]).Let us denote Γ := t Γ f ◦ Γ p ∈ A ( X × ˆ X ) the natural correspondence from X to ˆ X .Zero–cycles on X and ˆ X can be related as follows: Proposition 5.3.
There is an isomorphism of Chow motives
Γ : t ( X ) ∼ = t ( ˆ X ) in M rat (with inverse given by d t Γ , where d is the order of G ). In particular, the homomorphisms f ∗ p ∗ : A ( X ) −→ A ( ˆ X ) ,p ∗ f ∗ : A ( ˆ X ) −→ A ( X ) are isomorphisms.Proof. As we have seen, X satisfies B ( X ) and has finite–dimensional motive. Moreover, thegeneralized Hodge conjecture holds for X [Shio83]. The Proposition now follows from theproof of [Lat16b, Corollary 29(i)]. (cid:3) Thanks to Proposition 5.3, much information can be transported from X to ˆ X , and vice versa.For example, the fact that B ( X ) holds implies B ( ˆ X ) , because h ( ˆ X ) = t ( ˆ X ) ⊕ h ( C ) ⊕ M j L ( m j ) in M rat , where C is a (not necessarily connected) curve. Likewise, the fact that X has finite–dimensionalmotive implies that ˆ X has finite–dimensional motive.Alternatively, B ( ˆ X ) can be proven by invoking the main result of [Tan11], and the finite-dimensionality of the motive of ˆ X can also be derived from [Via11, Example 3.15] and the factthat ˆ X is rationally dominated by a product of curves (as X is). Lemma 5.4.
Let X be the Fermat quintic in P . Then the dimension of H tr ( X ) is . OISIN’S CONJECTURE FOR ZERO–CYCLES ON CALABI–YAU VARIETIES AND THEIR MIRRORS 17
Proof.
Take the order automorphism that permutes the coordinates of P . This descends to X and commutes with the elements of the group G of order . Therefore, there exists anorder automorphism of the mirror ˆ X acting on the four dimensional space of degree rationalcohomology. This space splits into four eigenspaces of such an automorphism, namely H ( ˆ X, Q ) = V ( η ) ⊕ V ( η ) ⊕ V ( η ) ⊕ V ( η ) , where η is a primitive fifth root of unity. Up to renaming the primitive root of unity, we canassume that H , ( ˆ X ) ⊕ H , ( ˆ X ) ≃ V ( η ) ⊕ V ( η ) , which is not defined over the field of rationalnumbers. Therefore, by Proposition 3.3 we have that dim H tr ( ˆ X ) = 4 . As the isomorphism ofHodge structures induced by Γ yields an isomorphism between H tr ( ˆ X ) and H tr ( X ) the Lemmais proved. (cid:3) Proposition 5.5.
The hypotheses of Theorem 4.12 hold for the following Calabi–Yau folds:(1) the Fermat quintic F and its mirror;(2) the Fermat hypersurface x + x + x + x + x = 0 in weighted projective space P (1 , and its mirror;(3) the Fermat hypersurface x + x + x + x + x = 0 in weighted projective space P (1 , , and its mirror;(4) the Fermat hypersurface x + x + x + x + x = 0 in weighted projective space P (1 , ) and its mirror;(5) the example Y ′ in [NvG95] .Proof. The claim follows for the Fermat quintic due to Lemma 5.4 and the fact that Fermathypersurfaces have finite dimensional motive by Lemma 5.2. For examples (2), (3), (4), noticethat they are dominated by Fermat hypersurfaces. The fact that dim H tr ( X ) ≤ is established in [KY08, Examples 5.3, (c), (d) and Table 4]. As for the mirror partners, one candirectly check that the hypotheses of Theorem 4.12 are verified.Let us describe briefly the example in [NvG95]. Take the (2 , , , complete intersection in P with homogeneous coordinates ( Y : Y : Y : Y : X : X : X : X ) given by Y = 2( X X + X X ) , Y = 2( X X + X X ) ,Y = 2( X X + X X ) , Y = 2( X X − X X ) . As proved in [NvG95], Proposition 2.8, the dimension of H ( X, Q ) is . Moreover, the Re-mark on page 69 loc. cit. shows that Y ′ has finite dimensional motive because there exists adominant rational map between C × E × E and Y , where E is the elliptic with complex mul-tiplication of order and C is a genus curve. In other words, Y ′ is dominated by curves, so ithas finite dimensional motive. (cid:3) Remark 5.6.
In [BvGK12] the authors show that there exist families of quintics in four di-mensional projective space such that their H , is isomorphic to that of the Fermat quintic: see[BvGK12], Section 3.3. Also, one of these examples has finite dimensional motive, namely: x x + x x + x + x + x = 0 because the Shioda-Katsura map shows that it is rationally dominated by a product S × C , where S is a Fermat quintic and C is some quintic curve. It follows that Theorem 4.12 also applies tothis quintic. Remark 5.7.
Notice that the N -maximality is also connected to modularity conditions. Forinstance, Hulek and Verrill in [HV06] investigate Calabi-Yau threefolds over the field of ratio-nal numbers that contain birational ruled elliptic surfaces S j for j = 1 , . . . , b , where b is thedimension of H , ( X ) . As they show, this is equivalent to the N -maximality. Under theseassumptions, the L -function of X factorizes as a product of the L -functions of the base ellipticcurves of the birational ruled surfaces and the L -function of the weight modular form associatedwith the -dimensional Galois representation given by the kernel U of the exact sequence: → U → H et ( X, Q l ) → ⊕ H et ( S j , Q l ) → . In [HV06], Section 3, examples of this type of Calabi–Yau varieties are given; however, wedo not know whether they have finite dimensional motive.5.3.
Example of Dimension of Fermat type: strong version. The main result of this para-graph is the following.
Proposition 5.8.
Let X be the hypersurface { [ x : x : x : x : x ] | x + x + x + x + x = 0 } in weighted projective space P (1 , , , , . Conjecture 1.2 holds for X .Proof. It is easy to check that X is a smooth Calabi-Yau variety. Moreover, it can be realized asa degree finite covering of P branched over the Fermat sextic surface. As such, X has an order automorphism, say τ . This also shows that is rationally dominated by a product of curves;hence it has finite dimensional motive. It remains to prove the N -maximality stated in Theorem4.1. This is proven in [M98, Section 8.3.1, Example 1], and also follows readily from [KY08,Example 5.3, (b)]; we propose a more direct proof:We observe that X can be thought of as the quotient of the degree Fermat threefold { Y + Y + Y + Y + Y = 0 } in four dimensional projective space by the action of the group generatedby the automorphism [ Y : Y : Y : Y : Y ] → [ Y : Y : Y : Y : − Y ] . The Hodge numbers of X are given by ( h , ( X ) , h , ( X )) = (1 , . As explained in [KY],the (topological) mirror of X can be described as follows. Take the group b G := (cid:8)(cid:0) ε i , ε i , ε i , ε i , ε i (cid:1) : i + i + i + i + 2 i ≡ mod (cid:9) /H, where H is a diagonal copy of Z / Z that acts trivially on weighted projective space P (1 , , , , . OISIN’S CONJECTURE FOR ZERO–CYCLES ON CALABI–YAU VARIETIES AND THEIR MIRRORS 19
Let us take into account the polynomials(3) X I =( i ,i ,i ,i ,i ) C I x i x i x i x i x i + λx x x x x where λ varies in A , the sum ranges over all solutions of the equation i + i + i + i + 2 i ≡ mod and C I are generic complex numbers. The vanishing of these polynomials define a pencilof varieties X ′ λ in P (1 , , , , that is b G -invariant. Notice that the members of it are smoothfor a generic choice of λ because they do not contain the singular point of weighted projectivespace. A mirror family of X can be found analogously to that of the mirror Fermat quintic bytaking the quotient of the pencil (3) by the group b G and, after that, by taking a crepant resolution.Let us denote by b X a crepant resolution of X ′ .Now, let us take into account the order four automorphism τ of projective space P (1 , , , , given by [ x , x , x , x , x ] → [ x , x , x , x , x ] . An easy computation shows that τ belongsto the normalizer of b G in the group of automorphisms of P (1 , , , , . Moreover, there existcomplex numbers C I such that X ′ is invariant with respect to τ . Finally, for such a choicethe fixed locus of b G is invariant with respect to the τ -action because τ normalizes b G . Since τ permutes the homogeneous coordinates of P (1 , , , , , it extends to all the members of themirror family, which by definition means that τ is maximal. Moreover, a direct computationshows that any λ is mapped to itself. The space of invariants of H , ( X ) with respect to the b G -action is thus one-dimensional; hence τ induces the identity on H , ( b X ) ⊕ H , ( b X ) . It remainsto understand the action induced by τ on H , ( b X ) ⊕ H , ( b X ) . For this purpose, we recall thata generator of H , ( b X ) is a -form on X that is invariant with respect to b G - recall that b X isa crepant resolution of X ′ = X/ b G . More precisely, this -form can be described as a ratio inwhich the denominator is b G -invariant by definition and the numerator is given as follows: x dx ∧ dx ∧ dx ∧ dx − x dx ∧ dx ∧ dx ∧ dx + x dx ∧ dx ∧ dx ∧ dx − x dx ∧ dx ∧ dx ∧ dx + 2 x dx ∧ dx ∧ dx ∧ dx . It is easy to check that this polynomial is mapped to its opposite by the induced action of τ .Therefore, the action on the group H , ( b X ) ⊕ H , ( b X ) is the opposite of the identity.To recap, the action of b τ on the space H ( ˆ X, Q ) induces a splitting into two eigenspacesof dimension two, one with eigenvalue +1 and one with eigenvalue − . This shows the N -maximality for the Calabi-Yau threefold b X and accordingly, for X because their H tr ’s are iso-morphic via an isomorphism of Hodge structures. (cid:3) Remark 5.9.
This example is not new; yet the proof of the N -maximality is more geometricthan those in [KY08] and [M98]. In the former reference, the authors prove the maximality bydescribing two Fermat motives.5.4. Examples of Dimension of Borcea-Voisin type: strong version. Let E be the ellipticcurve given by the equation y = x − . This curve admits an order three automorphism h ( x, y ) = ( ωx, y ) , where ω is a primitive third root of unit. Now, take S to be a K surface withan order three automorphism g such that the second cohomology group with rational coefficients splits as the trannscendental T S and the Neron Severi group such that H , ( S ) ⊆ T S and therank of N S ( S ) is . Moreover, the Neron Severi group coincides with the subspace of invariantclasses of H ( S, Q ) with respect to the action of g . In particular g is antisymplectic. Such a K3surface exists as shown in [BvGK12, p. 280].The product S × E admits the order three automorphism g × h . Assume that the action of g on the period of S is given by multiplication by ω (if not, just take the inverse of g ). Notice thatthe fixed point locus of g consists of isolated points and (smooth) rational curves.Denote by X a resolution of the (singular) quotient S × E by the group generated by theautomorphism g × h . By the description of the fixed locus of g × h , the third cohomology groupof X with rational coefficients is the invariant part of H ( S × E, Q ) , which is isomorphic to H ( S, Q ) ⊗ H ( E, Q ) . To prove the N -maximality, we check the equivalent condition that H , ( S × E ) ⊕ H , ( S × E ) is defined over the field of rational numbers. By K¨unneth formula,we have H , ( S × E ) ⊕ H , ( S × E ) ≃ H , ( S ) ⊗ H , ( E ) ⊕ H , ( S ) ⊗ H , ( E ) ,H , ( S × E ) ⊕ H , ( S × E ) ≃ H , ( S ) ⊗ H , ( E ) ⊕ H , ( S ) ⊗ H , ( E ) ⊕ H , ( S ) ⊗ H , ( E ) ⊕ H , ( S ) ⊕ H , ( E ) . The space H , ( S × E ) ⊕ H , ( S × E ) is defined over the rational field because it can bedefined as the subspace of invariants with respect to the action of the isomorphism ( g × h ) ∗ on H ( S × E, Q ) . Indeed, the action of this isomorphism is trivial on H , ( S ) ⊗ H , ( E ) ⊕ H , ( S ) ⊗ H , ( E ) . As for H , ( S × E ) ⊕ H , ( S × E ) , the action is by multiplication by ω, ω , ω , ω on H , ( S ) ⊗ H , ( E ) , H , ( S ) ⊗ H , ( E ) , H , ( S ) ⊗ H , ( E ) , H , ( S ) ⊕ H , ( E ) respectively, because the action of g on H , ( S ) is trivial.5.5. The Fermat 4fold: strong version.
We already know that every Fermat hypersurface { P x di = 0 } ⊂ P n has finite-dimensional motive.As Lefschetz standard conjecture holds for hypersurfaces and the hypothesis e N H ( X ) = H ( X ) also holds for a 4-dimensional hypersurface, in order to prove Theorem 4.1 we are leftwith the e N -maximality. Proposition 5.10.
The Fermat sextic fourfold is e N -maximal.Proof. We will use that(a) The Fermat sextic surface S ⊂ P is ρ -maximal (cf. [Beau14, Corollary 1].(b) The Fermat sextic 4-fold X ⊂ P is e N -maximal, i.e. e N H ( X ) ⊗ C = H , ( X ) (cf.[Mov, Corollary 15.11.1]).Consider the dominant rational morphism φ : S × S X It yields a surjective morphism of Hodge structures:(4) φ ∗ : H tr ( S × S ) ։ H tr ( X ) . Now H tr ( S × S ) ⊂ H tr ( S ) × H tr ( S ) . By item (a) above H tr ( S ) ⊗ C = H , ( S ) ⊕ H , ( S ) . OISIN’S CONJECTURE FOR ZERO–CYCLES ON CALABI–YAU VARIETIES AND THEIR MIRRORS 21
This, together with (4), implies that ( H tr ( X ) ⊗ C ) ⊂ H , ( X ) ⊕ H , ( X ) ⊕ H , ( X ) . By item (b) we see that there exists a non–empty Zariski open τ : U ⊂ X (defined as the com-plement of the span of the codimension cycle classes in H ( X, Q ) ) such that H , ( X ) maps to under the restriction map τ ∗ : H ( X, C ) → H ( U, C ) . This implies that τ ∗ (cid:0) H tr ( X ) ⊗ C (cid:1) ⊂ τ ∗ (cid:0) H , ( X ) ⊕ H , ( X ) (cid:1) , and so the restriction τ ∗ (cid:0) H tr ( X ) ⊗ C (cid:1) has dimension at most . On the other hand, by definitionof H tr () , we have that τ ∗ : H tr ( X ) → H ( U ) is an injection. Therefore, we conclude that dim (cid:0) H tr ( X ) ⊗ C (cid:1) = 2 , i.e. X is N –maximal.To establish the e N –maximality, it remains to show that the inclusion e N H ( X ) ⊂ N H ( X ) is an equality. Here, we again use the dominant rational map φ . The indeterminacy of the map φ is resolved by the blow–up ^ S × S with center C × C (where C ⊂ S is a curve). It thus sufficesto prove equality e N H ( ^ S × S ) = N H ( ^ S × S ) . The blow–up formula gives an isomorphism H ( ^ S × S ) = H ( S × S ) ⊕ H ( C × C ) , and the second summand is entirely contained in e N . It thus suffices to prove equality(5) e N H ( S × S ) ?? = N H ( S × S ) . This readily follows from the N –maximality of S : indeed, there is a decomposition H ( S ) = T ⊕ N , where T := H tr ( S ) is such that T ⊗ C = H , ⊕ H , . This induces a decomposition H ( S × S ) = T ⊗ T ⊕ N ⊗ T ⊕ T ⊗ N ⊕ N ⊗ N ⊕ H ( S ) ⊗ H ( S ) ⊕ H ( S ) ⊗ H ( S ) . All but the first summand are obviously contained in e N (because D × S satisfies the standardconjecture B , for any divisor D ⊂ S ). As for the first summand, we note that ( T ⊗ T ) C ⊂ H , ⊕ H , ⊕ H , , and so N ( T ⊗ T ) = ( T ⊗ T ) ∩ F = N ( T ⊗ T ) = e N ( T ⊗ T ) , since the Hodge conjecture is true for S × S [Shio79, Theorem IV]. This proves equality (5), andso the e N –maximality of X is established. (cid:3) To finish we observe that all the hypotheses of Theorem 4.1 are satisfied for a Fermat sexticfourfold, hence Conjecture 1.2 holds for it.5.6.
Examples of Dimension .Proposition 5.11 (Cynk–Hulek [CH07]) . Let E be an elliptic curve with an order automor-phism, and let n be a positive integer. There exists a Calabi–Yau variety X of dimension n , whichis rationally dominated by E n , and which has dim H n ( X ) = 2 if n is even, and dim H ntr ( X ) = 2 if n is odd.Proof. This is [CH07, Theorem 3.3]. The construction is also explained in [HKS06, section5.3]. (cid:3)
Proposition 5.12.
Let X be a Calabi–Yau variety as in proposition 5.11, of dimension n ≤ .Then conjecture 1.2 is true for X .Proof. We check all conditions of Theorem 4.1 are satisfied. Point (i) is obvious, as X is ratio-nally dominated by a product of curves. Point (ii) is taken care of by Proposition 5.11. Point (iii)is proven (in a more general set–up) in [Lat16d, Proof of Corollary 4.1]. (cid:3)
6. Q
UESTIONS
Question 6.1.
Let F d denote the Calabi–Yau Fermat hypersurface of degree d in P d − , i.e. x d + x d + · · · + x dd − = 0 . The variety F d is e N –maximal for d = 4 and for d = 6 . Are these the only two values of d forwhich F d is e N –maximal ?We suspect this might be the case (by analogy with the ρ –maximality of Fermat surfaces in P :as remarked in [Beau14] , the only ρ –maximal Fermat surfaces are in degree and ), but wehave no proof. Question 6.2.
Let { X λ } denote the Dwork pencil of Calabi–Yau quintic threefolds x + x + · · · + x + λx x x x x = 0 . As we have seen, the central fibre X has dim H tr ( X ) = 4 . Are there values of λ where dim H tr ( X λ ) drops to ? Are these values dense in P ?Also, can one somehow prove finite–dimensionality of the motive for non–zero values of λ ?(This seems difficult: as noted in [KY08, Remark 4.3] , the varieties X λ are not dominated by aproduct of curves outside of λ = 0 .) Acknowledgements.
We wish to thank Lie Fu, Bert van Geemen, Hossein Movasati, RobertoPignatelli and Charles Vial for useful and stimulating exchanges related to this paper.
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