On the universal CH

DOI 10.4171/JEMS/702

Claire Voisin

On the universal CH

group of cubic hypersurfaces

Received July 27, 2014

Abstract. We study the existence of a Chow-theoretic decomposition of the diagonal of a smooth cubic hypersurface, or equivalently, the universal triviality of its CH₀ group. We prove that for odd-dimensional cubic hypersurfaces or for cubic fourfolds, this is equivalent to the existence of a cohomological decomposition of the diagonal, and we translate geometrically this last condition.

For cubic threefoldsX, this turns out to be equivalent to the algebraicity of the minimal classθ⁴/4! of the intermediate JacobianJ (X). In dimension 4, we show that a special cubic fourfold with discriminant not divisible by 4 has universally trivial CH₀group.

Keywords. Cubic hypersurfaces, decomposition of the diagonal, stable rationality

1. Introduction

LetXbe a smooth rationally connected projective variety overC. Then CH₀(X) = Z, as all points ofX are rationally equivalent. However, ifLis a field containingC, e.g.

a function field, the group CH₀(X_L)can be different fromZ. As explained in [4], the group CH₀(X_L)is equal toZfor any fieldLcontainingCif and only if, forL=C(X), the diagonal (or generic) pointδ_Lis rationally equivalent overLto a constant pointx_Lfor some (in fact any) pointx∈X(C). Following [4], we will then say thatXhas universally trivialCH0group. Observe that, on the other hand, the equality

δ_L=x_L in CH₀(X_L)=CHⁿ(X_L), n=dimX,

is, by the localization exact sequence applied to Zariski open sets ofX×Xof the form U×X, equivalent to the vanishing in CHⁿ(U ×X)of the restriction of1X−X×x, whereUis a sufficiently small dense Zariski open subset ofXand1X ⊂X×Xis the diagonal ofX. This provides a Bloch–Srinivas decomposition of the diagonal

1_X=X×x+Z in CHⁿ(X×X), (1)

whereZ is supported onD×Xfor some proper closed subsetDofX. As in [29], we will call an equality (1) aChow-theoretic decomposition of the diagonal. So, having uni- versally trivial CH₀group is equivalent to admitting a Chow-theoretic decomposition of C. Voisin: Coll`ege de France, 3 rue d’Ulm, 75005 Paris, France; e-mail: [email protected] Mathematics Subject Classification (2010):14C15, 14E08, 14J70

the diagonal, but the second viewpoint is much more geometric, and leads to the study of weakened properties, like the existence of acohomological decomposition of the diago- nal, which is the cohomological counterpart of (1), studied for threefolds in [29]:

[1_X] = [X×x] + [Z] inH²ⁿ(X×X,Z), (2) whereZis supported onD×Xfor some proper closed algebraic subsetDofX. In these notions, integral coefficients are essential in order to make the property restrictive, as the existence of decompositions as above with rational coefficients already follows from the assumption that CH₀(X)=Z(see [7]).

Projective space has universally trivial CH₀group. It follows that rational or stably rational varieties admit a Chow-theoretic (and a fortiori cohomological) decomposition of the diagonal. More generally, if X is a unirational variety admitting a unirational parametrizationPⁿ99KXof degreeN, then there is a decomposition

N 1_X=N (X×x)+Z in CHⁿ(X×X), withZsupported onD×Xfor someD(X.

Note, however, that the existence of a decomposition of the diagonal is certainly not a sufficient condition for stable rationality, as there are surfaces of general type (hence very far from being rational or stably rational) which admit a Chow-theoretic decomposition of the diagonal (see Corollary2.2). It could also be the case that a smooth projective va- rietyXadmits unirational parametrizations of coprime degreesNi without being stably rational (although we do not know such examples). Nevertheless, the existence of a de- composition of the diagonal is a rather strong condition, and there are now a number of unirational examples where the non-existence provides an obstruction to rationality or stable rationality:

1) Examples of rationally connected varieties with no cohomological decomposition of the diagonal include varieties with non-trivial Artin–Mumford invariant (this is the tor- sion inH³(X,Z)or the second unramified cohomology group with torsion coefficients), or non-trivial third unramified cohomology groupH_nr³(X,Q/Z)(see [11] for examples), as both of these groups have to be 0 whenXhas a cohomological decomposition of the diagonal (see [29]).

2) Examples of rationally connected varieties with no Chow-theoretic decomposi- tion of the diagonal include varieties with non-trivial unramified cohomology groups H_nrⁱ(X,Q/Z)withi ≥ 4, as these groups have to be 0 when X has a Chow-theoretic decomposition of the diagonal (see [12]). We refer to [22] for such examples and to [33]

for the cycle-theoretic interpretation of this group in degree 4.

3) Furthermore, we proved in [28] that the non-existence of a decomposition of the diagonal is a criterion for (stable) irrationality which is actually stronger than those given by the non-triviality of unramified cohomology: For example, we show inloc. cit.that very general smooth quartic double solids do not admit a Chow-theoretic or even a co- homological decomposition of the diagonal, while their unramified cohomology vanishes in all positive degrees. We prove similar results for very general nodal quartic double solids with k ≤ 7 nodes, and in the case of very general double solids with exactly

seven nodes, we prove in [28] that the non-existence of a cohomological decomposition of the diagonal is equivalent to the non-existence of a universal codimension 2 cycle Z ∈ CH²(J (X)×X), where J (X)is the intermediate Jacobian of X. (J (X) is also known to be isomorphic to the group CH²(X)_homof codimension 2 cycles homologous to 0 onX.) Thus, in this case, the study of the decomposition of the diagonal led to the discovery of new stable birational invariants which are non-trivial for some unirational varieties.

The purpose of this paper is to investigate the existence of a decomposition of the diagonal for cubic hypersurfaces. Motivations for this study are the following problems:

1) It is well-known that 3-dimensional smooth cubics are irrational (see [10]), but they are not known not to be stably rational.

2) In dimension 4, some cubics are known to be rational and it is a famous open prob- lem to prove that there exist irrational cubic fourfolds. Some precise conjectures concern- ing the rationality of cubic fourfolds have been formulated and compared (see [15], [18], [2], [1]). All these conjectures develop the idea that if a cubic fourfold is rational, it is related in some way (Hodge-theoretic, categorical) to aK3 surface. An interesting com- putation has recently been made by Galkin and Shinder [13], who prove that a rational cubic fourfold has to satisfy the property that its variety of lines is birational to Hilb²(S) for someK3 surfaceS, unless a certain explicitly constructed non-zero element in the Grothendieck ring of complex varieties is annihilated by the class ofA¹, that is, provides a counterexample to the cancellation conjecture for the Grothendieck ring. (Note that such counterexamples are now known to exist by the work of Borisov [9].)

A priori, the existence of a cohomological decomposition is much weaker than the existence of a Chow-theoretic one. Our first result, which is unconditional for cubic four- folds and for odd-dimensional cubics, is the following:

Theorem 1.1. LetX be a smooth cubic hypersurface. Assume H^∗(X,Z)/H^∗(X,Z)_alg has no 2-torsion (this holds for example if dimX is odd ordimX ≤ 4, or X is very general of any dimension). ThenXadmits a Chow-theoretic decomposition of the diago- nal(equivalently,CH0(X)is universally trivial)if and only if it admits a cohomological decomposition of the diagonal.

HereH^∗(X,Z)alg ⊂ H^∗(X,Z)is the subgroup of classes of algebraic cycles. For any odd degree and odd dimension smooth hypersurface in projective space, the quotient groupH^∗(X,Z)/H^∗(X,Z)_alghas no 2-torsion. For cubic hypersurfaces, the first example where we do not know if the assumption is satisfied is 6-dimensional cubics and degree 6 integral cohomology classes on them.

The proof of Theorem 1.1 uses the fact that Hilb²(X) is birationally a projective bundle overX, a property which is also crucially used in the recent paper [13].

One consequence of this result, established in Section5, concerns the following no- tion:

Definition 1.2. (i) If Y ⊂ X is a closed algebraic subset of a variety defined over a fieldK, we say that CH₀(Y )→CH₀(X)isuniversally surjectiveif CH₀(Y_L)→CH₀(X_L) is surjective for any fieldLcontainingK.

(ii) Theessential CH0-dimensionof a variety X is the minimal integerk such that there exists a closed algebraic subset Y ⊂ X of dimension k such that CH0(Y ) → CH₀(X)is universally surjective.

Theorem 1.3. The essentialCH₀-dimension of a very generaln-dimensional cubic hy- persurface overCis eithernor0.

More precisely, Theorem 5.2 proves the result above for a smooth cubic hyper- surface of dimension n with no 2-torsion in Hⁿ(X,Z)/Hⁿ(X,Z)_alg and such that End_HSHⁿ(X,Q)_prim =Q. In dimension 4, we get further precise consequences, for ex- ample we prove that a cubic fourfold which is special in the sense of Hassett [15], with discriminant not divisible by 4, has universally trivial CH₀group.

The rest of the paper focuses on the cohomological decomposition of the diagonal.

We first investigate the existence of a cohomological decomposition of the diagonal for varieties whose non-algebraic cohomology is supported in middle degree, like complete intersections. We prove the following result:

Theorem 1.4. LetXbe a smooth projective variety such thatH^∗(X,Z)has no torsion.

Assume thatH²ⁱ(X,Z)is generated overZby algebraic cycles for2i 6= n = dimX.

ThenXadmits a cohomological decomposition of the diagonal if and only if there exist varietiesZiof dimensionn−2, correspondences0i ∈CHⁿ⁻¹(Zi×X), and integersni

with the property that forα, β ∈Hⁿ(X,Z), X

i

n_ih0_i^∗α, 0_i^∗βi_Z

i = hα, βi_X. (3)

Note that the condition (3) presents obvious similarities to the one considered in [23] by Shen, who studied the case of cubic fourfolds. It is however weaker in several respects:

the integersn_i do not need to be positive, and the correspondences0_i do not need to factor through the variety of lines. Finally, the condition formulated by Shen is only con- jecturally a necessary condition for rationality, while our condition is actually a necessary condition for the triviality of the universal CH0group, hence a fortiori for (stable) ratio- nality.

Remark 1.5. Concerning the second assumption in Theorem1.4, it is satisfied by uni- ruled threefolds by [31], but it is not clear that it is satisfied by Fano complete intersections in any dimension. The groupH²ⁱ(X,Z),2i6=n, is equal toZby Lefschetz hyperplane restriction theorem, but Koll´ar [17] exhibits examples of hypersurfaces where this group is not generated by an algebraic class for 2i > n. It is not known if such Fano examples can be constructed.

The case of rationally connected threefoldsXis also particularly interesting. In this case, we complete the results of [29] by proving the following result. LetJ (X)be the interme- diate Jacobian ofX. It is isomorphic as a group to CH²(X)_homvia the Abel–Jacobi map.

It is canonically a principally polarized abelian variety, the polarization being determined by the intersection pairing onH³(X,Z)/torsion∼=H¹(J (X),Z). Letθ ∈H²(J (X),Z) be the class of the Theta divisor ofJ (X).

Theorem 1.6 (see also Theorem4.1). LetXbe a rationally connected threefold. ThenX admits a cohomological decomposition of the diagonal if and only if the following three conditions are satisfied:

(i) H³(X,Z)has no torsion.

(ii) There exists a universal codimension2cycle inX×J (X).

(iii) The minimal classθ^g−1/(g−1)!onJ (X),dimJ (X)=g, is algebraic, that is, the class of a1-cycle inJ (X).

The main new result in this theorem is the fact that condition (iii) above is implied by the existence of a cohomological decomposition of the diagonal. In particular, it is a necessary condition for stable rationality. This can be seen as a variant of the Clemens–

Griffiths criterion which can be stated as saying that a necessary criterion for rationality of a threefold is that the minimal classθ^g−1/(g−1)!onJ (X)be the class of aneffective curve inJ (X).

Note that examples of unirational threefolds not satisfying (i) were constructed by Artin and Mumford [3], and examples of unirational threefolds not satisfying (ii) were constructed in [28]. It is not known if examples not satisfying (iii) exist. More generally, it is not known if there exists any principally polarized abelian variety(A, 2)such that the minimal classθ^g−1/(g−1)!is not algebraic onA, whereg = dimA. Notice that for many Fano threefolds, the intermediate JacobianJ (X)is a Prym variety, so the class 2θ^g−1(g−1)!is known to be algebraic. In the case of cubic threefolds, the algebraicity ofθ⁴/4!is a classical completely open problem. Combining the theorems above, we get in this case:

Theorem 1.7. LetX be a smooth cubic threefold. Then X has universally trivialCH0

group if and only if the classθ⁴/4!onJ (X)is algebraic. This happens(at least)on a countable union of closed subvarieties of codimension≤3of the moduli space ofX.

The paper is organized as follows: Theorem1.1is proved in Section2; Theorem1.4is proved in Section3; and Theorem1.6is proved in Section4. In Section5, we come back to the case of cubic hypersurfaces, where we prove Theorem5.2and establish further results, particularly in dimension 4.

2. Chow-theoretic and cohomological decomposition of the diagonal

In this section we prove Theorem1.1. In the case of cubic hypersurfaces of dimension

≤4, a shorter proof will be given, which uses the following result of independent interest:

Proposition 2.1. LetXbe a smooth projective variety. IfXadmits a decomposition of the diagonal modulo algebraic equivalence, that is,

1X−X×x=Z inCH(X×X)/alg, (4) withZ supported onD×X for some proper closed algebraic subsetD ( X, then X admits a Chow-theoretic decomposition of the diagonal.

Proof. We use the fact proved in [26], [27] that cycles algebraically equivalent to 0 are nilpotent for the composition of self-correspondences. We write (4) as

1_X−X×x−Z=0 in CH(X×X)/alg (5)

and apply the nilpotence result mentioned above. This provides

(1_X−X×x−Z)^◦N=0 in CH(X×X) (6)

for some largeN. As1X−X×xis a projector andZ◦(X×x)=0, this gives

1_X−X×x−W◦Z =0 in CH(X×X), (7)

for some cycleW onX×X. AsW ◦Zis supported onD×X, for someD (X, this

concludes the proof. ut

Corollary 2.2. LetSbe a surface of general type withCH₀(S)=ZandTors(H^∗(S,Z))

=0(for example the Barlow surface[5]). ThenShas universally trivialCH₀group.

Proof. (See [4] for a different proof.) Aspg(S)=q(S)=0 by Mumford’s theorem [21], the cohomologyH^∗(S,Z)is generated by classes of algebraic cycles. AsH^∗(S,Z)has no torsion, the cohomology ofS×Sadmits a K¨unneth decomposition with integral co- efficients, so that we can write the class of the diagonal ofSas

[1_S] =X

i

[α_i] ⊗ [β_i] inH⁴(S×S,Z), (8) whereα_i,β_i are algebraic cycles onSwith dimα_i +dimβ_i =2. Clearly,[α_i] ⊗ [β_i] = [α_i×β_i]is supported overD×SwithD(Swhen dimα_i <2, so that (8) provides in fact a cohomological decomposition of1_S:

[1S] = [S×s] + [Z] inH⁴(S×S,Z), (9) whereZis a cycle supported overD×S, for someD(S. Next, as CH₀(S×S)=Z, codimension 2 cycles onS×Swhich are cohomologous to 0 are algebraically equivalent to 0 by [7]. Thus the cycle0:=1_S−S×s−Zis algebraically equivalent to 0 onS×S.

The surfaceSthus admits a decomposition of the diagonal modulo algebraic equivalence,

and we then apply Proposition2.1. ut

For a smooth projective varietyX, we denote byX^[2]the second punctual Hilbert scheme ofX. It is smooth, obtained as the quotient of the blow-upX^×XofX×Xalong the diagonal by its natural involution. Letµ : X×X 99K X^[2]be the natural rational map andr:X^×X→X^[2]be the quotient morphism. We start with the following result:

Lemma 2.3. LetX be a smooth projective variety of dimensionn. Then there exists a codimensionncycleZinX^[2]such thatµ^∗Z=1_XinCHⁿ(X×X).

Proof. Let E1 be the exceptional divisor over the diagonal of the blow-up map τ : X^×X → X ×X. The key point is that there is a (non-effective) divisor δ on X^[2]

such thatr^∗δ=E₁. It follows that

r^∗δⁿ=E₁ⁿ in CHⁿ(X^×X).

Now we use the fact that

τ∗((−1)ⁿ⁻¹E₁ⁿ)=1X in CHⁿ(X×X),

which givesµ^∗((−1)ⁿ⁻¹δⁿ)=τ∗(r^∗((−1)ⁿ⁻¹δⁿ))=1_Xin CHⁿ(X×X). ut Corollary 2.4. Any symmetric codimension ncycle onX×Xis rationally equivalent toµ^∗0 for a codimensionncycle0onX^[2].

Proof. Indeed, we can write Z = Z₁+Z₂whereZ₁ is a combination of irreducible subvarieties ofX×X invariant under the involutioniofX×X, andZ₂is of the form Z₂⁰ +i(Z₂⁰), where the diagonal does not appear inZ₂⁰. WriteZ1 = n11_X+Z₁⁰, with Z₁⁰ =P

jnjZ⁰_1,j, theZ_j⁰ being invariant underibut different from1X. ThenZ_1,j⁰ is the inverse image of a subvarietyZ_1,j⁰⁰ ofX⁽²⁾. LetZg⁰⁰_1,jbe the proper transform ofZ_1,j⁰⁰ under the Hilbert–Chow mapX^[2]→X⁽²⁾. Then clearlyµ^∗(Zg⁰⁰_1,j)=Z_1,j⁰ in CHⁿ(X×X), so

µ^∗X

j

n_jZg⁰⁰_1,j

=Z₁⁰ in CHⁿ(X×X).

Next, letZ₂⁰ be the image ofZ₂⁰ inX^[2]byµ. Then clearly

µ^∗(Z⁰₂)=Z₂⁰ +i(Z₂⁰)=Z2 in CHⁿ(X×X).

Thus

Z=n₁1_X+Z⁰₁+Z₂=n₁1_X+µ^∗X

j

n_jZg⁰⁰_1,j

+µ^∗(Z₂⁰).

Finally, we use Lemma2.3to conclude. ut

We next have:

Lemma 2.5. SupposeXadmits a cohomological decomposition of the diagonal

[1X−x×X] = [Z] inH²ⁿ(X×X,Z), (10) whereZis a cycle supported onD×Xfor some proper closed algebraic subsetDofX.

ThenXadmits a cohomological decomposition of the diagonal

[1X−x×X−X×x] = [W] inH²ⁿ(X×X,Z), (11) whereWis a cycle supported onD×Xfor someD(X, andWis a symmetric cycle.

Proof. Let us denote by ^t0 the image of a cycle0 under the involutioni of X ×X.

Formula (10) gives as well

[1X−X×x] = [^tZ] inH²ⁿ(X×X,Z), and

[(1_X−X×x)◦(1_X−x×X)] = [^tZ◦Z] inH²ⁿ(X×X,Z). (12) To conclude, we observe that the cycle ^tZ ◦Z is not only supported onD ×X and invariant under the involution, but it even satisfies the stronger property of being rationally equivalent to a symmetric cycle inX ×X. This last fact is indeed clear if there is no excess formula in the intersection product defining^tZ◦Z; next it is easy to check that, by choosing representatives forZsupported inD×Xbut in general position otherwise, the intersection ofp₁₂^∗ Zandp₂₃^∗(^tZ)inX×X×Xis a proper intersection. Finally, the left-hand side in (12) is equal to[1X−X×x−x×X](we assume heren >0), which

concludes the proof. ut

The following proposition is a key point in our proof of Theorem1.1.

Proposition 2.6. Let X be a smooth odd degree complete intersection in pro- jective space. If X admits a cohomological decomposition of the diagonal, and H^2∗(X,Z)/H^2∗(X,Z)_alghas no2-torsion, then there exists a cycle0∈CHⁿ(X^[2])with the following properties:

(i) µ^∗0=1_X−x×X−X×x−WinCHⁿ(X×X)withW supported overD×X, for some proper closed algebraic subsetD(X.

(ii) [0] =0inH²ⁿ(X^[2],Z).

The proof of this proposition will use a few lemmas concerning the cohomology ofX^[2]

whenXis the projective space or a smooth complete intersection in projective space. We give here an elementary and purely algebraic proof. These results can be obtained as well as an application of [8] or [20] (cf. [25]), but those papers are written in a topologist’s language and it is not obvious how to translate them into the concrete statements be- low. With a better understanding of those papers, our arguments would presumably prove Proposition2.6for a smooth projective varietyXsuch thatH^∗(X,Z)is torsion free and H^2∗(X,Z)/H^2∗(X,Z)_alghas no 2-torsion.

LetXbe a smooth projective variety and let

jE,X:E1,X ,→X^[2], iE,X:E1,X →X^×X, τE,X:E1,X →X

be respectively the inclusion of the exceptional divisor over the diagonal inX^[2], its in- clusion inX^×X, and its natural morphism toX. Letδ ∈PicX^[2]be the natural divisor such that 2δ = E. Thenδ_E := δ_|E1,X is the line bundleO_E1,X(−1)of the projective bundleτ_E,X:E_1,X∼=P(T_X)→X.

Lemma 2.7. Assume thatX =Pⁿand leth =c1(O_Pn(1)). For any cohomology class a∈H^∗(E1,Pⁿ,Z), one has(jE,Pⁿ)∗a∈2H^∗+2((Pⁿ)^[2],Z)if and only if

a= X

i,m≥0

α_i,mδⁱ_E·τ_E,^∗

Pⁿh^m·(τ_E,^∗

Pⁿh−δ_E)^mmod 2H^∗(E_1,Pn,Z), (13) where theαi,mare integers.

Proof. To see that the condition is sufficient (and also to get a nice interpretation of the condition), we observe that for any smooth subvariety6 ⊂ Pⁿ of codimensionm, we have the inclusion 6^[2] ⊂ (Pⁿ)^[2] and the classb := [δ·6^[2]] ∈ H^4m+2((Pⁿ)^[2],Z) satisfies 2b= [E_1,Pⁿ·6^[2]], so that

2b=(j_E,Pⁿ)∗([6^[2]]_|E

1,Pn) inH^4m+2((Pⁿ)^[2],Z).

Next we observe that6^[2]∩E_1,Pⁿ is equal toP(T₆) ⊂ P(T_Pⁿ). Take now for 6 a P^n−m ⊂ Pⁿ. Then the class ofP(T₆) ⊂P(T_Pⁿ)isτ_E,^∗

Pⁿh^m·(τ_E,^∗

Pⁿh−δ_E)^m. We have thus proved that(j_E,Pⁿ)∗(τ_E,^∗

Pⁿh^m·(τ_E,^∗

Pⁿh−δ_E)^m)is divisible by 2 inH^∗((Pⁿ)^[2],Z), and hence so is the class

(j_E,Pn)∗(δⁱ_E·τ_E,^∗

Pⁿh^m·(τ_E,^∗

Pⁿh−δ_E)^m)=δⁱ·(j_E,Pn)∗(τ_E,^∗

Pⁿh^m·(τ_E,^∗

Pⁿh−δ_E)^m) for anyi≥0.

In the other direction, it is better to see(Pⁿ)^[2]as aP²-bundle over the Grassmannian G(2, n+1), namely, if π₁ : P → G(2, n+1)is the universalP¹-bundle, it is clear that(Pⁿ)^[2]is isomorphic to the second symmetric productπ₂:P₂→G(2, n+1)ofP overG(2, n+1). Furthermore,E_1,Pⁿ ⊂(Pⁿ)^[2]identifies with the Veronese embedding P ⊂ P₂. Write P = P(E)with polarization H = τ_E,^∗

Pⁿh; thenP₂ = P(S²E)with polarizationH₂, and clearly

j_E,^∗

PⁿH₂=2H inH²(P ,Z) (14) sincejE,Pⁿ :P →P2is the Veronese embedding. The cohomology ofP decomposes as

H^∗(P ,Z)=π₁^∗H^∗(G(2, n+1),Z)⊕H·π₁^∗H^∗−2(G(2, n+1),Z).

We claim that modulo 2, the set of classesaas in (13) is exactlyπ₁^∗H^∗(G(2, n+1),Z/2Z).

Letl =c1(E)andc=c2(E)be the two generators ofH^∗(G(2, n+1),Z). It is easy to check thatδ=H2−π₂^∗linH²(P2,Z). Restricting this equality toP, by (14) we get

δ_E =π₁^∗lmod 2H²(P ,Z). (15)

Finally, we haveπ₁^∗c=H·(π₁^∗l−H )inH^∗(P ,Z), hence by (15) we get π₁^∗c=τ_E,^∗

Pⁿh·(τ_E,^∗

Pⁿh−δE)mod 2H^∗(P ,Z), (16) which together with (15) proves the claim. Having this, the fact that condition (13) is sufficient tells us that

(j_E,Pⁿ)∗◦π₁^∗(H^∗(G(2, n+1),Z))⊂2H^∗(P₂,Z),

and in order to prove that it is necessary, we need to prove that the set of classesz ∈ H^∗(P ,Z)such that(jE,Pⁿ)∗z ∈2H^∗(P2,Z)is equal toπ₁^∗(H^∗(G(2, n+1),Z))mod- ulo 2H^∗(P ,Z). Equivalently, we have to prove that ifz ∈ H^∗(G(2, n+1),Z) sat- isfies(j_E,Pn)∗(H ·π₁^∗z) ∈ 2H^∗(P₂,Z), then z ∈ 2H^∗(G(2, n+1),Z). But for any α∈H^∗(G(2, n+1),Z), we have

hα, zi_G(2,n+1)= hπ₁^∗α, H·π₁^∗zi_P = hπ₂^∗α, (jE,Pⁿ)∗(H ·π₁^∗z)i_P

2,

which implies thathα, zi_G(2,n+1)is even since(j_E,Pⁿ)∗(H·π₁^∗z)is divisible by 2. Hence zis divisible by 2 by Poincar´e duality onG(2, n+1). ut Lemma 2.8. LetX⊂P^Nbe a smooth odd degree complete intersection of dimensionn.

(i) For any integerm≥0, the class(j_E,X)∗(τ_E,X^∗ h^m·(τ_E,X^∗ h−δ_E)^m)∈H^4m+2(X^[2],Z) is equal to2δ· [6^[2]_l ], where6_m ⊂Xis the smooth proper intersection ofXwith a linear space of codimensionminP^N.

(ii) For any integral cohomology class a ∈ H²ⁿ⁻²(E1,X,Z), one has (jE,X)∗a ∈ 2H²ⁿ(X^[2],Z)if and only if

a = X

i,m≥0, i+2m=n−1

αi,mδ_Eⁱ ·τ_E,X^∗ h^m·(δE−τ_E,X^∗ h)^mmod 2H^∗(E1,X,Z), (17) where theα_i,mare integers.

Proof. (i) This has already been proved in the case of P^N and follows from the fact that τ_E,X^∗ h^m ·(τ_E,X^∗ h−δ_E)^m ∈ H^4m(E_1,X,Z)is the class of P(T_6m) = E_1,6m in P(T_X)=E_1,X.

(ii) In the case where X ,→^jX P^N has odd dimension, observe that the mapj_X∗ : H^∗(X,Z/2) → H^∗+2k(P^N,Z/2),k := N −n, is injective sinceX has odd degree.

It follows as well that if we denote by ˜_X : P(T_X) → P(T_PN)the natural map, then

˜_X∗:H^∗(P(T_X),Z/2)→H^∗+4k(P(T_PN),Z/2)is also injective. Now, let a∈H²ⁿ⁻²(E_1,X,Z)=H²ⁿ⁻²(P(T_X),Z)

be such that (jE,X)∗a ∈ 2H²ⁿ(X^[2],Z). Then ˜X∗a ∈ H^∗(E_1,PN,Z) satisfies (j_E,PN)∗(˜X∗a) ∈ 2H^∗((P^N)^[2],Z). Thus we conclude by Lemma 2.7 that the class

˜_X∗amod 2 belongs to the subgroup ofH^∗(E_1,PN,Z/2)generated by theδⁱ_E·τ_E,^∗

Pⁿh^m· (δE−τ_E,^∗

Pⁿh)^mwithi, m≥0. It easily follows thatamod 2 belongs to the subgroup of H^∗(E_1,X,Z/2)generated by theδ_Eⁱ ·τ_E,X^∗ h^m·(δ_E−τ_E,X^∗ h)^mwithi, l ≥ 0, since the class ofE_1,XinE_1,PN is equal modulo 2 toτ_E,^∗

Pⁿh^N−n·(δ_E−τ_E,^∗

Pⁿh)^N−n.

IfXhas even dimension, then the mapsjX∗:Hⁿ(X,Z/2)→ H^n+2k(P^N,Z/2)and

˜X∗ : H²ⁿ⁻²(P(TX),Z/2)→ H^2n−2+4k(P(T_Pⁿ),Z/2)are no longer injective, but their kernels are equal respectively toHⁿ(X,Z/2)_primandδ^n/2−1_E ·τ_E,X^∗ Hⁿ(X,Z/2)_prim. The proof above shows that ifa∈H²ⁿ⁻²(E_1,X,Z)satisfies(j_E,X)∗a∈2H²ⁿ(X^[2],Z), then a∈δ_E^n/2−1·τ_E,X^∗ Hⁿ(X,Z/2)_primmodulo the subgroup ofH^∗(E_1,X,Z/2)generated by

theδⁱ_E·τ_E,X^∗ h^m·(δE −τ_E,X^∗ h)^m withi, m ≥ 0. By (i), it thus suffices to prove that a classa∈δ_E^n/2−1·τ_E,X^∗ Hⁿ(X,Z/2)_primsatisfying(j_E,X)∗a =0 inH²ⁿ(X^[2],Z/2)must be 0. This is proved as follows: By a monodromy argument, either any classa∈δ_E^n/2−1· τ_E,X^∗ Hⁿ(X,Z/2)primsatisfies this property, or no non-zero class satisfies it. Assume that the first possibility occurs. Then we get a contradiction as follows: Let

a =δ^n/2−1_E ·τ_E,X^∗ α, b=δ_E^n/2−1·τ_E,X^∗ β∈H²ⁿ⁻²(E_1,X,Z), withα, β ∈Hⁿ(X,Z)_prim. Then

hj_1,X∗a, j_1,X∗bi

X^[2] = −2hα, βi_X. (18)

If both (j1,X)∗α and (j1,X)∗β are divisible by 2 in H²ⁿ(X^[2],Z), then h(j1,X)∗a, (j1,X)∗bi_X[2] is divisible by 4, hencehα, βi_Xis divisible by 2 by (18). Thus, under our assumption, the intersection pairing modulo 2 would be identically 0 onHⁿ(X,Z/2)_prim. AsXhas odd degree, the intersection pairing is non-degenerate onHⁿ(X,Z/2)_prim, and

we get a contradiction. ut

Proof of Proposition2.6. LetX be an odd degree complete intersection in P^N which admits a cohomological decomposition of the diagonal. We know by Lemma 2.5that there is a symmetric cycleWsupported onD×X,D(X, such that[1X−x×X−X×x]

= [W]inH²ⁿ(X×X). Corollary2.4provides a cycle0₀∈CHⁿ(X^[2])such thatµ^∗0₀= 1_X−x×X−X×x−Win CHⁿ(X×X), which is property (i). It remains to see that we can modify0₀keeping property (i) and imposing condition (ii), namely

[00] =0 inH²ⁿ(X^[2],Z).

We know thatµ^∗[00] = 0 inH²ⁿ(X×X,Z), which implies thatr^∗[00]vanishes in H²ⁿ(X^×X\E₁,Z). Thus there is a cohomology classβ ∈H²ⁿ⁻²(E₁,Z)such that

i_E∗β =r^∗[0₀] inH²ⁿ(X^×X,Z),

where we come back to the notationiE:E1→X^×X,jE :E1→X^[2]for the natural inclusions of the exceptional divisor over the diagonal ofX. This implies thatjE∗β is divisible by 2 inH²ⁿ(X^[2],Z). Indeed, we have

jE∗β=r∗(iE∗β)=r∗(r^∗([00]))=2[00] inH²ⁿ(X^[2],Z). (19) According to Lemma2.8(ii), one then has

β = X

i≥0, i+2m=n−1

α_iδ_Eⁱ (h−δ_E)^mh^m+2γ inH²ⁿ⁻²(E₁,Z), (20) which by Lemma2.8(i) gives

jE∗β=2 X

i≥0, i+2m=n−1

αiδⁱ⁺¹[6_m^[2]]

inH²ⁿ(X^[2],Z)modjE∗(2H^∗(E1,Z)) (21) for some integersα_i. By (19), we thus have

2 X

i≥0, i+2m=n−1

α_iδⁱ⁺¹[6^[2]_m] +j_E∗(γ )

=2[0₀] inH²ⁿ(X^[2],Z) (22)

for some integral homology classγ ∈H²ⁿ⁻²(E,Z). It is proved in [25, Theorem 2.2] that the cohomology ofX^[2]has no 2-torsion whenXis an odd degree complete intersection in projective space, so (22) gives

X

i≥0, i+2m=n−1

α_iδⁱ⁺¹[6^[2]_m] +j_E∗(γ )= [0₀] inH²ⁿ(X^[2],Z). (23)

We now observe thatβ is algebraic inH²ⁿ⁻²(E₁,Z), which easily follows from the fact thatiE∗βis algebraic inH²ⁿ(X^×X,Z)by recalling thatX^×X,Zis simply the blow- up ofX×X along its diagonal. We thus conclude from (20) that 2γ is algebraic, and by our assumption thatH^2∗(X,Z)/H^2∗(X,Z)alghas no 2-torsion,γ is algebraic, that is, γ = [0⁰]for some0⁰∈CHⁿ⁻¹(E₁). Thus we have

X

i≥0, i+2m=n−1

α_iδⁱ⁺¹[6_m^[2]] +j_E∗[0⁰] = [0₀] inH²ⁿ(X^[2],Z). (24)

We have µ^∗δⁿ = ±[1X]and, for any z ∈ CHⁿ⁻¹(E1),µ^∗(jE∗z) = N 1X for some N ∈ Z. Asµ^∗[00] = 0, we can thus assume, up to modifying0⁰but without changing µ^∗00, that in formula (24),αn−1=0 and0⁰satisfiesτE∗0⁰=0, henceµ^∗(jE∗(0⁰))=0 in CHⁿ(X×X). Next, form >0,µ^∗δⁱ⁺¹[6^[2]_m]is supported overD×Xfor some proper closed algebraic subsetD⊂X. It follows that the cycle

0:=0₀− X

i≥0, i+2m=n−1

α_iδⁱ⁺¹[6_m^[2]] −j_E∗(0⁰)

is cohomologous to 0 onX^[2], and satisfiesµ^∗0=µ^∗0₀+Z⁰whereZ⁰∈CHⁿ(X×X)

is supported overD×X. ut

We now consider the case whereXis a smooth cubic hypersurface inPⁿ⁺¹. We then have the following description ofX^[2], which is also used in [13]. We denote below byF (X) the variety of lines ofX. Let

P = {([l], x):x ∈l, l⊂X}

be the universalP¹-bundle, with first projectionp : P → F (X), and letP2 → F (X) be theP²-bundle defined as the symmetric product ofP overF (X). There is a natural embeddingP2 ⊂X^[2]which maps each fiber ofP →F (X), which is the second sym- metric product of a line inX, isomorphically onto the set of subschemes of length 2 ofX contained in this line. Letp_X:P_X→Xbe the projective bundle with fiber overx ∈X the set of lines inPⁿ⁺¹passing throughx. Note thatP is naturally contained inP_X, as one sees by considering the second projectionq:P →X.

Proposition 2.9. (i) The rational map8:X^[2]99KP_Xwhich to an unordered pair of pointsx, y ∈Xnot contained in a common line ofXassociates the pair([l_x,y], z), wherel_x,yis the line inPⁿ⁺¹generated byxandy, andz∈Xis the residual point of the intersectionl_x,y∩X, is desingularized by the blow-up ofP₂inX^[2].

(ii) The induced morphisme8 : Xg^[2] → P_XidentifiesXg^[2] with the blow-upPf_XofP inP_X.

(iii) The exceptional divisors of the two mapsXg^[2]→X^[2]andPf_X→P_Xare identified by the isomorphisme8⁰:Xg^[2]∼=Pf_Xof (ii).

Proof. (i) There is a morphism from X^[2] to the GrassmannianG(1, n+1) of lines inPⁿ⁺¹, which tox +y associates the linehx, yi. Letπ : Q→ X^[2] be the pull-back of the naturalP¹-bundle onG(1, n+1). Letα : Q → Pⁿ⁺¹be the natural map. Then α⁻¹(X)is a reducible divisor inQ, which is generically of degree 3 overX^[2], with one componentD₁which is finite of degree 2 overX^[2], parametrizing the pairs(x, x+y), and a second componentDwhich is of degree 1 overX^[2]and parametrizes generically the pairs(z, x+y). The divisorDis isomorphic to X^[2] away from P₂. OverP₂, the restrictedP¹-bundleQ_P2 is contained inα⁻¹(X)but not inD₁, so it is contained inD.

We claim that D is smooth and identifies with the blow-up of X^[2] along P2. In- deed, this simply follows from the fact that the divisorDis the zero-set of a sectionsof O_Q(3)(−D1)onQ. The line bundleO_Q(3)(−D1)onQhas degree 1 along the fibers of Q→X^[2], soR⁰π∗O_Q(3)(−D1)is a rank 2 vector bundleEonX^[2]such thatQ=P(E).

The sectionsprovides a sections⁰ofEand one easily checks thatP2⊂X^[2]is scheme- theoretically defined as the zero-locus ofs⁰. This implies thatDis the blow-up ofX^[2]

alongP₂and the smoothness ofP₂implies the smoothness ofD. The claim is thus proved.

On the other hand, the rational map8clearly pulls back to a morphism onD, so (i) is proved.

(ii)&(iii) If the length 2 subscheme Z = x +y (or Z = (2x, v) with v tangent toX atx) does not belong toP2, then the linelx,y (or lx,v) is not contained inX, the morphism8is well defined atZ and its image is a pair([l], u)wherelis a line passing throughuand is not contained inX. At such a point([l], u)ofPX,8⁻¹is well-defined, and associates to each([l⁰], u⁰)in a neighborhood of([l], u)inPXthe residual scheme ofu⁰inl⁰∩X. This proves (iii).

It remains to understand what happens along the exceptional divisorQ_P2 ofD. Now we haveQ_P2 = {(u, x+y,[l]): l ⊂ X, x+y ∈ l⁽²⁾, u ∈ l}. By definition,8emaps such a triple to the pair (u,[l]), which by definition belongs to P_X. Furthermore, the fiber of8eover(u,[l])whenl ⊂X, that is, when(u,[l])∈P, identifies with the plane l⁽²⁾ ∼= P². Thus 8e⁻¹(P )is equal to the smooth irreducible hypersurface QP₂ in the smooth variety D, and this implies thate8factors through a morphismf : D → PfX

which has to be an isomorphism, since it cannot contract any curve; indeed, otherwise the contracted curve would be a curve in a fiberP² as described above, so the whole corresponding P²would be contracted by f, hence also all deformations of thisP² in D=Xg^[2]. But then the divisorQP₂ would be contracted byf, while its image has to be

the exceptional divisor ofPfX→PX. ut

We now first give the proof of Theorem 1.1in the case of cubics of dimension≤ 4, because the argument is shorter in this case.

Proof of Theorem1.1forn≤4. LetXbe a smooth cubic hypersurface of dimension≤4 and assumeXadmits a cohomological decomposition of the diagonal. The assumptions

of Proposition2.6are satisfied byX, since the integral cohomology of a smooth cubic hypersurface has no torsion and the integral Hodge conjecture is proved in [32] for cubic fourfolds. With the notation introduced previously, there exists by Proposition2.6a cycle 0∈CHⁿ(X^[2])such that

µ^∗0=1_X−x×X−X×x−W in CHⁿ(X×X), (25) withW supported overD×Xfor someD(X, and[0] =0 inH²ⁿ(X^[2],Z).

By Proposition2.9, the blow-upσ : Xg^[2] → X^[2]ofX^[2]alongP₂identifies viae8 with a blow-up of the projective bundleP_XoverX. Furthermore, the exceptional divisor of8e:Xg^[2]→P_Xis also the exceptional divisor ofσ :Xg^[2]→X^[2], hence maps viaσ toP₂⊂X^[2]. It follows that the pull-backσ^∗(0)of the cycle0toXg^[2]can be written as

σ^∗(0)=0₁+0₂, (26)

where01and02are cohomologous to 0,01is a cycle cohomologous to 0 on the excep- tional divisor ofe8 : Xg^[2] → P_X, and02is the pull-back of a cycle0₂⁰ cohomologous to 0 onPX. As the exceptional divisor ofe8equals the exceptional divisor ofσ, it follows from (26), by applyingσ∗, that

0=i_P2∗(0⁰₁)+8^∗(0⁰₂) in CH(X^[2]), (27) where0₁⁰ is a cycle cohomologous to 0 onP₂. Herei_P2 denotes the inclusion map ofP₂ inX^[2]. It is known that for a smooth cubic hypersurface of dimension≤ 4, cycles ho- mologous to 0 are algebraically equivalent to 0. For codimension 2 cycles, this is proved by Bloch and Srinivas [7] and is true more generally for any rationally connected vari- ety; for 1-cycles on cubic fourfolds, this is proved in [24], and this is true more generally for 1-cycles on Fano complete intersections of index≥ 2. The result then also holds for cycles on a projective bundle over a cubic of dimension≤ 4. Thus0⁰₂is algebraically equivalent to 0 and thus we conclude that

0=i_P2∗(0⁰₁) in CH(X^[2])/alg (28) for somen-cycle0₁⁰ homologous to 0 onP2. We now apply the following result:

Lemma 2.10. Let X be an n-dimensional smooth cubic hypersurface and Z be an n-cycle homologous to0onP₂

i_P2

,→ X^[2]. Thenµ^∗(i_P2∗Z)∈ CHⁿ(X×X)is supported onD×Xfor some proper closed algebraic subsetDofX.

Proof. Recall thatP2is the union of the symmetric productsL⁽²⁾over all linesL⊂X.

As before, denote by

q :P →X, p:P →F

the natural maps, and byq₂the natural mapP×_FP →X×Xinduced byq. We will also denote byπ :P ×_F P →F the map induced byp. Viaπ,P ×_FP is aP¹×P¹-bundle

overF. LetH := c1(O_X(1)) ∈ CH¹(X)and leth = q^∗H ∈ CH¹(P )be its pull-back toP. For a cycleZsupported onP2, we have

µ^∗(i_P2∗Z)=q_2∗(r^0∗Z) in CH(X×X)

wherer⁰is the quotient mapP ×_F P →P2. LetT :=r^0∗Z ∈CH(P ×_F P ). This is a cycle homologous to 0 onP ×_FP, and thus it can be written as

T =h₁h₂π^∗α+h₁π^∗β+h₂π^∗γ+π^∗ζ in CH(P ×_F P ), (29) for some cyclesα, β, γ , ζ homologous to 0 onF, withh_i =pr^∗_i hfori = 1,2, pr_i : P ×_F P →P being thei-th projection.

We now push forward these cycles toX×Xviaq2and observe that the three cycles q_2∗(π^∗ζ ), q_2∗(h₁h₂π^∗α), q_2∗(h₁π^∗β)

are cycles supported on D × X for some D ( X. Indeed, for the two last ones, this is due to the projection formula (and the equality h₁ = q₂^∗(H₁), where H₁ :=

pr^∗₁H ∈ CH¹(X×X)), and for the first one, this is becauseq_2∗(π^∗ζ )is supported on q(p⁻¹(Suppζ ))×X. Nowζ is a (n−2)-cycle, soq(p⁻¹(Suppζ ))is a proper closed algebraic subset ofX. It remains to examine the cycleq2∗(h2π^∗γ ). We observe now that the diagonal1P ⊂P×_FP ⊂P×P is a divisordinP×_FP whose class is of the form d =h₁+h₂+π^∗λfor some divisor classλ∈CH¹(F ). Furthermore, we obviously have q₂(1_P)⊂1_X. Thus we can write

h₂π^∗γ =(d−h₁−π^∗λ)π^∗γ in CH(P ×_F P ).

Hence

q_2∗(h₂π^∗γ )=q_2∗(dπ^∗γ )−q_2∗(h₁π^∗γ )−q_2∗(π^∗(λγ )) in CH(X×X).

As already explained, the cyclesq2∗(h1π^∗γ )andq2∗(π^∗(λγ ))are supported overD×X for someD (X. Finally, the last cycleq2∗(dπ^∗γ )has to be 0 in CH(X×X). Indeed, this is ann-cycle ofX×Xwhich is supported on the diagonal, hence proportional to it,

and also cohomologous to 0. ut

Combining (25), (28) and Lemma2.10, we conclude that

1_X=x×X+X×x−W⁰ in CH(X×X)/alg, (30) whereW⁰is supported onD⁰×Xfor someD⁰(X. In conclusion,Xadmits a decomposi- tion of the diagonal modulo algebraic equivalence, and we can now apply Proposition2.1 to conclude thatXadmits a Chow-theoretic decomposition of the diagonal. ut We now turn to the general case.

Proof of Theorem1.1for generaln. LetXbe a smooth cubic hypersurface such that the groupH^2∗(X,Z)/H^2∗(X,Z)_alg has no 2-torsion. Then Propositions2.6and2.9apply.

Next, if we examine the proof in the case of dimension≤ 4, we see that the only place

where we used the fact that dimX ≤ 4 is in the analysis of the term 8^∗(0⁰₂), where 0⁰₂is cohomologous to 0 onPX. We directly used the fact that this term is algebraically equivalent to 0, which we do not know in higher dimension. The following provides an alternative argument which works also in higher dimension:

Lemma 2.11. Let X be a smooth cubic hypersurface of dimension n ≥ 2. Let Z be ann-cycle cohomologous to0onP_X. Then 3µ^∗(8^∗(Z)) ∈ CHⁿ(X×X)is rationally equivalent to a cycle supported onD×Xfor someD(X.

Proof. Recall thatp_X : P_X → X is thePⁿ-bundle over X with fiber overx ∈ X the set of lines inPⁿ⁺¹throughx. Let us denote byl ∈ CH¹(P_X)the class ofO_PX(1)(we choose forO_PX(1)the pull-back of the Pl¨ucker line bundle on the Grassmannian of lines G(1,Pⁿ⁺¹)). The cycleZcan be written as

Z=p^∗_XZ₀+lp_X^∗Z₁+ · · · +lⁿp^∗_XZ_n, (31) whereZ_iare cycles of codimensionn−ionX. Note that the cyclesZ_iare all homologous to 0. As dimX ≥2, we have CH₀(X)_hom =0, and thusZ₀=0 in CH₀(X). Hence (31) shows thatZ =l·Z⁰for someZ⁰ ∈CH(P_X). Let9 :=8◦µ:X×X99KP_Xand let 9e: ^

X^×X→PXbe the desingularization of9obtained by blowing up first the diagonal ofX, and then the inverse image ofP₂ ⊂X^[2](see Proposition2.9). Letτ˜ :X^^×X → X×Xbe the composition of the two blow-ups. There are two exceptional divisors ofτ˜, namelyE₁andE_P2. We thus have (using the fact that9efactors throughX^[2])

9e^∗(l)=ατ˜^∗(H₁+H₂)+βE₁+γ E_P2,

where the coefficientsα, β, γ can be explicitly computed but this is not useful here.

It follows that

9^∗(Z)=9^∗(l·Z⁰)= ˜τ∗(e9^∗(l·Z⁰))= ˜τ∗(e9^∗(l)·9e^∗(Z⁰))

= ˜τ∗((ατ˜^∗(H1+H2)+βE₁+γ E_P2)·e9^∗(Z⁰)) in CH(X×X). (32) Now, we develop the last expression and observe again that sinceτ˜∗(βE₁·9e^∗(Z⁰))is supported on the diagonal ofX, it must be proportional to1_X, hence in fact identically 0 as it is cohomologous to 0. Next, the cycleτ˜∗(γ E_P2·9e^∗(Z⁰))comes from ann-cycleZ⁰⁰ homologous to 0 onP ×_FP, i.e.

τ˜∗(γ E_P2·9e^∗(Z⁰))=q2∗(Z⁰⁰), (33) whereq₂ :P ×_F P → X×Xis introduced above. We can then apply Lemma2.10to conclude that the cycleτ˜∗(γ E_P2·9e^∗(Z⁰))is supported onD×Xfor someD(X. Thus we conclude from (32), the projection formula, and the analysis above that

9^∗(Z)=H₁·W₁+H₂·W₂+W in CH(X×X), (34) whereW is supported onD×Xfor someD(X, andW₁, W₂are cycles homologous to 0 on X×X. It thus suffices to show that cycles0 onX×X of the formH₁·W₁