Hodge structures through an \'etale motivic point of view

Abstract

We define the category of \'etale Chow motives as the \'etale analogue of Grothendieck motives and proved that it embeds in DMeˊt(k)\text{DM}_{\text{\'et}}(k). This construction provides a characterization of the generalized Hodge conjecture in terms of an \'etale analogue of it. Finally we study the non-algebraic classes in the Atiyah-Hirzebruch, Benoist-Ottem and K\'ollar counter-examples in terms of Lichtenbaum classes and we give a characterization of their preimages in the Lichtenbaum cohomology groups.

Full text

arXiv:2212.02128v1 [math.AG] 5 Dec 2022
HODGE STRUCTURES THROUGH AN ´
ETALE MOTIVIC POINT OF VIEW
IV ´
AN ROSAS SOTO
Abstract. We define the category of ´etale Chow motives as the ´etale analogue of Grothendieck
motives and proved that it embeds in DM´et (k). This construction provides a characterization of
the generalized Hodge conjecture in terms of an ´etale analogue of it. Finally we study the non-
algebraic classes in the Atiyah-Hirzebruch, Benoist-Ottem and K´ollar counter-examples in terms
of Lichtenbaum classes and we give a characterization of their preimages in the Lichtenbaum
cohomology groups.
Contents
1. Introduction 1
Conventions 4
Acknowledgements 4
2. ´
Etale Chow groups and Lichtenbaum cohomology 4
2.1. ´
Etale Chow groups 4
2.2. Lichtenbaum cohomology groups 5
2.3. Category of ´etale Chow motives 7
2.4. ´
Etale Chow motives 9
3. Hodge and Generalized Hodge conjecture 10
3.1. Hodge conjecture and Lichtenbaum cohomology 10
3.2. Examples 14
3.3. Generalized Hodge conjecture and Lichtenbaum cohomology 19
3.4. Bardelli’s example 23
References 23
1. Introduction
Let Xbe a smooth projective variety over C,k∈Nand consider the cycle class map ck:
CHk(X)→H2k
B(X, Z(k)) where Z(k) = (2πi)kZwhose image is a subgroup of the Hodge classes
Hdg2k(X, Z). The integral Hodge conjecture asks whether or not this map is surjective. Putting
n= dimC(X), then for k= 0 and k=nthe conjecture is immediately true and also for k= 1 by the
Lefschetz (1,1) theorem, but for k= 2 the statement is not true as is shown by the counterexamples
given by Atiyah and Hirzebruch in [AH62] (a torsion class which is not algebraic) and by K´ollar in
[BCC92] (a non-algebraic non-torsion class) respectively; nevertheless, with rational coefficients the
validity of the statement regarding the surjectivity of the cycle class map is still an open question
and it is known as the Hodge conjecture. In a more general and ambitious framework, there exists
another conjecture, called the generalized Hodge conjecture, which deals with sub-Hodge structures
of smooth projective varieties in different weights and levels. To be more precise the conjecture for
weight kand level k−2c(or equivalently for weight kand coniveau c) says that for any rational
sub-Hodge structure H⊂Hk(X, Q) of level at most k−2cthere exists a closed subvariety Z ֒→X
of codimension ≥csuch that
H⊂im nHk−2c(e
Z, Q(−c)) γ∗
−→ Hk(X , Q)o
Date: December 2022.
1

2 IV ´
AN ROSAS SOTO
where γ∗=i∗◦d∗,i∗is the Gysin map associated to the inclusion i:Y ֒→Xand d:e
Z→Zis a
resolution of singularities.
Although the use of rational coefficients in the statement of both conjectures is necessary, in
recent years Rosenschon and Srinivas proved that the validity of the Hodge conjecture with rational
coefficients is equivalent to an ´etale version of it with integral coefficients, using Lichtenbaum
cohomology groups, denoted by CHk
L(X), as presented in [RS16, Theorem 1.1]. For this, they
constructed a L-cycle class map ck
L: CHk
L(X)→H2k
B(X, Z(k)) such that im(ck
L)⊂Hdg2k(X, Z).
This new approach gives that the restriction CHk
L(X)tors →H2k
B(X, Z(k))tors is surjective in an
unconditional way.
In the present article we study an ´etale (or Lichtenbaum) version of the generalized Hodge con-
jecture using Lichtenbaum cohomology groups, and ask whether or not it is possible to give an
equivalence between the generalized Hodge conjecture with Q-coefficients and its ´etale integral ver-
sion. With this purpose we revisited the equivalence regarding the validity of the Hodge conjecture
given in [RS16]. Also we study counter-examples to the integral Hodge conjecture where the non-
algebraic class could arise as a torsion or torsion-free class, and giving a description of them using
Lichtenbaum cohomology groups.
We start by looking at the two different definitions for the ´etale analogue of Chow groups:
Using the triangulated category of motives as in [CD16] and Lichtenbaum cohomology groups as in
[RS16]. For the first case, we set the functorial behaviour of the ´etale Chow groups with respect to
projective morphisms. These functorial properties allows us to define correspondences in the ´etale
setting and a composition law as in the classical case. After that we present some of the principal
properties of Lichtenbaum cohomology and we make the link between the two definitions of ´etale
Chow group. With the the properties that we give we construct the category of ´etale Chow motives,
denoted by Chow´et(k), fits in the following commutative diagram:
Chow(k)op DM(k)
Chow´et(k)op DM´et (k)
Φ
Φ´et
where Chow(k) is the category of Chow motives, DM(k) and DM´et(k) are the triangulated cate-
gories of motives over kand its ´etale counterpart respectively, and the horizontal arrows are full
embedding.
After that we give a refined version of [RS16, Theorem 1.1]. In order to prove and establish this
version, we use a generalized version of Roitman’s theorem stating that for a smooth and pro jective
complex varieties:
Proposition (see Proposition 3.1.5).Let Xbe a smooth projective variety over C, and fix an integer
ksuch that 0≤k≤dimC(X). Then the induced map of torsion groups Φk
Ltors :CHk
L(X)homtors →
Jk(X)tors is an isomorphism.
where Jk(X) is the k-th intermediate Jacobian of Xand Φk
L: CHk
L(X)hom →Jk(X) is the
Lichtenbaum Abel-Jacobi map. With this generalized version of Roitman’s theorem we are able to
state and prove the refined version of [RS16, Theorem 1.1]:
Proposition (see Proposition 3.1.7).Let Xbe a complex smooth projective variety and consider
a sub-Hodge structure W⊂H2k
B(X, Z(k)) of type (k, k). Then Wis L-algebraic, i.e. W⊂im(ck
L),
if and only if W⊗Qis algebraic.
Concerning the counter-examples, in claims 3.2.1 and 3.2.2 we give an explicit description of the
torsion classes which arise as counter-examples to the integral Hodge conjecture given in [AH62] and
[BO20] respectively. Since in both cases the class that is not algebraic is a torsion class, the main
result that we used is the fact that for Lichtenbaum cohomology with finite coefficients we have the
isomorphism Hm
L(X, Z/ℓr(n)) ≃Hm
´et (X, µ⊗n
ℓr) which is a consequence of the Bloch-Kato conjecture
proved by Voevodsky (see [CD16, Section 4] for an argument in terms of rigidity of ´etale motives).
We need to remark that the way we use the rigidity theorem are different in both cases: in the

HODGE STRUCTURES THROUGH AN ´
ETALE MOTIVIC POINT OF VIEW 3
first case we consider two things, that the counter-example comes from a Godeaux-Serre variety
X, so there is a Serre spectral sequence associated fibration BG →Y→X, and the Steenrod
operations for ´etale cohomology. For the second case, which comes from the product of a very
general curve Cof genus ≥1 and a smooth Enriques surface S, we used the fact that Br(S) = Z/2
and the K¨unneth formula for integral and finite coefficients. After that, in Proposition 3.2.5 we
study general properties of the Lichtenbaum cohomology groups of smooth hypersurfaces in Pn+1
C
obtaining that their higher Brauer groups are zero and consequently CHk
L(X)⊗Z/ℓr≃H2k
´et (X, µ⊗k
ℓr).
This allows us in remark 3.2.7 to give a better description of the Lichtenbaum classes for the K´ollar
counter-example and stating the differences with motivic cohomology and the failure of the Hodge
conjecture with integral coefficients.
Starting from [RS16, Remark 5.1.a] we use the ´etale analogue of the generalized Hodge conjecture
given there in order to study the classical version. In Proposition 3.3.6 we give a complete proof of
the equivalence between the different versions of the generalized Hodge conjecture (usual case and
Lichtenbaum) in weight 2k−1 and level 1, result that was stated in the same remark in [RS16]. For
that, we split the proof in two parts: in the first, we prove that the L-generalized Hodge conjecture
in weight 2k−1 and level 1 is equivalent to the fact that a part of the Hodge conjecture for the
product of X×Cis true for all smooth and projective curve C, after that we invoke Proposition
3.1.7. To finalize, our main results are the following:
First, we obtain a characterization of the generalized conjecture (for all X∈SmProjC) given
in [RS16] which follows the idea of the classical case, that is, in term of realization of motives
previously defined in section 2 and the Hodge conjecture:
Theorem (see Theorem 3.3.8).The Lichtenbaum generalized Hodge conjecture for all X∈SmProjC
holds if and only if the following two conditions hold:
•the Lichtenbaum Hodge conjecture holds,
•a homological ´etale motive is effective if and only if its Hodge realization is effective.
With this, we obtain as a corollary the following equivalence:
Corollary (see Corollary 3.3.9).The generalized Hodge conjecture with Q-coefficients holds if and
only if the generalized integral L-Hodge conjecture holds.
The article is organized as follows: In the first two subsections of section 2 we recall the different
definitions that we can use as the ´etale analogue of Chow groups, using the category of ´etale
motives as in [CD16] and Lichtenbaum cohomology groups, stating the similarities between the two
definitions. In subsections 1.3 and 1.4 we present the correspondences with their classical operation
and the construction of the category of Chow ´etale motives as in the case of Chow motives, for
instance see [MNP13].
In sub-section 3.1 we mainly focus on the Hodge conjecture and its generalized version. We start
by some reminders and definitions of classical Hodge theory. For the complex case we give the proof
of the generalized Roitman theorem and with this result we show that if we restrict to a sub-Hodge
structure W⊂H2k
B(X, Z(k)) and ask whether or not W⊗Qis algebraic in the usual sense, it
is equivalent to ask if Wis L-algebraic. Following the revisit of the classical Hodge conjecture
through an ´etale point of view, in sub-section 3.2 we continue with the explicit description of the
torsion classes that are not algebraic in the classical sense, for the counterexamples presented in
[AH62] and [BO20]. After that the main topic in sub-section 3.2.3 is the study of the Lichtenabum
cohomology groups for hypersurfaces, giving general results for them and by that, we explain the
torsion-free counter-example of K´ollar given in [BCC92].
With respect to the generalized Hodge conjecture, in sub-section 3.3 we show several equivalences
between the classical case and the L-version (involving Lichtenbaum cohomology and integral Hodge
structures) in different weights and levels using characterizations through the Hodge conjecture
(´etale and classical setting) and the effectiveness of motives. In the last subsection we mention the
consequences of the equivalence between the classical and ´etale version of the generalized Hodge
conjecture regarding Bardelli’s example in [Bar91].

4 IV ´
AN ROSAS SOTO
Conventions
For a field kwe denote the n-dimensional k-projective space as Pn
kand SmProjkis the category of
smooth and projective reduced k-schemes. Let Gbe an abelian group, ℓa prime number and r≥1,
then we denote G[ℓr] := {g∈G|ℓr·g= 0},G{ℓ}:= SrG[ℓr], Gtors denotes the torsion sub-group
of Gand Gfree := G/Gtors its torsion free quotient. The prefix “L-” indicates the respective version
of some result, conjecture, group, etc. in the Lichtenbaum setting. Hi
B(X, Z(n)) denotes the Betti
cohomology groups of X.
Acknowledgements
The author thanks to his advisor Fr´ed´eric D´eglise and Johannes Nagel for their suggestions, useful
discussions and the time for reading this article. This work is supported by the EIPHI Graduate
School (contract ANR-17-EURE-0002). We thank the French “Investissements d’Avenir” project
ISITE-BFC (ANR-15-IDEX-0008) and the French ANR project “HQ-DIAG” (ANR-21-CE40-0015).
2. ´
Etale Chow groups and Lichtenbaum cohomology
We have an ´etale analogue of the Chow groups, which leads us to the construction of the category
of ´etale Chow motives with integral coefficients. In section 1.1 we recall the definition of the ´etale
Chow groups and present correspondences in this setting. It should be noted that the category that
we construct cannot be defined as the subcategory of DM´et (k) generated by elements of pure weight
0, in the sense of Bondarko, contrary to the Nisnevich case, for the definition of weight structure
see [Bon14, Section 1] and [Bon14, Theorem 2.1.1]. For a detailed explanation of the non-existence
of the weight structure see [CD16, Remark 7.2.26].
2.1. ´
Etale Chow groups. In this subsection we use the category of ´etale motives, since we do
not mention much more details about the construction and/or functorial behaviour of the category,
for further details about these properties we refer the reader to [Ayo14] and [CD16]. Let kbe a
field and let Rbe a commutative ring. We denote the category of effective motivic ´etale sheaves
with coefficients in Rover the field kas DMeff
´et (k, R). If we invert the Lefschetz motive, we then
obtain the category of motivic ´etale sheaves denoted by DM´et(k, R). If no confusion arises we denote
DM(eff)
´et (k) := DM(eff)
´et (k, Z). One defines the ´etale motivic cohomology group of bi-degree (m, n)
with coefficients in a commutative ring Ras
Hm
M,´et(X , R(n)) := HomDM´et (k,R)(M´et(X), R(n)[m]).
where M´et(X) = ρ∗M(X) with ρis the canonical map associated to the change of topology ρ:
(Smk)´et →(Smk)Nis which induces an adjunction ρ∗:= Lρ∗: DM(k)⇄DM´et(k) : Rρ∗=: ρ∗. In
particular we define the ´etale Chow groups of codimension nas the ´etale motivic cohomology in
bi-degree (2n, n) with coefficients in Z, i.e.
CHn
´et(X) : = H2n
M,´et(X, Z(n))
= HomDM´et (k)(M´et (X),Z(n)[2n]).
Remark 2.1.1.Let kbe a field and let ℓbe a prime number different from the characteristic of k.
By the rigidity theorem for torsion motives, see [CD16, Theorem 4.5.2], we have an isomorphism
Hm
M,´et(X, Z/ℓr(n)) ≃Hm
´et (X, µ⊗n
ℓr).
2.1.1. Gysin morphism and functoriality properties. With respect to functoriality properties of the
´etale Chow groups we should mention that we can recover well-known properties analogous to that
of classical Chow groups, such as pull-back and proper pushforwards of cycles. In particular, we
get a degree map. All these properties will arise from the properties of the category DM´et (k) (resp.
DM(k)) and the covariant functor M´et(−) (resp. M(−)).
Let us recall that the canonical map ρ: (Smk)´et →(Smk)Nis induces an adjunction of triangu-
lated categories
ρ∗: DMgm(k)⇄DMgm,´et(k) : ρ∗,

HODGE STRUCTURES THROUGH AN ´
ETALE MOTIVIC POINT OF VIEW 5
which leads us to express the ´etale Chow groups in terms of morphism in the category DM(k) as
follows
Hm
M,´et(X , Z(n)) := HomDM´et(k)(M´et(X),Z´et (n)[m])
≃HomDM(k)(M(X), ρ∗Z´et (n)[m]).
Proposition 2.1.2. The comparison map
σm,n :Hm
M(X, Z(n)) →Hm
M,´et(X, Z(n))
coming from the adjunction of triangulated categories, is compatible with pullbacks, pushforward
and intersection products.
Proof. Consider the adjunction of triangulated categories
ρ∗: DM(k)⇄DM´et (k) : ρ∗
where ρ∗is the functor induced by the ´etale sheafification and ρ∗is the right adjoint which is a
forgetful functor which forgets that the complexes are ´etale. As we have said, the cycle class map
is obtained by the following use of the adjunction
HomDM(k)(M(X),Z(n)[m]) ρ∗
−→ HomDM´et (k)(ρ∗M(X), ρ∗Z(n)[m])
where Z(n) is the motivic complex of twist n and M(X) is the triangulated motive associated with
X. By adjunction we have
HomDM´et (k)(ρ∗M(X), ρ∗Z(n)[m]) ≃HomDM(k)(M(X), ρ∗ρ∗Z(n)[m])
so we obtain a canonical map Z(n)→ρ∗ρ∗Z(n) = ρ∗Z´et(n) given by the unit transformation
associated to the adjunction. Now, the functorial properties of maps f:X→Ycome from the
(covariant) functorial properties of the motive M(X) and the existence of Gysin maps, for more
details about the existence of Gysin morphisms we refer to [Deg12] and [Deg08]. To be more precise:
Let f:Y→Xbe a morphism of relative dimension d, then we have induced commutative squares
M(X)(d)[2d]M(Y)M(Y)M(X)
M´et (X)(d)[2d]M´et(Y)M´et (Y)M´et(X)
f!
ρ∗ρ∗
f∗
ρ∗ρ∗
f!f∗
which induce the pullback and pushforward for proper morphisms. In fact, any morphism of
motivic complexes like the one given by the adjunction will yield a morphism of cohomology theory
compatible both with pullbacks and pushforward.
Finally, we need to prove the compatibility with respect to products. This property comes from
the fact that we have a quasi-isomorphism
Z(i)⊗Z(j)∼
−→ Z(i+j)
and that the functor ρ∗is monoidal, i.e. ρ∗(M⊗N)≃ρ∗(M)⊗ρ∗(N), therefore we also obtain
that
Z(i)´et ⊗Z(j)´et
∼
−→ Z(i+j)´et
For intersection products the remaining part is to consider that the product comes from the oper-
ation α·β= ∆∗(α⊗β). 
2.2. Lichtenbaum cohomology groups. We consider a second notion of the ´etale version of
Chow groups, which is the well known Lichtenbaum cohomology groups, groups defined by the
hypercohomology of the ´etale sheafification of Bloch’s complex sheaf. These groups are character-
ized by Rosenschon and Srinivas in [RS16] using ´etale hypercoverings. In this context, we consider
Smkas the category of smooth separated k−schemes over a field k. We denote zn(X, •) the cycle
complex of abelian groups defined by Bloch
zn(X, •) : · · · → zn(X, i)→ · · · → zn(X, 1) →zn(X, 0) →0

6 IV ´
AN ROSAS SOTO
where the differentials are given by the alternating sum of the pull-backs of the face maps and
whose homology groups define the higher Chow groups CHn(X, 2n−m) = Hm(zn(X , •)).
Let us recall that zn(X, i) and the complex zn(X, •) is covariant functorial for proper maps and
contravariant functorial for flat morphisms between smooth k-schemes, see [Blo86, Proposition 1.3],
therefore for a topology t∈ {flat,´et,Nis,Zar}we have a complex of t-presheaves zn(−,•) : U7→
zn(U, •). In particular the presheaf zn(−, i) : U7→ zn(U, i) is a sheaf for t∈ {flat,´et,Nis,Zar},
see [Ge04, Lemma 3.1], and then zn(−,•) is a complex of sheaves for the small ´etale, Nisnevich and
Zariski sites of X. We set the complex of t-sheaves
RX(n)t= (zn(−,•)t⊗R) [−2n]
where Ris an abelian group and for our purposes we just consider t= Zar or ´et and then we
compute the hypercohomology groups Hm
τ(X, RX(n)τ). For example, setting t= Zar and R=Z
the hypercohomology of the complex allows us to recover the higher Chow groups CHn(X, 2n−m)≃
Hm
Zar(X, Z(n)). We denote the motivic and Lichtenbaum cohomology groups with coefficients in R
as
Hm
M(X, R(n)) = Hm
Zar(X, R(n)), Hm
L(X, R(n)) = Hm
´et (X, R(n))
and in particular we set CHn
L(X) = H2n
L(X, Z(n)). Let π:X´et →XZar be the canonical morphism
of sites, then the associated adjunction formula ZX(n)→Rπ∗π∗ZX(n) = Rπ∗ZX(n)´et induces
comparison morphisms
Hm
M(X, Z(n)) κm,n
−−−→ Hm
L(X, Z(n))
for all bi-degrees (m, n)∈Z2. We can say more about the comparison map, due to [Voe11, Theorem
6.18], the comparison map κm,n :Hm
M(X, Z(n)) →Hm
L(X, Z(n)) is an isomorphism for m≤n+ 1
and a monomorphism for m≤n+ 2.
In some cases it is possible to obtain more information about the Lichtenbaum cohomology
groups and the comparison map between them and higher Chow groups. For instance there is a
quasi-isomorphism AX(0)´et =A, the latter as an ´etale sheaf, thus we obtain that the Lichtenbaum
cohomology agrees with the usual ´etale cohomology, i.e. Hm
L(X, A(0)) ≃Hn
´et(X, A) for all m∈Z≥0
and in particular CH0
L(X) = Zπ0(X). In the next step, n= 1, since there is a quasi-isomorphism of
complexes ZX(1)´et ∼Gm[−1] we obtain the following isomorphisms
CH1(X)≃CH1
L(X) = Pic(X)
H3
L(X, Z(1)) ≃H3
´et(X, Gm[−1]) = Br(X)
where Pic(X) and Br(X) are the Picard and Grothendieck-Brauer groups of Xrespectively. In
fact bi-degree (n, 1) by [VSF00, Corollary 3.4.3] there exists an isomorphism Hn
M(X, Z(1)) ≃
Hn−1
Zar (X, Gm) because the quasi-isomorphism ZX(1) ∼Gm[−1] also holds in Zariski topology. As
a particular case consider H3
M(X, Z(1)) ≃H2
Zar(X, Gm) = 0 because Hm
M(X, Z(n)) = 0 if m > 2n
where the Grothendieck-Brauer group of Xis not always zero (for instance consider Xan Enriques
surface).
In bi-degree (4,2) the comparison map is known to be injective but in general not surjective; we
have a short exact sequence
0→CH2(X)κ2
−→ CH2
L(X)→H3
nr(X, Q/Z(2)) →0,
where H3
´et(Q/Z(2)) is the Zariski sheaf associated to U7→ H´et(U, Q/Z(2)) and unramified part is
the global section H3
nr(X, Q/Z(2)) = Γ(X, H3
´et(Q/Z(2))), for a proof we refer to [Ka12, Proposition
2.9]. If k=Cthe latter group surjects onto the torsion of the obstruction, in codimension 4, to the
integral Hodge conjecture, i.e.
H3
nr(X, Q/Z(2)) ։Hdg4(X, Z)/im c2: CH2(X)→H4
B(X, Z(2))tors
and then in general is not zero and the comparison map κ2is not surjective, for more details see
[C-TV12, Th´eor`eme 3.7].

HODGE STRUCTURES THROUGH AN ´
ETALE MOTIVIC POINT OF VIEW 7
Remark 2.2.1.The adjunction formula for rational coefficients, the morphism QX(n)→Rπ∗QX(n)´et
turns out to be an isomorphism (see [Ka12, Th´eor`eme 2.6]), thus Hm
M(X, Q(n)) ≃Hm
L(X, Q(n)) for
all (m, n)∈Z2.
If Ris torsion then we can compute the Lichtenbaum cohomology as an ´etale cohomology. To
be more precise for a prime number ℓ,r∈N≥1 and R=Z/ℓrthen we have the following
quasi-isomorphisms
(Z/ℓr)X(n)´et
∼
−→ (µ⊗n
ℓrif char(k)6=ℓ
νr(n)[−n] if char(k) = ℓ
where νr(n) is the logarithmic Rham-Witt sheaf. After passing to direct limit we have also quasi-
isomorphisms
(Qℓ/Zℓ)X(n)´et
∼
−→ (lim
−→rµ⊗n
ℓrif char(k)6=ℓ
lim
−→rνr(n)[−n] if char(k) = ℓ
and finally set (Q/Z)X(n)´et =L(Qℓ/Zℓ)X(n)´et
∼
−→ Q/Z(n)´et. The following result is a well known
fact about the morphism ZX(n)→Rπ∗ZX(n)´et for n≥dim(X) over k=¯
k.
Lemma 2.2.2. [VSF00, Theorem 4.2],[Ge18, Section 2] Let Xbe a smooth projective variety of
dimension dover an algebraically closed field k. Then for n≥dthe canonical map π:X´et →XZar
the induced morphism between complexes of Zariski sheaves ZX(n)→Rπ∗ZX(n)´et is a quasi-
isomorphism.
Proof. Having that QX(n)→Rπ∗QX(n)´et is a quasi-isomorphism for all n∈N, then we only have
to focus on torsion coefficients. In characteristic zero this was already proved by Suslin in [VSF00,
Prop. 4.1, Thm. 4.2], an in general away from the characteristic of the field k.
For the general case, assume that n=dand l∈N. If khas positive characteristic then
by [Ge10, Lemma 2.4] for a constructible sheaf Fwe have that RHom(F,Z/l(d)[2d])[−1] ∼
=
RHom(F,Z(d)[2d]) and also there exists a perfect pairing of finite groups Ext1−m(F,ZX(d)[2d]) ×
Hm
c(X´et ,Z/l)) →Q/Zso this gives us an isomorphism H2d−m
M(X, Z/l(d))∗≃Hm
c(X´et ,Z/l). Since
Xis smooth, Poincar´e duality holds for ´etale cohomology, see [Mil80, Chapter VI §11], then
Hm
c(X´et ,Z/l))∗≃H2d−m
´et (X, Z/l(d)) and therefore we obtain the isomorphisms H2d−m
M(X, Z/l(d)) ≃
H2d−m
´et (X, Z/l(d)) ≃H2d−m
L(X, Z/l(d)). As in [VSF00, Theorem. 4.2] for a general n≥dwe use
the homotopy invariance of the higher Chow groups
H2d−m
M(X, Z/l(d)) ≃H2d−m
M(X×An
k,Z/l(d))
≃Hm
c(X×An
k,Z/l)∗
≃Hm−2(n−d)
c(X, Z/l(d−n))∗.
To conclude, we have a quasi-isomorphism (Z/l)X(n)→Rπ∗(Z/l)X(n)´et for all l∈Ntherefore
(Q/Z)X(n)→Rπ∗(Q/Z)X(n)´et as well. Thus from the commutative diagram
Hm−1
M(X, Q/Z(n)) Hm
M(X, Z(n)) Hm
M(X, Q(n)) Hm
M(X, Q/Z(n)) →
Hm−1
L(X, Q/Z(n)) Hm
L(X, Z(n)) Hm
L(X, Q(n)) Hm
L(X, Q/Z(n)) →
≃≃ ≃
we then conclude that Hm
M(X, Z(n)) ≃Hm
L(X, Z(n)). 
2.3. Category of ´etale Chow motives.
2.3.1. Correspondences. Proceeding in a similar way as in the case of pure motives, we need to
introduce the concept of correspondences which have a main role in the definition of the morphisms
in the category of ´etale Chow motives. For this construction we will use the notion of ´etale Chow
groups, but always keeping in mind the following: Consider a field kof characteristic exponent p
and a smooth k-scheme Xwhich is of finite type. For all bi-degree (m, n)∈Z2there exists a map
ρm,n
X:Hm
L(X, Z(n)) →Hm
M,´et(X, Z(n)) which is induced by the A1-localization functor of effective

8 IV ´
AN ROSAS SOTO
´etale motivic sheaves. If we tensor by Z[1/p], then ρm,n
Xbecomes an isomorphism, for a proof we
refer to [CD16, Theorem 7.1.2]. In particular the two definitions coincide in characteristic zero.
Definition 2.3.1. Let Xand Ybe smooth projective varieties. An ´etale correspondence from X
to Yof degree ris defined as follows: if Xis purely of codimension d
Corrr
´et (X, Y ) = CHr+d
´et (X×Y).
For the general case
Corrr
´et (X, Y ) =
n
M
i=1
CHr+di
´et (Xi×Y)
where X=`n
i=1 Xiand diis the dimension of Xi.
For α∈Corrr
´et(X, Y ) and β∈Corrs
´et(Y , Z) we define the composition β◦α∈Corrr+s
´et (X, Z ) of
correspondences as
β◦α= (pr13)∗(pr∗
12α·pr∗
23β)
where pr12 :X×kY×kZ→X×kY(similar definition for pr23 and pr13 with the respective change
in the projection’s components).
Proposition 2.3.2. The composition of correspondences is an associative operation.
Proof. To see that this operation is associative, we recall Gysin morphism for ´etale motives. Con-
sider X,Yand Ssmooth schemes over ksuch that there exists a cartesian square of smooth
schemes
(1)
X×SY Y
X S
q
gf
p
with pand qare projective morphism and dim(X/S) = dim(X×SY /Y ), thus by [Deg08, Proposition
5.17] we have the following commutative diagrams
(2)
M(X×SY)(−n)[−2n]M(Y) CHi+n
´et (X×SY) CHi
´et(Y)
M(X)(−n)[−2n]M(S) CHi+n
´et (X) CHi
´et(S)
g∗
q∗
f∗
q∗
p∗
g∗
p∗
f∗
where n= dim(X/S).
Consider the following commutative diagram
X×Y×Z×W X ×Y×Z
X×Z×W X ×Z
prXY ZW
XY Z
prXY ZW
XZW prX Y Z
XZ
prXZW
XZ
by (2) we have that prXY ZW
XY Z ∗prXY Z W
XZ W ∗=prX Y Z
XZ ∗prX ZW
XZ ∗, so the rest of the proof is
similar to the proof of [Ful84, 16.1.1.(a)]. 
Remark 2.3.3.•The composition of correspondences on Corr´et (X, X ) induces a binary op-
eration which gives to it a ring structure. In general it is not a commutative operation.
•Let Xbe a smooth projective scheme of dimension n, then the ´etale cycle ∆´et
X, induced by
the diagonal, is the identity for the composition operation, i.e. for α∈Corrr
´et(X, Y ) and
β∈Corrr
´et(Y , X) we obtain that α◦∆´et
X=αand ∆´et
X◦β=β.

HODGE STRUCTURES THROUGH AN ´
ETALE MOTIVIC POINT OF VIEW 9
2.3.2. Operation of correspondences. It is possible to define the addition and product of correspon-
dences in the following way: suppose that α∈Corr´et (X, X ) and β∈Corr´et (Y, Y ), then we define
the element α+βas the element resulting from the following operation on cycles:
CH´et(X×X)⊕CH´et(Y×Y)֒→CH´et XaY×XaY
(α, β)7→ (i1)∗α+ (i2)∗β
where i1:X×X ֒→(X`Y)×(X`Y) is the usual closed immersion map (similar definition for
i2and Y). In a similar way we define the tensor product of correspondences as
CH´et(X×X)⊗CH´et(Y×Y)→CH´et (X×Y×X×Y)
(α, β)7→ pr∗
XX α·pr∗
Y Y β
where prXX :X×Y×X×Y→X×X, similar definition for prY Y . Both of this structures will
play a big role for the definition of operations in the category of Chow ´etale motives. Another
important operation is the transposition of cycles
Definition 2.3.4. Let Xand Ybe smooth projective varieties and let τ:X×Y→Y×Xwhich
permutes the components (x, y)7→ (y, x). Let Γ∈CHn
´et (X×Y), we define the transpose cycle as
Γt:= τ∗(Γ).
2.3.3. Action on cycles and cohomology groups. Let Xand Ybe smooth projective varieties. For
a correspondence Γ ∈Corrr
´et(X, Y ) we define the action Γ∗: CHi
´et(X)→CHi+r
´et (Y) as
Γ∗Z= prY∗(Γ ·pr∗
X(Z)) ∈CHi+r
´et (Y)
for Z∈CHi
´et(X). Here arises the necessity of working with ´etale Chow groups, because of its
functoriality properties for proper maps, instead of Lichtenbaum cohomology. In order to use
an action considering Lichtenbaum cohomology, it would be necessary to invert the exponential
characteristic of the base field.
Let Γ ∈Corrr
L(X, Y ) be a correspondence of degree r, then we have an operation on Betti (with
integral coefficients) and ℓ-adic cohomology Γ∗:Hi(X)→Hi+2r(Y) defined as in the following
expression
Γ∗z:= prY∗(cL(Γ) ∪pr∗
X(z))
with z∈Hi(X). As we will see in the following sections and subsections this notion of actions will
be the cornerstone to have a well defined Hodge conjecture and generalized Hodge conjecture in
the Lichtenbaum setting.
2.4. ´
Etale Chow motives. Let SmProjkbe the category of smooth projective varieties over k.
We construct the category of effective Lichtenbaum motives over k, denoted by Choweff
´et (k), as
follows:
•The elements are tuples (X, p) where Xis a smooth projective variety and p∈Corr0
´et(X, X )
is an idempotent element, i.e. p◦p= 0.
•Morphism (X, p)→(Y , q) are the elements of the form f=q◦g◦pwhere g∈Corr0
´et(X, Y ),
therefore
HomChoweff
´et (k)((X, p),(Y , q)) = q◦Corr0
´et(X, Y )◦p
Finally, the category Chow´et (k) of Chow ´etale motives is defined in the following way: the objects
are triplets (X, p, m) where Xis a smooth pro jective variety, pis a correspondence of degree 0 and
idempotent and m∈Z. The morphisms (X, p, m)→(Y, q, n) are defined as
HomChow´et(k)((X , p, m),(Y, q, n)) = q◦Corrn−m
´et (X, Y )◦p
As in the theory of Chow motives, for ´etale motives there is an obvious fully faithful functor
Choweff
´et (k)֒→Chow´et (k).

10 IV ´
AN ROSAS SOTO
We define a functor h´et : SmProjop
k→Chow´et(k) as
h´et : SmProjk→Chow´et(k)
X7→ h´et (X) := (X, idX,0)
Xf
−→ Y7→ h´et (Y)h´et (f)
−−−−→ h´et(X)
where idXis the element that acts as the identity on the correspondences from Xto itself and
h´et (f) = κ([Γt
f]).
Proposition 2.4.1. Similar to the theory of pure Chow motives, there exists a fully faithful em-
bedding functor F:Chow´et (k)op ֒→DM´et (k)
Proof. Let X, Y, Z ∈SmProjk. The map ǫX,Y : HomDM´et(k)(M(X), M (Y)) ≃
−→ Corr0
´et(Y , X) is an
isomorphism, which can be obtained with the same arguments as in [MVW06, Proposition 20.1].
The compatibility with respect composition is obtained as in [Jin16, Theorem 3.17] using [Jin16,
Proposition 2.39]. 
3. Hodge and Generalized Hodge conjecture
3.1. Hodge conjecture and Lichtenbaum cohomology. Fix an integer k∈Z. An integral pure
Hodge structure Hof weight kis a finitely generated Z-module such that H⊗C=Lp+q=kHp,q
where Hp,q is a complex vector space with Hq,p =Hp,q . For m∈Zwe denote as Z(m) the Tate
Hodge structure of weight −2mwhose Hodge decomposition is concentrated in bi-degree (−m, −m).
For a pure Hodge structure Hof weight kits Tate twist H(m) is defined to be the tensor product
H⊗ZZ(m) which is a Hodge structure of weight k−2mand its decomposition is
H(m)⊗ZC=M
p+q=k−2m
H(m)p,q =M
p+q=k−2m
Hp−m,q−m
If Xis a complex smooth projective variety we denote by Hdg2n(X, Z) the Hodge classes of X
of weight 2n, defined as
Hdg2n(X, Z) := nα∈H2n
B(X, Z(n)) ρ(α)∈FnH2n(X, C)o
where ρ:H2n
B(X, Z)→H2n(X, C) and FpH2n(X, C) = Li≥pHi,2n−i(X). Note that by def-
inition H2n
B(X, Z)tors ⊂Hdg2n(X, Z). The image of the cycle class map to Betti cohomology
cn: CHn(X)→H2n
B(X, Z(n)) is contained in Hdg2n(X, Z). We denote as HCn(X) the following
statement:
Conjecture 3.1.1 (Hodge conjecture with integral coefficients).For a complex smooth projective
variety Xand n∈Nthe image of the cycle class map cn:CHn(X)→H2n
B(X, Z(n)) is Hdg2n(X, Z).
If we replace in the conjecture Zby Q, we will denote as HCn(X)Q. For n6= 1 it is known that the
Hodge conjecture (with integral coefficients) does not hold, even if we work with torsion free classes.
We define the obstruction to the integral Hodge conjecture as Z2i(X) := Hdg2i(X, Z(i))/im(ci).
In [AH62] it is proved that for every prime number pthere exists a smooth variety Xsuch that
Z4(X)[p]6= 0.
This conjecture can be stated in terms of motives as well, using the Hodge realization to char-
acterize the validity of the conjecture for the category SmProjC
Proposition 3.1.2. Consider k=Cand ρHthe Hodge realization for Chow(C)(with rational
coefficients), then HC(X)Qholds for all X∈SmProj if and only if ρHis a full functor.
Before going into the proof of the equivalences of the weaker version of the equivalence between
the Hodge conjecture with rational coefficients and the Lichtenbaum Hodge conjecture let us recall
the definitions of Deligne cohomology and intermediate Jacobians and some facts about them. Fix
an integer k≥0 and let us denote the k−th intermediate Jacobian Jk(X) as the complex torus
Jk(X) := H2k−1(X, C)/(FkH2k−1(X, C)⊕H2k−1(X, Z)).

HODGE STRUCTURES THROUGH AN ´
ETALE MOTIVIC POINT OF VIEW 11
Consider the Deligne complex Z(p)Dof a complex manifold Xdefined as
0→Z(p)→ OX→Ω1
X→...→Ωp−1
X→0.
We then define the Deligne cohomology groups as the hypercohomology groups of the Deligne
complex i.e.
Hk
D(X, Z(p)) := Hk
an(X, Z(p)D).
which yields an exact sequence relating Hodge classes and intermediate Jacobians
0→Jk(X)→H2k
D(X, Z(k)) →Hdg2k(X, Z)→0.
Remark 3.1.3.The definition of intermediate Jacobians can be extended to pure Hodge structures
of odd weight. Assume that His a Hodge structure of weight 2k−1 then we define the complex
torus Jk(H) := HC/(FkH⊕H). This construction is functorial with respect to morphisms of Hodge
structures. For more details about these facts see [Voi02, Remarque 12.3] and [PS08, Section 3.5].
There exist maps ck
D: CHk(X)→H2k
D(X, Z(k)) and Φk
X: CHk(X)hom →Jk(X) called the
Deligne cycle class and the Abel-Jacobi map respectively. There is a useful relation between
the Deligne cycle class map, the Abel-Jacobi map and the cycle class map given by the following
commutative diagram with exact rows:
0 CHk(X)hom CHk(X)Ik(X) 0
0Jk(X)H2k
D(X, Z(k)) Hdg2k(X, Z) 0.
Φk
X
ck
ck
Dinto
For Lichtenbaum cohomology groups we have analogous maps, ck
L,D : CHk
L(X)→H2k
D(X, Z(k))
and Φk
X,L : CHk
L(X)hom →Jk(X) (the construction of the first one is done in [RS16, Theorem 4.4])
which fit in a similar commutative diagram as the one given before.
Remark 3.1.4.Let ℓbe a prime number and r∈N. Notice that the exact triangle 0 →Ω≤n−1[−1] →
ZD(n)→Z(n)→0 induces maps cm,n
D,B :Hm
D(X, Z(n)) →Hm
B(X, Z(n)) which fit in the following
commutative diagram
0Hm−1
D(X, Z(n)) ⊗Z/ℓrHm−1
D(X, Z/ℓr(n)) Hm
D(X, Z(n))[ℓr] 0
0Hm−1
B(X, Z(n)) ⊗Z/ℓrHm−1
B(X, Z/ℓr(n)) Hm
B(X, Z(n))[ℓr] 0
βD
≃cm,n
D,B
β
where βDis the morphism induced by the exact triangle 0 →ZD(n)·ℓr
−−→ ZD(n)→(Z/ℓr)D(n)→0.
Also we obtain another commutative diagram:
0Hm−1
L(X, Z(n)) ⊗Z/ℓrHm−1
´et (X, µ⊗n
ℓr)Hm
L(X, Z(n))[ℓr] 0
0Hm−1
D(X, Z(n)) ⊗Z/ℓrHm−1
D(X, Z/ℓr(n)) Hm
D(X, Z(n))[ℓr] 0
0Hm−1
B(X, Z(n)) ⊗Z/ℓrHm−1
B(X, Z/ℓr(n)) Hm
B(X, Z(n))[ℓr] 0.
≃cm,n
D,L
βD
≃cm,n
D,B
β
By the snake lemma the arrows
Hm−1
L(X, Z(n))/ℓr→Hm−1
D(X, Z(n))/ℓrand Hm−1
D(X, Z(n))/ℓr→Hm−1
B(X, Z(n))/ℓr
are injective while the arrows
Hm
L(X, Z(n))[ℓr]→Hm
D(X, Z(n))[ℓr] and Hm
D(X, Z(n))[ℓr]→Hm
B(X, Z(n))[ℓr]
are surjective. Also the image of the composite of the right vertical arrows is equal to the image of
cm,n
Lrestricted to ℓr-torsion elements.

12 IV ´
AN ROSAS SOTO
Proposition 3.1.5. Let Xbe a smooth projective variety over C, and fix an integer ksuch that 0≤
k≤dimC(X). Then the induced map of torsion groups Φk
Xtors :CHk
L(X)homtors →Jk(X)tors
is an isomorphism.
Proof. Consider the following commutative diagram with exact rows
0 CHk
L(X)hom CHk
L(X)Ik
L(X) 0
0Jk(X)H2k
D(X, Z(k)) Hdg2k(X, Z) 0
ck
D,L|hom
ck
L
ck
D,L into
Since CHk
L(X)hom ⊗Q/Z= 0 by [RS16, Proposition 5.1 (b)] and Jk(X)⊗Q/Z= 0 because Jk(X)
is divisible, we have then a commutative diagram
(3)
0CHk
L(X)homtors CHk
L(X)tors Ik
L(X)tors 0
0Jk(X)tors H2k
D(X, Z(k))tors Hdg2k(X, Z)tors 0
ck
D,L|hom ck
D,L into
Since CHk
L(X)tors ≃H2k
D(X, Z(k))tors by [RS16, Proposition 5.1 (a)] and the map CHk
L(X)tors →
H2k(X, Z)tors is surjective (see [RS16, Remark 3.2]), the middle arrow is an isomorphism as well
as the right one. Therefore the left arrow is an isomorphism. 
Remark 3.1.6.Notice that if we set k= dim(X), by Proposition 2.2.2 Chow groups and Lichnte-
baum cohomology coincide. Then we recover the classical Roitman’s theorem
CHhom
0(X)tors ≃AlbX(C)tors.
We say that W⊂H2k
B(X, Z(k)) is a sub-Hodge structure if Wis s sub-lattice of H2k
B(X, Z(k))
such that it has an induced Hodge decomposition WC=Lp+q=2kWp,q with Wp,q =WC∩Hp,q . Let
W⊂H2k
B(X, Z(k)) be sub-Hodge structure, we define the partial Hodge conjecture with rational
coefficients related to Was the following statement: for every element α∈Wthere exists N∈N
and an algebraic cycle eα∈CHk(X) such that c(eα) = Nα. It is clear that for W= Hdg2k(X, Z)
we recover the usual Hodge conjecture. For a fixed Wwe denote the previous statement by
HCk(X, W )Q.
Similarly we denote by HCk
L(X, W )Zthe statement that for every element of α∈Wthere exists
a Lichtenbaum cycle eα∈CHk
L(X) such that cL(eα) = α. Then, inspired by the proof of [RS16,
Theorem 1.1], we obtain the following result:
Proposition 3.1.7. Let Xbe a complex smooth projective variety and let W⊂H2k
B(X, Z(k)) be a
sub-Hodge structure. Then HCk
L(X, W )Zholds if and only if HCk(X , W )Qholds.
Proof. Let W⊂H2k
B(X, Z(k)) be a sub-Hodge structure and let ck
L: CHk
L(X)→H2k
B(X, Z(k))
be the Lichtenbaum cycle class map constructed in [RS16] (similarly we can consider the classical
cycle class map ck: CHk(X)→H2k
B(X, Z)). Set CHk
W,L(X) := (ck
L)−1(W) as the preimage of Win
CHk
L(X). It is easy to see that CHk
L(X)hom ֒→CHk
W,L(X). Following with this notation, we will
denote Ik
W,L(X) := im(ck
L)∩W, therefore Wis algebraic if and only if Zk
W,L(X) := W/I k
W,L(X) = 0.
In this the classical case this is equivalent to say that Ik
W(X)Qis algebraic if and only if Zk
W(X) is
a finite group. Since Ik
W(X)⊂Ik
W,L(X) we have an exact sequence
0→Ik
W,L(X)/Ik
W(X)→Zk
W(X)→Zk
W,L(X)→0
Denote π:H2k
D(X, Z(k)) →Hdg2k(X, Z) the surjective map coming from the short exact
sequence of Deligne-Beilinson cohomology, intermediate Jacobian and Hodge classes and denote
H2k
W,D(X, Z(k)) := π−1(W). Then we have the following commutative diagram:

HODGE STRUCTURES THROUGH AN ´
ETALE MOTIVIC POINT OF VIEW 13
0 CHk
L(X)hom CHk
W,L(X)Ik
W,L(X) 0
0Jk(X)H2k
W,D(X, Z(k)) W0
ck
D,L|hom
ck
L
ck
D,L|W−1
Linto
Since CHk
L(X)hom ⊗Q/Z= 0 by [RS16, Proposition 5.1 (b)] and Jk(X) is divisible, then we
obtain the commutative diagram but with the torsion part of each group
0CHk
L(X)homtors CHk
W,L(X)tors Ik
W,L(X)tors 0
0Jk(X)tors H2k
W,D(X, Z(k))tors Wtors 0
ck
D,L|hom
ck
L
ck
D,L|W−1
Linto
Due to the surjectivity of CHk
L(X)tors →H2k(X, Z)tors, the right vertical arrow is an isomorphism.
If we can prove that the arrow in the middle is surjective, then we can conclude with similar
arguments as in [RS16], but this comes from the fact that CHk
L(X)homtors ≃Jk(X)tors by
lemma 3.1.5, and therefore ck
D,L|hom induces an isomorphism in the torsion part.
Since we have an isomorphism CHk
W,L(X)tors ≃H2k
W,D(X, Z)tors, we obtain a commutative
diagram
(4)
0Ators A A ⊗QA⊗Q/Z0
0Btors B B ⊗QB⊗Q/Z0
≃ck
D,L|Ainto
where A= CHk
W,L(X) and B=H2k
W,D(X, Z(k)) and A⊗Q/Z֒→B⊗Q/Zis an injection by remark
3.1.6. We can split diagram (4) into two diagrams with short exact sequences as rows:
0Ators A Afree 0
0Btors B Bfree 0
≃ck
D,L f
and
(5)
0Afree A⊗QA⊗Q/Z0
0Bfree B⊗QB⊗Q/Z0
fck
D,L into
The cokernel from the induced map A⊗Q→B⊗Qis torsion free as a quotient of Q-vector
spaces. Thus from diagram (5), we obtain that coker(f) is torsion free because it injects into a
torsion free group, which, implies that coker(ck
D,L|A) is torsion free and, along with the divisibility
of Jk(X), so Zk
W,L(X) as well.
The remaining part of the proof consists in proving that Ik
W,L(X)/Ik
W(X) is a torsion group,
but this comes from the fact that Ik
W,L(X) and Ik
W(X) have the same Z−rank and therefore the
quotient should be a finite group, so
Zk
W(X)⊗Q= 0 ⇐⇒ Zk
W,L(X)⊗Q= 0 ⇐⇒ Zk
W,L(X) = 0.

Remark 3.1.8.Consider X∈SmProjCof dimension dand let us consider H2d
B(X×X, Z) modulo
torsion. As we consider it modulo torsion, we apply the K¨unneth H2d
B(X×X, Z)≃LH2d−i
B(X, Z)⊗
Hi
B(X, Z) and let ∆i∈H2d−i
B(X, Z)⊗Hi
B(X, Z) be the i-th component of the diagonal. Consider

14 IV ´
AN ROSAS SOTO
Wibe the sub-Hodge structure generated by ∆i, thus by Proposition 3.1.7 Wiis L-algebraic if and
only if Wi⊗Qis algebraic, therefore the rational K¨unneth conjecture for Xholds if and only if the
K¨unneth components are L-algebraic.
3.2. Examples. In the present subsection we study two counterexamples of the Hodge conjecture
with integral coefficients, which are the ones presented in [AH62] and [BO20]. Both cases deal with
torsion Hodge classes that do not come from algebraic cycles, but the construction of the examples
are different, while the first example uses arguments of K-theory the second one uses degeneration
arguments.
3.2.1. Atiyah-Hirzebruch’s countexample. Let us start by giving a quick overview of the construc-
tion of Atiyah-Hirzebruch’s counterexample presented in [AH62], which consists in a smooth projec-
tive quotient variety with a non-algebraic torsion class, being constructed using Steenrod algebra of
cohomology groups and classifying spaces. By [AH62, Theorem 6.1], if a class α∈H2p
B(X, Z) is alge-
braic then Sqi( ¯α) = 0 for all iodd prime, where ¯αis the reduction mod 2 and Sqi:H2p
B(X, Z/2) →
H2p+i
B(X, Z/2) is the i-th the Steenrod operation. Also considering [AH62, Proposition 6.6], for
every finite group Gand r∈N≥1there exists a complete intersection variety Ywith dimC(Y) = r
and Gacting freely on Y, such that for the Godeaux-Serre type variety X=Y/G, the group
cohomology Hi(G, Z) is a direct factor of Hi
B(X, Z) for all i≤r.
As a particular case, consider G=Z/2×Z/2×Z/2 and r= 7, thus there exists X∈SmProjC
such that Hi(G, Z)֒→Hi
B(X, Z) as a direct factor for i≤7. As H∗(Z/2,Z/2) ≃Z/2[u] with
deg(u) = 1, the K¨unneth formula shows that H∗(G, Z/2) ≃Z/2[u1, u2, u3]. Consider the element
u1u2u3∈H3(G, Z/2) and β(u1u2u3) =: α∈H4(G, Z)֒→H4
B(X, Z), where βis the Bockstein’s
morphism β:H3(G, Z/2) →H4(G, Z), and the following commutative diagram
H3(G, Z/2) H4(G, Z)H4
B(X, Z(2))
H4(G, Z/2) H4
B(X, Z/2(2))
H7(G, Z/2) H7
B(X, Z/2(4)).
β
Sq1red2
Sq3S q3
A direct computation gives that Sq3(Sq1(u1u2u3)) = S q3(¯α)6= 0 ∈H7
B(X, Z/2(4)), and conse-
quently αis a 2-torsion class which is not algebraic. However, we have a short exact sequence
0→J2(X)→H4
D(X, Z(2)) g
−→ Hdg4(X, Z)→0
which after tensoring by Z/2, and considering that J2(X) is divisible, induces a short exact sequence
or torsion groups
0→J2(X)[2] →H4
D(X, Z(2))[2] g
−→ Hdg4(X, Z)[2] →0
and therefore the composite map CH2
L(X)[2] →H4
D(X, Z(2))[2] →Hdg4(X, Z)[2] is surjective.
Specifically, we have the following result, which gives an explicit representative of the Lichtenbaum
class that maps to α.
Claim 3.2.1. Let Xbe a Godeaux-Serre variety as the one described previously. Then there exists
a class x∈CH2
L(X)[2] such that c2
L(x) = αand red2(x) = ¯α∈ker H4
´et (X, µ⊗2
2)→H5
L(X, Z(2)).
Also there exists x∈CH2
L(X)which maps to αand is characterized as the image of u1u2u3∈
H3(G, Z/2) in CH2
L(X).
Proof. Let Xbe a smooth projective quotient variety coming from the action of G= (Z/2)3
over Y, with Yfulfilling the above hypotheses (a complete intersection variety of dimension 7).
We consider the fibration Y→X→BG with its associated the Serre spectral sequence Ep,q
2=
Hp(BG, H q(Y, Z/2)) =⇒Hp+q(X, Z/2), where the differentials are graded derivations. Since Yis

HODGE STRUCTURES THROUGH AN ´
ETALE MOTIVIC POINT OF VIEW 15
a smooth complete intersection variety then
Hq
´et(Y , Z/2) ≃(Z/2,if qeven and q6= 7
0,if qodd and q6= 7,
therefore if q6= 7, the terms of the Serre spectral sequence are either Hp(B G, Z/2) ∼
=Hp(G, Z/2) or
0. Notice that due to the structure of the Ep,q
2-terms we have isomorphisms Ep,q
2≃Ep,q
3for q < 7.
Since d3is a graded derivation, then d3:E0,2
3≃Z/2→E3,0
3is the trivial map, so E3,0
∞≃E3,0
3
and therefore 0 →E3,0
3→H3
´et(X, Z/2), which gives us the existence of an injection H3(G, Z/2) ֒→
H3
B(X, Z/2(2)) ≃H3
´et(X, Z/2). Consider u1u2u3∈H3
B(X, Z/2(2)) and the following short exact
sequence
0→J2(X)→H4
D(X, Z(2)) g
−→ Hdg4(X, Z)→0,
and let β(u1u2u3) = α∈H4
B(X, Z(2))[2] be the non-algebraic torsion class. It can be lifted to
an element αD∈g−1(α)⊂H4
D(X, Z(2))[2] because J2(X) is divisible. It follows from [RS16,
Proposition 5.1 (a)] that it has a unique preimage ex∈CH2
L(X)[2]. Consider the exact triangle
ZX(2)´et
·2
−→ ZX(2)´et →(Z/2)X(2)´et
+1
−−→ and the following commutative diagram with exact rows
obtained from it together with the Deligne cycle class map
... CH2
L(X)H4
´et(X, µ⊗2
2)H5
L(X, Z(2)) ...
. . . H4
D(X, Z(2)) H4
B(X, Z/2(2)) H5
D(X, Z(2)) ...
red2
≃
It is an immediate consequence that red2(CH2
L(X)) = ker H4
´et(X, µ⊗2
2)→H5
L(X, Z(2))and
red2(ex) = ¯α. Take again the element u1u2u3∈H3
B(X, Z/2(2)) and consider its image in CH2
L(X)
via the map p:H3
B(X, Z/2(2)) ∼
−→ H3
´et(X, µ⊗2
2)→CH2
L(X) (map which is surjective over the
2-torsion of CH2
L(X)), denoted by x=p(u1u2u3)∈CH2
L(X)[2]. The last assertion to be proved
is that c2
L(x) = αand c2
L,D(x)∈g−1(α). For that, considering the morphisms and commutative
diagrams of remark 3.1.4, we get the following commutative diagram:
0 CH2
L(X, 1) ⊗Z/2H3
´et(X, µ⊗2
2) CH2
L(X)[2] 0
0H3
D(X, Z(2)) ⊗Z/2H3
D(X, Z/2(2)) H4
D(X, Z(2))[2] 0
H3
B(X, Z/2(2)) H4
B(X, Z(2))[2] 0.
≃c2
D,L
βD
≃c2
D,B
β
Notice that the image of the map c2
D,B restricted to 2-torsion classes is isomorphic to the image of
grestricted to such classes thus c2
L(x) = β(u1u2u3) = α.
3.2.2. Benoist-Ottem counterexample. Let us recall [BO20, Theorem 0.1]. Let Sbe an arbitrary
but fixed complex Enriques surface and let g≥1 be an integer. Then if Bis a very general smooth
projective complex curve of genus g, the integral Hodge conjecture for codimension 2 cycles does not
hold on the product B×S, and the non-algebraic class is a 2-torsion class. Since the non-algebraic
cycle is torsion, it comes from a Lichtenbaum class. In the sequel we give an explicit construction
of such a Lichtenbaum cycle.
Let C∈SmProjCbe a connected curve of genus g≥1 and Sbe the previous Enriques surface,
we have then a cycle class map c2
L: CH2
L(C×S)→H4
B(C×S, Z(2)). As is mentioned in [BO20,
Proposition 1.1], H2,0(C×S) = 0 because H1,0(S) = H2,0(S) = 0, thus the validity of the L-Hodge
conjecture for C×Srelies on the surjectivity of the map c2
L: CH2(C×S)→H4
B(C×S, Z(2)). Since
H∗
B(C, Z) is torsion free, we have the following isomorphism obtained from the K¨unneth formula:
H4
B(C×S, Z(2)) ≃H0(C, Z)⊗H4(S, Z)⊕H1(C, Z)⊗H3(S, Z)⊕H2(C, Z)⊗H2(S, Z),

16 IV ´
AN ROSAS SOTO
where H0(C, Z)⊗H4(S, Z) is algebraic and by the Lefschetz (1,1) theorem H2(C, Z)⊗H2(S, Z) is
algebraic as well, so L-algebraic. In particular if there exists a non-algebraic class, it should come
from H1(C, Z)⊗H3(S, Z).
Consider the exact sequence of abelian groups
0→Z(1) ·2
−→ Z(1) →Z/2(1) →0
which induces a short exact sequence
0→H1
B(C, Z(1)) ⊗Z/2→H1
B(C, Z/2(1)) →H2
B(C, Z(1))[2] →0.
In the case of Lichtenbaum cohomology the sequence of complexes of ´etale sheaves
0→ZC(1)´et
·2
−→ ZC(1)´et →(Z/2)C(1)´et →0
induces a short exact sequence
0→H1
L(C, Z(1)) ⊗Z/2→H1
L(C, Z/2(1)) →CH1
L(C)[2] →0.
Moreover H1
L(C, Z(1)) ⊗Z/2 = 0 because H1
L(C, Z(1)) ≃C∗is divisible, and H1
L(C, Z/2(1)) ≃
H1
´et(C, µ2)≃H1
B(C, Z/2) because of the comparison theorem of cohomologies of complex va-
rieties, and because the cohomology groups of a smooth and projective curve are torsion free
H2
B(C, Z(1))[2] = 0, therefore CH1
L(C)[2] ≃H1
B(C, Z(1)) ⊗Z/2. For the Enriques surface Scon-
sider the short exact sequence
0→H2
L(S, Z(1)) ⊗Z/2→H2
L(S, Z/2(1)) →H3
L(S, Z(1))[2] →0
where H3
L(S, Z(1))[2] = Br(S)[2] ≃Br(S)≃Z/2 (see [Bea09, 2]) and H2
L(S, Z/2(1)) ≃H2
´et(S, µ2).
We have a composite map
p:H1
´et(C, µ2)⊗H2
´et(S, µ2)֒→H3
´et(C×S, µ⊗2
2)→H4
L(C×S, Z(2))[2]
where the first inclusion is the one given by the K¨unneth formula with finite coefficients and the
second map is obtained from the short exact sequence
0→CH2(C×S, 1) ⊗Z/2→H3
´et(C×S, µ⊗2
2)→H4
L(C×S, Z(2))[2] →0(6)
induced by the exact triangle ZC×S(2)´et
·2
−→ ZC×S(2)´et →(Z/2)C×S(2)´et
+1
−−→.
Finally, we need to find an element which is not contained in the image of the induced injection
H3
B(C×S, Z(2))⊗Z/2֒→H3
´et(C×S, µ⊗2
2). So we can take c∈H1
´et(C, µ2) and denote by bS∈Br(S)
the non-zero element of the Brauer group of S. We then fix an element s∈H2
´et(S, µ2) such that s
maps to bSthrough the map H2
´et(S, µ2)→Br(S) and define γc:= p(c⊗s)∈CH2
L(C×S)[2]. In
the following result we give an explicit description of the Lichtenbaum classes that maps to a given
element of H1
B(C, Z)⊗H3
B(S, Z) in terms of γc.
Claim 3.2.2. Let Sbe an Enriques surface and Cbe a smooth projective and connected curve
curve of genus g≥1, let ec⊗e
bS∈H1(C, Z)⊗H3(S, Z)be an arbitrary class and let c∈H1
B(C, Z/2)
be the reduction mod 2 of ec. Then c2
L(γc) = ec⊗e
bS.
Proof. Let pr1:C×S→Cand pr2:C×S→Sbe the canonical projections and consider
the induced pull-backs and Bockstein homomorphisms, then we have the following commutative
squares
Hi
B(C, Z/2) Hi+1
B(C, Z)Hj
B(S, Z/2) Hj+1
B(S, Z)
Hi
B(C×S, Z/2) Hi+1
B(C×S, Z)Hj
B(C×S, Z/2) Hj+1
B(C×S, Z).
pr∗
1
β
pr∗
1pr∗
2
β
pr∗
2
β β
From now on fix i= 1, j= 2 together with a= pr∗
1(c) and b= pr∗
2(s) in order to have a∪b=c⊗s.
Since Bockstein homomorphisms satisfy derivation properties (see [Hat02, Section 3.E]), it follows

HODGE STRUCTURES THROUGH AN ´
ETALE MOTIVIC POINT OF VIEW 17
that
β(a∪b) = β(a)∪b−(−1)deg(a)a∪β(b)
=β(pr∗
1(c)) ∪b−(−1)deg(a)a∪β(pr∗
2(s))
= pr∗
1(β(c)) ∪b−(−1)deg(a)a∪pr∗
2(β(s))
=c×[S]∪[C]×e
bS=c⊗e
bS∈H1
B(C, Z)⊗H3
B(S, Z)⊂H4
B(C×S, Z)[2].
As c⊗s /∈im H3
B(C×S, Z)⊗Z/2→H3
B(C×S, Z/2), then neither can be lifted to H3
D(C×
S, Z(2)) ⊗Z/2 nor H3
L(C×S, Z(2)) ⊗Z/2 thus γc6= 0 for all c6= 0 ∈H1
B(C, Z/2). Therefore from
the commutative diagrams of Remark 3.1.4, we obtain c2
L(γc) = β(c⊗s) giving the characterization
of the preimages of the elements in H1
B(C, Z)⊗H3
B(S, Z). 
Remark 3.2.3.By the same kind of arguments as in [BO20, Proposition 1.1], mutatis mutandis, we
can obtain an equivalence between the action of Lichtenbaum correspondences and the L-Hodge
conjecture, i.e. the Lichtenbaum Hodge conjecture holds for codimension 2-cycles in the product
if and only if for every c∈H1(C, Z/2) there exists a correspondence Z∈CH2
L(C×S) such that
Z∗α=c, where α∈H1
B(S, Z/2) is the class corresponding to the degree 2 ´etale cover of Sby a
K3-surface. Since c2
L(γc) = c⊗e
bSwe have that
γ∗
cα= pr1∗(c2
L(γc)∪pr∗
2(α))
= pr1∗(pr∗
1(c)∪pr∗
2(e
bS)∪pr∗
2(α))
= pr1∗(pr∗
1(c)∪pr∗
2(e
bS∪α)).
By Poincar´e duality we have that e
bS∪αis a non-zero element in H4(S, Z/2), then γ∗
cα=c.
3.2.3. K´ollar counterexample. As we said, the situation with the Hodge conjecture does not improve
if we consider just the free part of the cohomology, due to K´ollar’s example, provided in [BCC92].
We will start by giving some general facts about smooth hypersurfaces. Consider a smooth hyper-
surface X⊂Pn+1
Cof degree d. By the Lefschetz hyperplane theorem ([Voi02, Th´eor`eme 13.23]) if
k < n then we have the isomorphism
Hk
B(Pn+1,Z)i∗
−→ Hk
B(X, Z)
and if k=n, the map i∗is an injection. Here H2k
B(Pn+1,Z)≃ZHkwith H=c1(OPn+1 (1)) and
H2k+1
B(Pn+1,Z) = 0. Since Betti cohomology groups of hypersurfaces (with integral coefficients)
are torsion free, by Poincar´e duality we obtain the isomorphisms
H2k
B(X, Z)∗≃H2(n−k)
B(X, Z).
In particular if 2k > n then H2k
B(X, Z)≃Zαwhere hα, hn−ki= 1 with h·,·i being the intersection
product and h=c1(OX(1)) = HX.
Remark 3.2.4.By the Lefschetz hyperplane section in ´etale cohomology (see [Mil80, Chapter VI,
§7]) the map Hi(X, µ⊗k
ℓr)i∗
−→ Hi+2(Pn+1
C, µ⊗k+1
ℓr) is an isomorphism if i > n and a surjection if
i=n.
In the following proposition, we give characterizations for some of the Lichtenbaum cohomology
groups of a smooth hypersurfaces Xin Pn+1
Cand study the close relation with the ´etale cohomology
groups with finite coefficients.
Proposition 3.2.5. Let i:X ֒→Pn+1
Cbe a smooth and projective hypersurface of degree dand let
j: (U:= Pn+1
C\X)→Pn+1
Cbe the open complement. Let kbe an integer with 0≤k≤nsuch that
2k /∈ {n−1, n}, then:
(1) The higher Brauer group Brk(X) := H2k+1
L(X, Z(k)) and the torsion group H2k+1
D(X, Z(k))tors
are trivial.
(2) If 2k > n then CHk+1 (U)≃Z2k(X)and CHk+1
´et (U)≃Z2k
L(X)and in particular the group
CHk+1
L(U)is trivial.

18 IV ´
AN ROSAS SOTO
Proof. Let us start with the first statement. For that, and since our base field is of characteristic
zero, fix an arbitrary prime number ℓand an (also arbitrary) natural number r. As in [RS16,
Proposition 3.1] consider the long exact sequence
...→CHk
´et(X)·ℓr
−−→ CHk
´et(X)→H2k
´et (X, µ⊗k
ℓr)→Brk(X)·ℓr
−−→ Brk(X)→H2k+1
´et (X, µ⊗k
ℓr)→...
By assumption 2k+ 1 6=nand hence H2k+1
´et (X, µ⊗k
ℓr) = 0, therefore the map Brk(X)·ℓr
−−→ Brk(X)
is surjective for all prime number ℓand all r, thus Brk(X) is divisible and torsion group. For the
remaining part, we consider the commutative diagram given in remark 3.1.4 and the short exact
sequence
0→Jk(X)→H2k
D(X, Z(k)) →Hdg2k(X, Z)→0.
Since Jk(X) is divisible, then H2k
D(X, Z(k))/ℓr≃Hdg2k(X, Z)/ℓr. Under the conditions for kwe
have the isomorphisms H2k
B(X, Z(k)) ≃Hdg2k(X, Z(k)) and since H2k+1
B(X, Z) = 0 then by the
diagram of remark 3.1.4 we conclude that H2k+1
D(X, Z(k))[ℓr] = 0.
Now consider the short exact sequence
0→CHk
L(X)/ℓr→H2k
´et (X, µ⊗k
ℓr)→Brk(X)[ℓr]→0
so Z/ℓr→Brk(X)[ℓr] is surjective and taking the direct limit, we obtain a surjection Q/Z։
Brk(X), but as Brk(X)≃(Q/Z)r(for the structure of Lichtenbaum cohomology see [Ge18, Theorem
1.1]) for some r∈Nwe have that r= 0 or 1. Since I2k(X)6= 0 we have I2k
L(X)6= 0 and hence
there are isomorphisms CHk
L(X)⊗Q/Z≃I2k
L(X)⊗Q/Z≃Q/Zso Brk(X) = 0.
For part 2. consider the localization sequence for Chow groups and its ´etale analogue. By
functoriality of the comparison map with Gysin morphisms and pull-backs we have the following
commutative diagram:
CHk+1(U, 1) CHk(X) CHk+1 (Pn+1
C) CHk+1(U) 0
CHk+1
L(U, 1) CHk
L(X) CHk+1
L(Pn+1
C) CHk+1
L(U) 0.
∂i∗j∗
≃
∂´et i∗j∗
Notice that the map CHk+1
L(Pn+1
C)j∗
−→ CHk+1
L(U) in general is not surjective, but in this case it is
as a consequence of part 1.By the functorial properties of the usual cycle class map we have the
following commutative square
CHn−k(X) CHn−k(Pn+1
C)
H2(n−k)(X, Z)H2(n−k)(Pn+1
C,Z)
i∗
cn−k≃
i∗
where if 2(n−k)< n then the map i∗:H2(n−k)(X, Z)→H2(n−k)(Pn+1
C,Z) is an isomorphism,
therefore i∗CHk(X)≃im(ck), with this and the previous commutative diagrams which relates
localization exact sequences it is easy to see that
CHk+1(U)≃coker(i∗)
≃H2(n−k)(X, Z)/im(cn−k) = Z2k(X).
For the ´etale case we proceed in the same way. The last part of the second statement is due to the
fact that i∗CHk
L(X)≃CHk+1
L(Pn+1
C)≃Ztherefore j∗: CHk+1
L(Pn+1
C)→CHk+1
L(U) has trivial
image, thus we conclude that CHk+1
L(U) injects into Brk(X), which is trivial by the first point. 
Corollary 3.2.6. Let X⊂Pn+1
Cbe a smooth projective hypersurface of degree dand let kbe an
integer such that 2k /∈ {n−1, n}. Then we have the following

HODGE STRUCTURES THROUGH AN ´
ETALE MOTIVIC POINT OF VIEW 19
(1) For all prime numbers ℓand all rthe cycle class map ck
L,ℓr:CHk
L(X)→H2k
´et (X, µ⊗k
ℓr)is
surjective.
(2) For all prime numbers ℓ, all rand ksuch that 2k /∈ {n−1, n, n + 1}the pairing
CHn−k
L(X)/ℓr⊗CHk
L(X)/ℓr→CHn
L(X)/ℓr≃Z/ℓr
is non-degenerate. The result also holds for the particular case of n= 3 and k= 2.
(3) If 2k > n there exists a class z∈CHk
L(X)such that i∗(z) = Hk+1 ∈CHk+1(Pn+1
C)where
Hk+1 is the generator of CHk+1(Pn+1
C). Furthermore ck
ℓr(z)and ck
L(z)are the generators
of the groups H2k
´et (X, µ⊗k
ℓr)and H2k
B(X, Z(k)) respectively.
Proof. This is a direct consequence of Proposition 3.2.5. Fix arbitrary prime and natural numbers
denoted ℓand rrespectively. For point 1. use the long exact sequence
...→CHk
L(X)·ℓr
−−→ CHk
L(X)→H2k
´et (X, µ⊗k
ℓr)→Brk(X)·ℓr
−−→ Brk(X)→...
since Brk(X) = 0 we obtain the surjectivity. The second part follows from the vanishing of Brk(X).
Because of that we obtain an isomorphism CHk
L(X)⊗Z/ℓr≃
−→ H2k
´et (X, µ⊗k
ℓr) and the same for
codimension n−k. Thus the non-degeneracy comes from Poincar´e duality in ´etale cohomology. For
the case n= 3 and k= 2, use that CH1
L(X)≃CH1(X)≃Z·c1(OX(1)), then CH1
L(X)⊗Z/ℓr≃
H2
´et(X, µℓr). While CH2
L(X)⊗Z/ℓr≃H4
´et(X, µ⊗2
ℓr) by Corollary 3.2.6.1. The last assertion follows
from the localization sequence and the vanishing of Brk(X) and the compatibility of the cycle class
maps with respect to push-forwards. 
Remark 3.2.7.(1) Fix the parameters n= 3 and k= 2. Notice that the Hodge classes of X
are Hdg4(X, Z) = Zwhile the image of the cycle class map I4(X) = I⊂Zis determined
by the degrees of the curves C⊂X, i.e. I= gcd ({deg(C)|C⊂X})Z. The strategy for
the counter-example to the Hodge conjecture with integral values whose non-algebraic class
is as follows: Consider a very general hypersurface Xof degree d=sp3with pa prime
number ≥5, K´ollar proved that under these assumptions, for all curve C⊂Xits degree
deg(C) is divisible by pand therefore Z4(X) = Z/m 6= 0 with mdivisible by p. Notice
that if d > 6 the Griffiths-Harris conjecture would imply that m=d. Here the class that
is not algebraic is α, whereas dα =h2.
(2) We want to remark the differences between motivic and Lichtenbaum cohomology when
we work with finite coefficients. In general, for j∈N, the group H2j+1
M(X, Z(j)) = 0, so
CHj(X)⊗Z/ℓr≃
−→ H2j
M(X, Z/ℓr(j)) and by Bloch-Ogus we know that CHj(X)⊗Z/ℓr≃
Aj(X)⊗Z/ℓrwhere Aj(X) is the group of codimension jcycles of Xmodulo algebraic
equivalence. Again consider X⊂Pn+1
Cand kas in part 2 of Corollary 3.2.6. We obtain a
commutative diagram
H2k
M(X, Z/ℓr(k)) ⊗H2(n−k)
M(X, Z/ℓr(n−k)) H2n
M(X, Z/ℓr(n)) Z/ℓr
H2k
´et (X, µ⊗k
ℓr)⊗H2(n−k)
´et (X, µ⊗n−k
ℓr)H2n
´et (X, µ⊗n
ℓr)Z/ℓr
∪
≃
degℓr
∪trℓr
where the pairing in the lower row is non-degenerate because of Poincar´e duality, whereas
the one in the upper row could be degenerate as K´ollar’s example shows or as Griffiths-
Harris’ conjecture states. By Proposition 3.2.5 there is an isomorphism CHk
L(X)⊗Z/ℓr≃
−→
H2k
´et (X, µ⊗k
ℓr), thus (if 2k > n) we always have element of degree 1 in the Lichtenbaum
groups.
3.3. Generalized Hodge conjecture and Lichtenbaum cohomology. Let Hbe a pure Hodge
structure of weight nand let 0 6=HC=H⊗C=Lp+q=nHp,q . We say that His effective if and
only if Hp,q = 0 for p < 0 or q < 0. The level of lof His defined as l= max {|p−q| | Hp,q 6= 0}.
Let Xbe a smooth projective complex variety, we write GHC(n, c, X )Qfor the generalized Hodge
conjecture in weight nand level n−2c, where the conjectured result is the following:

20 IV ´
AN ROSAS SOTO
Conjecture 3.3.1 ([Gro69, Generalized Hodge conjecture]).For every Q−sub-Hodge structure
H⊂Hn(X, Q)of level n−2cthere exists a subvariety Y⊂Xof pure codimension csuch that H
is supported on Y, i.e. His contained in the image of
H⊂im nHl(e
Y , Q(−c)) γ∗
−→ Hn(X, Q)o
where γ∗=i∗◦d∗,i∗is the Gysin map associated to the inclusion i:Y ֒→Xand d:e
Y→Yis a
resolution of singularities.
There is an equivalent assertion of the generalized Hodge conjectures, in terms of algebraic cycles,
for a proof of which, we refer to [Sch89, Lemma 0.1].
Conjecture 3.3.2. If H⊂Hn(X, Q)is a Q-sub-Hodge structure of level l=n−2c, then
GHC(n, c, X)holds for Hif and only if there exists a smooth projective complex variety Yto-
gether with an element z∈Corrc(Y, X )such that His contained in z∗Hl(Y , Q), where z∗is given
by the formula
z∗(η) = prX∗(pr∗
Y(η)∪c(z)) .
Also notice that this conjecture, and similar to the Hodge conjecture, can be stated in term of
classical motives over C
Proposition 3.3.3. [Gro69, Page 301] The generalized Hodge conjecture for all X∈SmProjC
is equivalent to the following statement: the Hodge conjecture holds and an homological motive is
effective if and only if its Hodge realization is effective.
Based on the previous reformulation of the generalized Hodge conjecture, the authors Rosen-
schon and Srinivas proposed the following variant of the generalized Hodge conjecture for integral
coefficients, but using Lichtenbaum cohomology groups:
Conjecture 3.3.4 (L-Generalized Hodge Conjecture).Let Xbe a smooth projective complex va-
riety. If H⊂Hn(X, Z)is a Z-sub-Hodge structure of level l=n−2c, then GHCL(n, c, X )holds
for Hif and only if there exist a smooth projective complex variety Ytogether with an element
z∈Corrc
L(Y, X )such that His contained in z∗Hl(Y, Z).
For a smooth projective complex variety Xthe conjecture 3.3.4 is denoted by GHCL(n, c, X)Q.
In some particular cases it is known to be equivalent to GHC(n, c, X)Q. For instance if we consider
GHC(2k−1, k −1, X )Qin [Gro69, §2] it was mentioned that with this level and weights is related
with the usual Hodge conjecture:
Proposition 3.3.5. [Lew99, Remark 12.30] Let Xbe a smooth projective complex variety, then
GHC(2k−1, k −1, X )Qholds if and only if H2k−1(X, Q)⊗H1(Γ,Q)∩Hk,k (Γ ×X)is algebraic
for every smooth projective complex curve Γ.
The Lichtenbaum version of the previous result still holds as is stated in [RS16, Remarks 5.2]
whose proof uses similar arguments as the ones presented in [Lew99, Remark 12.30]. Before we go
into the proof of the proposition it is necessary to introduce some notations and conventions. First
Betti cohomology is considered modulo torsion. Denote as
H2k−1
L-alg (X, Z) := nσ∗:H1(Y, Z)→H2k−1(X , Z)|σ∈Corrk−1
L(Y, X ),dimY= 1o/tors
where Yis smooth and pro jective, and recall
H2k−1
max (X, Z) = the largest Z-sub HS in Hk,k−1(X)⊕Hk−1,k(X)∩H2k−1(X, Z)
the equality of both is equivalent to the generalized Hodge conjecture GHCL(2k−1, k −1, X ). Note
that H2k−1
L-alg (X, Z)⊗C=Hk,k−1
L-alg (X)⊕Hk−1,k
L-alg (X) because of the Hodge decomposition. Also there
exists a partial version of the previous result, which asks whether or not a sub-Hodge structure
W⊂H2k−1(X, Z) is contained in the image of the action of some Lichtenbaum correspondence
over cohomology groups.
In the following proposition, we characterize this partial ´etale version of the generalized Hodge
conjecture of a Hodge structure of weight 2k−1 and level 1, give a general description of the
GHCL(2k−1, k −1, X ) and its equivalence to GHC(2k−1, k −1, X)Q:

HODGE STRUCTURES THROUGH AN ´
ETALE MOTIVIC POINT OF VIEW 21
Proposition 3.3.6. Let X∈SmProjC,k∈N≥1and let W⊂H2k−1(X, Z)be a sub Hodge structure
of level 1. Then:
(i.) there exist Y∈SmProjCand a Lichtenbaum correspondence z∈CHdY+1
L(Y×X)such that
W⊂z∗H1(Y, Z)if and only if for all curves C∈SmProjCthe Hodge classes Hk,k (C×
X)∩H1(C, Z)⊗Ware algebraic.
(ii.) In particular GHCL(2k−1, k −1, X )holds if and only if for all curves C∈SmProjCthe
Hodge classes Hk,k(C×X)∩H1(C, Z)⊗H2k−1(X, Z)are L-algebraic, i.e.
Hk,k (C×X)∩H1(C, Z)⊗H2k−1(X, Z)⊂im nck
L:CHk
L(C×X)→H2k
B(X, Z(k))o.
(iii.) GHC(2k−1, k −1, X )Qholds if and only if GHCL(2k−1, k −1, X )holds.
Proof. For (i.), let W⊂H2k−1(X, Z) be a sub Hodge structure of weight 2k−1 and level 1 and
assume that there exists Y∈SmProjCand a Lichtenbaum correspondence z∈CHdY+k−1
L(Y×X)
(correspondence of degree k−1) such that W⊂z∗H1(Y , Z). Consider Ca smooth complex projec-
tive curve and consider an element h∈H1(Y, Z)⊗W∩Hk,k (C×X)≃HomHS (H1(C, Z), W ).
Let h∗:H1(C, Z)→Wbe the map of Hodge structures induced by h. Define V= ker(z∗), which
by the theory of Hodge structures, is a Hodge structure itself. We know that the image of im(z∗) is
a Hodge structures of the same weight (see [Voi02, Lemme 7.23 et 7.25]), then H1(Y, Z) = V⊕R
where R=V⊥. Then we have a morphism λ:= (z∗|R)−1W: im(z∗)∩W⊂H2k−1(X, Z)→R
which fits into the following commutative diagram
H1(C, Z)Wim(z∗)∩W H1(Y, Z)
W W.
h∗
h∗
λ
z∗
The map obtained obtained from the upper arrows is induced by a Hodge class by Lefschetz (1,1)
and therefore his algebraic.
Conversely, suppose that for all smooth and projective curve Cthe Hodge classes Hk,k (C×
X)∩H1(C, Z)⊗Ware algebraic. Let W⊂H2k−1(X, Z) be a sub-Hodge structure of level 1
and notice that Whas a decomposition as W⊗C=Wk,k−1⊕Wk−1,k . Then its associated k-th
intermediate Jacobian is of the form Jk(W) = Wk−1,k /W which is an abelian variety. Since Jk(W)
is a complex torus, then its holomorphic tangent bundle is Wk−1,1and the fundamental group is
isomorphic to the lattice W, thus π1(Jk(W)) ≃H1(Jk(W),Z) = W. Set m= dim(Jk(W)) then
H2m−1(Jk(W),C) = Hm−1,m(Jk(W)) ⊕Hm,m−1(Jk(W)) and
Hm−1,m(Jk(W)) ≃H1,0(Jk(W))∗
=H0(Jk(W),Ω1
Jk(W))∗
≃H0(Jk(W),Ω1
Jk(W))∗
≃H0(Jk(W), T ∗
Jk(W))∗≃Wk−1,k .
then H2m−1(Jk(W),C) = Wk−1,k ⊕Wk,k−1.
Taking hyperplane sections of Jk(W) and applying Bertini’s theorem, we find a smooth projective
curve Γ ⊂Jk(W) and a surjective map H1(Γ,Z)→H1(Jk(W),Z)≃W. Also by Poincar´e
duality H1(Γ,Z)≃H1(Γ,Z) so we have a surjective map f:H1(Γ,Z)→W. Since the map
fis a morphism of Hodge structures, then it is an element in Hk,k(Γ ×X)∩H1(Γ,Z)⊗W
which by hypothesis is L-algebraic. Therefore there exists a class z∈CH2k
L(Γ ×X) such that
W⊂z∗H1(Γ,Z)⊂H2k−1(X, Z).
The statement (ii.) is a direct consequence of (i.) taking W=H2k−1(X, Z) and the maximal sub
Hodge structure of it H2k−1
max (X, Z). For (iii.) notice that for a complex smooth projective curve C
the Betti cohomology groups are torsion free. Thus K¨unneth formula holds for the product C×X
and then
Hk,k (C×X)∩H1(C, Z)⊗H2k−1(X, Z)⊂Hk,k (C×X)∩H2k(C×X, Z) = Hdg2k(C×X, Z).

22 IV ´
AN ROSAS SOTO
Invoking Proposition 3.1.7 Hk,k (C×X)∩H1(C, Z)⊗H2k−1(X, Z)is L-algebraic if and only
if Hk,k (C×X)∩H1(C, Q)⊗H2k−1(X, Q)is algebraic in the usual sense, which gives us the
equivalences
GHC(2k−1, k −1, X )Qholds
⇐⇒ Hk,k (C×X)∩H1(C, Q)⊗H2k−1(X, Q)is alg. ∀curve C
⇐⇒ Hk,k (C×X)∩H1(C, Z)⊗H2k−1(X, Z)is L-alg. ∀curve C
⇐⇒ GHCL(2k−1, k −1, X ) holds.

In the sequel, we give more subtle relations between the Hodge conjecture and the generalized
one, following the proof of the classical case given in [Fu12, Lemma 2.3]:
Lemma 3.3.7. Let Xbe a smooth projective variety of dimension nand H⊂Hk(X, Z)be a sub-
Hodge structure of coniveau at least cand assume that there exists a smooth projective variety Yof
dimension dY, such that H(c)is a sub-Hodge structure of Hk−2c(Y, Z). Then if HdY+c,dY+c(Y×
X)∩H2(dY+c)−k(Y, Z)⊗Hk(X, Z)is L-algebraic then generalized L-Hodge conjecture for H
holds.
Proof. Since torsion classes come from Lichtenbaum cycles, for simplicity we will neglect torsion
Hodge classes. Suppose that His a sub-Hodge structure of Hk(X, Z) of weight kand coniveau c.
We know that H(c) is still an effective Hodge structure, then there is a smooth projective variety Y
such that H(c) is a sub-Hodge structure of Hk−2c(Y , Z), which by polarization can be decomposed
as Hk−2c(Y, Z)≃H(c)⊕R. Consider f:Hk−2c(Y, Z)→Hk(X, Z) the morphism resulting from
the composition of the following maps
Hk−2c(Y, Z)pr1
−−→ H(c)id⊗Z(−c)
−−−−−−→ H ֒→Hk(X, Z)
Since HomHSZ(Hk−2c(Y, Z), H k(X, Z)) ≃H2(dY+c)−k(Y, Z)⊗Hk(X, Z) by the hypothesis that
HdY+c,dY+c(Y×X)∩H2(dY+c)−k(Y, Z)⊗Hk(X, Z)is L-algebraic we conclude that fcomes
from a Lichtenbaum algebraic cycle γ∈CHdY+c
L(Y×X) and H⊂γ∗Hk−2c(Y, Z). Thus the
generalized L-Hodge conjecture holds for H.
Using the same kind of arguments, and adding an hypothesis of effectiveness it is possible to
characterize the generalized Hodge conjecture in terms of the integral Hodge conjecture in the ´etale
setting.
Theorem 3.3.8. The Lichtenbaum generalized Hodge conjecture for all X∈SmProjCholds if and
only if the following two conditions hold:
•the Lichtenbaum Hodge conjecture holds,
•a homological ´etale motive is effective if and only if its Hodge realization is effective.
Proof. The generalized L-Hodge conjecture immediately implies the L-Hodge conjecture. Suppose
that Mhas an effective realization and let H:= ρH(M) be its associated Hodge structure of
weight nand coniveau c. By the generalized L-Hodge conjecture there exists Y∈SmProjCand
γ∈CHdY+c
L(Y×X) such that H⊂γ∗Hn−2c(Y, Z)⊂Hn(X, Z). The motive M(c) is effective in
h(Y), thus Mis effective because it is a subobject of the effective motive h(X).
Assume that the L-Hodge conjecture holds for every X∈SmProjCand that a homological motive
is effective if and only if its realization is effective. We can neglect torsion Hodge classes because they
always come from torsion algebraic cycles. Suppose that His a sub-Hodge structure of Hn(X , Z)
of weight nand coniveau c. We know that H(c) is still an effective Hodge structure. Then there
is a smooth projective variety Ysuch that H(c) is a sub-Hodge structure of Hn−2c(Y, Z) which by
polarization can be decomposed as Hn−2c(Y , Z)≃H(c)⊕R. Consider f:Hn−2c(Y, Z)→Hn(X, Z)
the morphism resulting from the composition of the following maps
Hn−2c(Y, Z)pr1
−−→ H(c)id⊗Z(−c)
−−−−−−→ H ֒→Hn(X, Z)

HODGE STRUCTURES THROUGH AN ´
ETALE MOTIVIC POINT OF VIEW 23
Since HomHSZ(Hn−2c(Y, Z), H n(X, Z)) ≃H2(dY+c)−n(Y, Z)⊗Hn(X, Z) by the assumption of the
Hodge conjecture and Proposition 3.1.7 we conclude that fcomes from a Lichtenbaum algebraic
cycle γ∈CHdY+c
L(Y×X) and H⊂γ∗Hn−2c(Y, Z). Thus the generalized L-Hodge conjecture
holds. 
Then we have the following corollary coming from the previous characterizations of the Gener-
alized Hodge conjecture (classical and Lichtenbaum setting)
Corollary 3.3.9. The generalized Hodge conjecture holds if and only if the generalized L-Hodge
conjecture holds.
3.4. Bardelli’s example. Let us recall the example presented in [Bar91] of a certain threefold
Xwhere GHC(3,1, X )Qholds. Let σ:P7→P7be the involution defined as σ(x0:... :x3:
y0,...,y3) = (x0:... :x3:−y0,...,−y3) and let X=V(Q0, Q1, Q2, Q3) be a smooth complete
intersection of four σ-invariant quadrics. There exists a smooth irreducible curve C, of genus 33,
obtained as the intersection of two nodal surfaces, and an ´etale double covering e
C→Csuch
that H1(e
C, Q)−→H3(X, Q)−is surjective, where the first group is the anti-invariant part of the
involution τ:e
C→e
Cassociated to the double covering and the later group is the anti-invariant
part associated to the involution σ. Notice that by [Bar91, Fact 2.4.1] if we assume that Xis
a very general threefold, then H3(X, Q)+and H3(X, Q)−are perpendicular with respect to the
cup product on H3(X , Q) and H3,0(X)⊂H3(X, C)+therefore H3(X, Q)−is a polarized Hodge
structure perpendicular to H3,0(X) i.e. a polarized sub-Hodge structure of H3(X, Q) of level 1.
The isogeny α: Prym( e
C→C)→J(X)−, where J(X)−is the pro jection of H1,2(X)−into J2(X),
is the correspondence that induces the isomorphism H1(e
C, Q)−→H3(X, Q)−, but in the case of
integral coefficients the image of the correspondence is a subgroup of index 2. From the previous
results we have the following equivalences:
GHC(3,1, X )Qholds for H3(X, Q)−
⇐⇒ H2,2(Γ ×X)∩H1(Γ,Q)⊗H3(X, Q)−is alg. ∀curve Γ
⇐⇒ H2,2(Γ ×X)∩H1(Γ,Z)⊗H3(X, Z)−is L-alg. ∀curve Γ
⇐⇒ GHCL(3,1, X) holds for H3(X , Z)−
so there exists a smooth projective curve Γ′and a correspondence z∈CH2
L(Γ′×X) such that
H3(X, Z)−⊂z∗H1(Γ′,Z).
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Institut de Math´
ematiques de Bourgogne, UMR 5584 CNRS, Universit´
e Bourgogne Franche-Comt´
e,
F-2100 Dijon, France
E-mail address:ivan-alejandro.rosas-soto@u-bourgogne.fr