The integral Hodge conjecture for the total space of one-parameter families of cubic threefolds

The integral Hodge conjecture for the total space of one-parameter families of cubic threefolds

Jean-Louis Colliot-Th´el`ene CNRS

Universit´e Paris-Sud ⊂Paris-Saclay Algebraic Geometry Conference Arnaud Beauville’s 70th birthday

Oct. 23-27, 2017, Beijing

X/Cprojective and smooth.

LetH_Hodge²ⁱ (X(C),Z(i))⊂H_Betti²ⁱ (X(C),Z(i)) be the inverse image of Hodge classes inH_Betti²ⁱ (X(C),Q(i)).

The cycle mapcli :CHⁱ(X)→H_Betti²ⁱ (X(C),Z(i)) lands in H_Hodge²ⁱ (X(C),Z(i)).

The (rational) Hodge conjecture asserts that the group Z²ⁱ(X) :=H_Hodge²ⁱ (X(C),Z(i))/Im(CHⁱ(X)) is a finite group, equal to the torsion subgroup of H_Betti²ⁱ (X(C),Z(i))/Im(CHⁱ(X)).

This is so fori = 1 (Lefschetz).

IfZ²ⁱ(X) = 0, one says that the integral Hodge conjecture holds for cycles of codimensioni onX, or in (cohomological) degree 2i. Atiyah and Hirzebuch gave examples withZ²ⁱ(X)6= 0.

In this talk we shall concentrate onZ⁴(X).

In that degree,Z⁴(X) is a birational invariant in X (Voisin).

Koll´ar gave examples of hypersurfacesX ⊂P⁴ of suitable (high) degree for whichZ⁴(X)6= 0.

ForX a threefold, Voisin (2006) proved Z⁴(X) = 0 ifX is uniruled orX is Calabi-Yau.

For rationally connected varieties, the rational Hodge conjecture for cycles of codimension 2 holds (Bloch-Srinivas). Voisin (2007) asked whetherZ⁴(X) = 0 holds for any such variety.

ForX of any dimension at least 6, a negative answer was given by CT-Voisin 2012, using work of CT-Ojanguren 1988.

The question whetherZ⁴(X) = 0 holds for rationally connected varieties of dimension 4 and 5 is open.

In dimension 4, the following special results are known :

TheoremThe integral Hodge conjecture for cycles of

codimension 2 holds for smooth projective fourfolds X with a dominant morphism X →Y to a smooth projective Y if : (i) dim(Y) = 2, and the generic fibre is a del Pezzo surface of degree d at least 5 (CT-Voisin 2012 when fibre is a quadric surface ; CAO Yang 2017).

(ii) dim(Y) = 1, and the generic fibre is a smooth complete intersection of two quadrics inP⁵ (Voisin 2010 when bad fibres have ord. quad. singularities ; CT 2012).

Theorem (Voisin)

The integral Hodge conjecture for cycles of codimension 2 holds for :

(a) (2007) smooth cubic hypersurfaces inP⁵;

(b) (2010, J. Alg. Geom. 2013) varieties X with a fibration X →Γ to a curveΓ, whose generic fibre is a smooth cubic in P⁴ and whose special fibres have ordinary quadratic singularities (implies a)).

In this talk, I will describe the proof of : Theorem (CT-Pirutka, 2017)

The integral Hodge conjecture for cycles of codimension 2 holds for fourfolds X with a fibration X →Γ to a curveΓ, fibration whose generic fibre is a smooth cubic inP⁴.

The improvement over Voisin’s result is that we impose no restriction on the singular fibres. Our proof combines geometric ideas in her paper with tools from algebraicK-theory (motivic homology, unramified cohomology).

Unramified cohomology of degree 3 and integral Hodge conjecture

F a field, char(F) = 0

X/F variety ,n∈N,n>0,i ∈N,j ∈Z. Let Hⁱ(µ^⊗jn ) be the Zariski sheaf associated to the presheafU 7→H_etⁱ (U, µ^⊗jn ).

Definition of unramified cohomology ofX

H_nrⁱ (X, µ^⊗j_n ) :=H_Zar⁰ (X,Hⁱ(µ^⊗j_n )) LetQ/Z(j) =colim_nµ^⊗jn .

AssumeX/F smooth. H_nr¹(X,Q/Z) =H_et¹(X,Q/Z).

H_nr² (X,Q/Z(1)) =Br(X).

We shall be concerned withH_nr³ (X,Q/Z(2)).

Theorem (Bloch-Ogus 1974, Gersten’s conjecture for ´etale cohomogy)For X/F smooth connected, with field of functions F(X), there is an exact sequence

0→H_nrⁱ (X, µ^⊗j_n )→Hⁱ(F(X), µ^⊗j_n )→ ⊕_x∈X(1)Hⁱ⁻¹(F(x), µ^⊗j−1_n ).

Here, for any point x of codimension 1 ofX, the map Hⁱ(F(X), µ^⊗j_n )→Hⁱ⁻¹(F(x), µ^⊗j_n ⁻¹)

is the residue map associated to the DVROX,x (with residue field κ(x)).

CorollaryFor X/F connected, smooth and proper, the group H_nrⁱ (X, µ^⊗j_n ) is an F -birational invariant, hence sometimes denoted H_nrⁱ (F(X)/F, µ^⊗jn ). If X is stably F -rational, then

Hⁱ(F, µ^⊗jn )→^' H_nrⁱ (F(X)/F, µ^⊗jn ).

ForX/F smooth and proper, H_nrⁱ (X, µ^⊗j_n )⊂Hⁱ(F(X), µ^⊗j_n ) may be computed by using all DVR’sR of F(X) withF ⊂R.

For algebraically closed fieldsk ⊂K, andX/k smooth and

projective, for torsion coefficients, one has the “rigidity theorem” : H_nrⁱ (X,•) =H_nrⁱ (XK,•) (CT, Jannsen ; Suslin’s method)

Using results of Merkurjev–Suslin, Rost, Voevodsky, one proves : Theorem (CT-Voisin 2012)Let X/Cbe a smooth projective variety. The quotient of H_nr³(X,Q/Z(2)) by its maximal divisible subgroup is finite and it is equal to the torsion subgroup of H_Betti⁴ (X(C),Z(2))/Im[CH²(X)], hence also to the torsion subgroup of Z⁴(X).

[Compare : the quotient ofH_nr²(X,Q/Z(1)) by its maximal divisible subgroup is finite and is equal to the torsion subgroup of H_Betti³ (X,Z).]

CorollaryIf X is a rationally connected variety, or a uniruled threefold, there is an isomorphism of finite groups

Z⁴(X)'H_nr³ (X,Q/Z(2)).

Starting from theorems obtained via geometry and Hodge theory, one gets results onH_nr³ (X,Q/Z(2)) such as its vanishing for uniruled threefolds.

Starting from results in algebraicK-theory (e.g. computation of unramified cohomology of quadrics over arbitrary ground fields, Kahn, Rost, Sujatha), one gets results on the integral Hodge conjecture.

This accounts for some of the results mentioned above.

Galois descent on Chow groups : some long exact sequences

LetF be a field,char.(F) = 0. Let F be an algebraic closure of F andg=Gal(F/F). For a Galois moduleM, letHⁱ(g,M) be its Galois cohomology groups.

LetX be a smooth, geometrically integralF-variety. We write X =X ×_F F.

Recall that Quillen has associated groupsK_i(A) (i ≥0) to any commutative ringA. On a given schemeX, sheafification of the Zariski presheafU 7→K_i(Γ(U,O_U)) defines the Zariski sheaf K_i on X. ForX smooth over a field, a celebrated formula, conjectured by Bloch and proved by Quillen, identifiesH_Zarⁱ (X,K_i) with the Chow groupCHⁱ(X) of codimensioni cycles modulo rational equivalence.

Theorem A .Let X/F be as above. Assume that H¹(X,OX) = 0 andPic(X) =NS(X) has no torsion. Assume X(F)6=∅. Then there is an exact sequence

0→Ker[CH²(X)→CH²(X)^g]→H¹(g,H¹(X,K₂))→ N(X)→Coker[CH²(X)→CH²(X)^g]→H²(g,H¹(X,K₂)), where

N(X) =Ker[H_nr³ (X,Q/Z(2))/H³(F,Q/Z(2))→H_nr³ (X,Q/Z(2))].

Special case of a general sequence without assumption on the Picard group. Parts of this sequence : Bloch, 1970’s ; CT-Sansuc and CT-Raskind 1980’s. Whole sequence :B. Kahn 1996 (via Lichtenbaum’sZ(2)’s) CT-Kahn 2013 and CT 2013.

The proof relies on theorems of Merkurjev and Suslin and on the Gersten conjecture (Quillen forK-theory, Bloch-Ogus for ´etale cohomology). Most of the sequence may be established in a pedestrian manner, combining 1993 results of B. Kahn with the Galois cohomology of the complex

K₂(F(X))→ ⊕

x∈X⁽¹⁾F(x)^×→ ⊕

x∈X⁽²⁾Z,

the homology groups of which are precisely the groupsHⁱ(X,K₂).

There is a natural map

Pic(X)⊗_ZF^× →H¹(X,K₂).

A special case of CT-Raskind results (1985) – building on works of Bloch, Merkurjev, Suslin – gives that this map is an isomorphism if X is rationally connected and Br(X) = 0. We thus get :

Theorem B.Let X/F be a smooth, projective, geometrically connected variety. Assume X(F)6=∅and :

(a) X rationally connected, hencePic(X)is a lattice ; (b)Br(X) = 0;

(c) H_nr³ (X,Q/Z(2)) = 0.

Then there is an exact sequence

0→Ker[CH²(X)→CH²(X)^g]−→H^{α 1}(g,Pic(X)⊗F^×)→

→H_nr³(X,Q/Z(2))/H³(F,Q/Z(2))→

→Coker[CH²(X)→CH²(X)^g]−→H^{β 2}(g,Pic(X)⊗F^×).

Corollary.Let K =C(Γ)be the function field of a curve. Let X/K be smooth, projective, geometrically rationally connected variety of dimension at most 3. AssumeBr(X) = 0. Then

H_nr³ (X,Q/Z(2))→^' Coker[CH²(X)→CH²(X)^g].

Corollary.Let K =C(Γ)be the function field of a curve, and let Y/K be a smooth, projective, geometrically rational surface, for example a smooth cubic surface inP³_K. Then

H_nr³ (Y,Q/Z(2)) = 0.

An aside on some birational invariants

Extending the ground fieldCof X to an arbitrary overfield F

Theorem B and rigidity of unramified cohomology imply : Theorem C.Let X/Cbe a connected smooth projective variety.

Assume

(a) X is rationally connected, hencePic(X) is a lattice.

(b)Br(X) = 0.

(c) H_nr³ (X,Q/Z(2)) = 0.

Then for any field F/C, with algebraic closure F , letting g=Gal(F/F), we have an exact sequence

0→H_nr³ (XF,Q/Z(2))/H³(F,Q/Z(2))→

→Coker[CH²(X_F)→CH²(X_F)^g]−→H^{β 2}(g,Pic(X)⊗F^×).

One way to look at this theorem is as an ordered search to establish the nonrationality ofX.

Indeed, if any of the hypotheses (a), (b) or (c) is not fulfilled, then X is not rational.

For any fieldF/C, the groupCoker[CH²(X_F)→CH²(X_F)^g] is a birational invariant ofX/C.

For any fieldF/C, the first two groups in the sequence are birational invariants ofX/C, and vanish ifX is rational.

Theorem D.Let X ⊂P⁵_Cbe a smooth Fano hypersurface. Assume that the integral Hodge conjecture holds for codimension 2 cycles on X . Then for any field F/C, one has

H³(F,Q/Z(2)) =H_nr³(F(X)/F,Q/Z(2)).

Proof. The hypothesis impliesH_nr³ (X,Q/Z(2)) = 0, hence (rigidity)H_nr³(X×_CF,Q/Z(2)) = 0. Theorem C then gives an embedding

H_nr³ (F(X)/F,Q/Z(2))/H³(F,Q/Z(2)),→Coker[CH²(X_F)→CH²(X_F)^g].

For codimension 2 cycles overX as above over an algebraically closed field, rational equivalence, algebraic equivalence and homological equivalence coincide. ThusCH²(X)→^' CH²(X_F).

HenceCH²(X_F)→CH²(X_F)^g is onto. Hence H³(F,Q/Z(2)) =H_nr³(F(X)/F,Q/Z(2)). QED

ForX ⊂P⁵_C a cubic hypersurface, the integral Hodge conjecture in degre 4 holds (Voisin 2007) and the conclusion of the above theorem is due to her (2015).

What aboutsmooth cubic hypersurfaces X ⊂P⁴_C ?

None is rational (Clemens-Griffiths) but could some, all, be stably rational ? Could none be ?

For any suchX and any fieldF/C, using Theorem C one may show :

H_nr³ (F(X)/F,Q/Z(2))/H³(F,Q/Z(2))→^' Coker[CH²(X_F)→CH²(X_F)^g].

Could this group be non zero for some suitableF (hence X not be stably rational) ?

The relevant fieldF to look at is the field of rational functions of the intermediate jacobianJ³(X). This is an abelian variety which parametrizes cycles of codimension 2 onX which are homologically equivalent to zero. There is an obvious class inCH²(X_F)^g and one wonders whether it comes fromCH²(X_F), thus defining on

X×J³(X) what C. Voisin (2014) calls a “universal codimension 2 cycle”. She shows that the existence of a universal codimension 2 cycle onX is actually equivalent to

H³(F,Q/Z(2)) =H_nr³(F(X)/F,Q/Z(2)) for any fieldF/C.

C. Voisin also shows that there exist cubic hypersurfacesX ⊂P⁴_C for whichH³(F,Q/Z(2)) =H_nr³ (F(X)/F,Q/Z(2)) for any field F/C, and indeed for which thestronger property

CH₀(X_F)'Z

holds for everyF/C (CH0-triviality, integral decomposition of the diagonal).

I have given a simple proof that this stronger property holds for any cubic hypersurfaceX ⊂P⁴_C given by a partially diagonal equation

f(x0,x1,x2) +g(x3,x4) = 0.

More generally, one has the following result, which applies for instance to the Fermat hypersurfacePn

i=0X_i³ = 0.

Theorem.Let X ⊂Pⁿ_C, n≥4, be a nonsingular cubic hypersurface given by a form

F(X₀, . . . ,X_n) =

r

X

i=1

G_i(X₀, . . . ,X_n)

where the G_i’s are forms in at most three explicit variables and two different Gi’s have no explicit variable in common. Then X is CH0-trivial : for any field extension F/Cthe degree map deg_F :CH₀(X_F)→Z is an isomorphism.

The proof uses :

Theorem.Let k be a field and X a smooth, projective, connected variety with a k-point. Assume H¹(X,O_X) = 0. If there exists a smooth, projective, connected curveΓwith a k-point and a k-morphismΓ→X such that over any overfield F of k the

induced map CH₀(Γ_F)→CH₀(X_F) is onto, then over any overfield F of k, the degree map degF :CH0(XF)→Zis an isomorphism : the k-variety X is universally CH₀-trivial.

End of the (long) aside on some birational invariants.

The title theorem and its proof

Theorem (CT-Pirutka, 2017)Let Γbe a smooth curve and K =C(Γ) its function field. The integral Hodge conjecture for cycles of codimension 2 holds for fourfolds Y/Cwith a fibration Y →Γwhose generic fibre X/K is a smooth cubic threefold in P⁴_K.

Such a varietyY is dominated by the product ofP³_C and a curve, hence the rational Hodge conjecture holds in degree 4

(Bloch-Srinivas). Using the CT-Voisin result

H_nr³ (X,Q/Z(2))/maxdiv '[H_Betti⁴ (X(C),Z(2))/Im(CH²(X))]tors, it is thus enough to proveH_nr³ (X,Q/Z(2)) = 0.

NowH_nr³ (Y,Q/Z(2))⊂H_nr³ (X,Q/Z(2)). The result thus follows from (i) in the following :

Theorem (CT-Pirutka 2017).Let K =C(Γ)be the function field of a complex curveΓ. Let X ⊂P⁴_K be a smooth cubic threefold over K . Then

(i) One has H_nr³(X,Q/Z(2)) = 0.

(ii) The map CH²(X)→CH²(X)^g is onto.

Over the cohomological dimension 1 fieldK =C(Γ), Theorem B gives

H_nr³(X,Q/Z(2))→^' Coker[CH²(X)→CH²(X)^g].

We shall prove thatCH²(X)→CH²(X)^g is onto.

Letk be a field of char. zero,k an algebraic closure,g=Gal(k/k).

LetX ⊂P⁴_k be a smooth cubic hypersurface, let X =X ×_k k.

Algebraic and numerical equivalence coincide onCH²(X). There is a natural exact sequence of Galois modules

0→CH²(X)_alg →CH²(X)→Z→0,

where the mapCH²(X)→Z is given by the intersection with a hyperplane.

The groupCH²(X) is then the disjoint union of the sets CH²(X)_d consisting of classes of degreed ∈Z.

The groupCH²(X)_alg =CH²(X)₀ is “the group of k-points of the intermediate jacobian ofX”. It can be given an algebraic structure via the Prym construction (Mumford, Murre, Beauville 1977) attached to a conic bundle structure on a blow-up ofX along a (general) line. This depends on the choice of a line onX, and such a line need not exist overk.

For cubic threefolds, another presentation ofCH²(X)_alg goes back to Murre and can be followed overk. LetS/k be the Fano variety of lines onX. It is a smooth projective surface.

LetV ⊂S ×_k X be the incidence variety consisting of pairs (L,x) withx ∈X lying in the lineL. One then has the Galois equivariant map

pS,∗◦p_X^∗ :CH²(X)→CH¹(S) =Pic_S/k(k).

LetJ :=Pic⁰_S/k.

Murre : This map induces an isomorphism of Galois modules CH²(X)alg

→' J(k).

For eachd, there exists a principal homogeneous spaceJ_d of J and an isomorphism of Galois setsCH²(X)d

→' Jd(k).

One wants to prove that ifk =C(Γ) is the function field of a complex curve, then each mapCH²(X)_d →CH²(X)^g_d is onto.

From this point on, the proof uses many geometric ingredients from the paper of C. Voisin (JAG 2013).

The traceh of a plane on X gives a class in CH²(X)₃. A simple argument (consider±c+nh) then shows that it is enough to prove the surjectivity statement ford = 5 andd = 6.

Some moduli spaces of curves of genus 1 lying on a cubic hypersurface X ⊂P⁴_k

Ford ≥1 letM_d,1(X) be the union of (reduced, closed) components of the Hilbert scheme ofX whose general point parametrizes smooth, connected curvesC ⊂X ⊂P⁴ of genus 1, of degreed, and which spanP⁴. Thek-points of these spaces

correspond to (possibly) reductible projective curves of degreed overk.M5,1(X) is geometrically irreducible (Markushevich, Tikhomirov ; Harris, Roth, Starr). ForX general,M6,1(X) is geometrically irreducible (Voisin).

One fixes desingularisations ˜M5,1(X)→M5,1(X) and

M˜_6,1(X)→M_6,1(X). Using the universal family of curves over M_d,1(X), ford = 5,6, one gets k-morphisms ˜M_d,1(X)→J_d. These will be called Abel-Jacobi morphisms.

Theorem (Iliev, Markushevich, Tikhomirov ford = 5 ; Voisin for d = 6)Let k =Cand X ⊂P⁴_C be a very general cubic threefold.

Then for d = 5 and d= 6 the Abel-Jacobi morphisms

M˜_d,1(X)→J_d are surjective and their generic fibre is geometrically rationally connected.

All the previous constructions were done over an arbitrary fieldk of characteristic zero.

One may globalize them over the parameter spaceP ⊂P³⁴_C of cubic threefolds and the universal familyX →P : at the generic point of P we get the above construction over the function fieldk =C(P).

One has the familyS →P of Fano surfaces of lines, global correspondences, components of the relative Hilbert scheme Hilb_X/P, irreducible familiesM_5,1 andM_6,1 overP of

nondegenerate curves of genus 1 and degreed = 5,6, with the propertyM_d,1,t ⊂M_d,1(X_t) for t∈P(k). One fixes

desingularizations ˜M_d,1 → M_d,1 for d = 5,6. One then finds globalized Abel-Jacobi morphisms (d = 5,6) overP :

ϕ_d : ˜M_d,1 → J_d =Pic^d_S/P

which are proper, surjective and whose geometric generic fibre is smooth and rationally connected (by the above theorem).

A theorem of A. Hogadi and C. Xu 2009 then guarantees that over any schematic pointx of the scheme J_d, with residue field κ(x), the fibre ofϕ_d contains aκ(x)-subvariety which is geometrically rationally connected. Ifx corresponds to a point of dimension 0 or 1 onJ_d, thenκ(x) =C orC(Γ) (Γ a curve). By the

Graber-Harris-Starr theorem, the fibre atx contains aκ(x)-point.

SinceX →P is the universal family of cubic fourfolds, we conclude : forK =C(Γ) the function field of a curve and any smooth projective cubic threefoldX ⊂P⁴_K, and d = 5,6, the map

CH_d²(X)(K)→J_d(X)(K) =CH_d²(X)^g is onto. QED.

Over finite fields

Finite fields may sometimes be viewed as “simpler” analogues of function fields in one variable overC. They both have Galois cohomological dimension 1.

Theorem (CT, Sansuc, Soul´e ; K. Kato) (1983)

Let X be a smooth projective connected surface over a finite field F. Then H_nr³ (X,Q`/Z`(2)) = 0.

This is one special result in higher class field theory (Bloch, Kato, Saito in the 80s, later Jannsen, Saito ...).

Theorem (CT-Kahn 2013)Let X be a smooth, projective, geometrically connected variety over a finite fieldF. Let G =Gal(F/F). Let M be the finite Galois module

M =⊕<sub>`</sub>H<sup>3</sup>(X,Z`(2))tors. (H³ is for H_et³ for`6=char(F)).

There is an exact sequence

0→Ker[CH²(X)→CH²(X)]→H¹(G,M)→

→ker[H_nr³ (X,Q`/Z`(2))→H_nr³ (X,Q`/Z`(2))]→

→Coker[CH²(X)→CH²(X)^G]→0.

The following theorem parallels the result by CT-Voisin overC. Theorem (Kahn ; CT-Kahn 2013)Let X/Fbe a smooth projective, geometrically connected variety over a finite fieldF. The quotient of H_nr³ (X,Q`/Z`(2)) by its maximal divisible subgroup is finite and is equal to the torsion subgroup of

cyc_X :Coker[CH²(X)⊗Z`→H<sup>4</sup>(X,Z`(2)].

If Tate conjecture holds (coefficientsQ`), that cokernel is torsion.

Threefolds over a finite field

Finite fieldsFand function fields C(Γ) in one variable over the complex field have some properties in common. Here is one more.

Theorem.Let X ⊂P⁴_F be a smooth cubic hypersurface over a finite fieldF of odd characteristic p. Let` be a prime,`6=p.

(i) The map CH²(X)[1/p]→CH²(X)^G[1/p] is an isomorphism.

(ii) One has H_nr³(X,Q`/Z`(2)) = 0.

(iii)The cycle map CH²(X)⊗Z`→H<sup>4</sup>(X,Z`(2)) is onto.

Proof. LetY denote the Fano surface of lines onX. It is a

smooth, geometrically integral surface, hence contains a zero-cycle of degree 1. For our statements, we may thus assume that there exists a line defined overFonX. Intersection with a hyperplane thus induces a split exact sequence of Galois modules :

0→CHf²(X)→CH²(X)→Z→0,

The universal line inducesCH0(Y)→CH²(X) and similarly for CHf² and for the groups over F.

The mapCHf0(Y)→CHf²(X) factorizes as a composite of isomorphismsCHf₀(Y)→^' Alb_Y(F)→^' CHf²(X) (the first one, Roitman ; the second one, identification of the intermediate jacobian).Higher class field theory of the surfaceY/F (Kato, Saito, 1980s) gives that the mapCHf0(Y)→CHf0(Y)^G is onto.

One then deduces thatCHf²(X)→CHf²(X)^G is onto. Hence CH²(X)→CH²(X)^G is onto.

From this one deducesH_nr³(X,Q`/Z`(2)) = 0 and then cyc_X :CH²(X)⊗Z` →H<sup>4</sup>(X,Z`(2)) onto.

One may also prove the above result by reduction to a conic bundle situation.

Theorem (Parimala & Suresh, 2012)Let Fbe a finite field Fof odd characteristic. Let X →S be a conic bundle, with X and S smooth, projective varieties overF, dim(X) = 3 and dim(S) = 2.

Then for any prime`6=char(F), H<sub>nr</sub><sup>3</sup> (X,Q`/Z`(2)) = 0.

The proof of this theorem relies onhigher class field theory of surfacesand on many delicate arguments, some of them inspired by work of D. Saltman on unramified cohomology.

Theorem (Pirutka)Let Fbe a finite field. Let X =C ×_F S be the product of a smooth, projective curve C/Fand a geometrically rational smooth projective surface S/F. Then for any prime

`6=char(F), H<sub>nr</sub><sup>3</sup> (X,Q`/Z`(2)) = 0.

[This would be obvious ifS was birational to P²_F overF.]

Questions (already raised in various conferences and papers) : (1) IsH_nr³ (X,Q`/Z`(2)) finite for any smooth projective variety X over a finite field ?

(2) IsH_nr³ (X,Q`/Z`(2)) = 0 for any smooth projective 3-fold or 4-foldX over a finite field ? (Not so for 5-folds – A. Pirutka) (3) Special case : LetX ⊂P⁵_F a smooth cubic hypersurface.

IsH_nr³(X,Q`/Z`(2)) = 0 ?

Iscyc_X :CH²(X)⊗Z`→H<sup>4</sup>(X,Z`(2)) onto ? Known forX =X ×_FF(Charles-Pirutka).