# Local-global principle and integral Tate conjecture for certain varieties

Zhiyu Tian  Beijing International Center for Mathematical Research
Peking University
100871, Beijing, China<EMAIL_ADDRESS>
###### Abstract.

We give a geometric criterion to check the validity of integral Tate
conjecture for one cycles on separably rationally connected fibrations over a
curve, and to check that the Brauer-Manin obstruction is the only obstruction
to local-global principle for zero cycles on a separably rationally connected
variety defined over a global function field.

We prove that the Brauer-Manin obstruction is the only obstruction to local-
global principle for zero cycles on all geometrically rational surfaces
defined over a global function field, and to Hasse principle for rational
points on del Pezzo surfaces of degree four defined over a global function
field of odd characteristic.

Along the way, we also prove some results about the space of one cycles on a
separably rationally connected fibration over a curve, which leads to the
equality of the coniveau filtration and the strong coniveau filtration
(introduced by Benoist-Ottem and Voisin) on degree $3$ homology of such
varieties.

###### Contents

  1. 1 Introduction
    1. 1.1 Local-global principle for zero cycles
    2. 1.2 Integral Tate conjecture
    3. 1.3 Coniveau and strong coniveau
    4. 1.4 Integral Tate conjecture for one cycles: arithmetic part
    5. 1.5 Algebraic equivalence
    6. 1.6 Structure of the paper
  2. 2 Space of one cycles
  3. 3 Lawson homology
  4. 4 Chow sheaves
  5. 5 Integral Tate conjecture and local-global principle for zero cycles
  6. 6 Examples

##  1\. Introduction

###  1.1. Local-global principle for zero cycles

Given a smooth projective variety defined over a global field, a natural and
important problem is to find criteria for the existence of rational points and
a description of the set of all rational points. Hasse principle and weak
approximation problem, or local-global principle, gives a characterization of
this set in terms of the adelic points. There are various obstructions for
local-global principle to hold, notably the so called Brauer-Manin
obstruction. A conjecture due to Colliot-Thélène states that for rationally
connected varieties defined over a global field, this is the only obstruction.
The study of zero cycles, as natural generalizations of rational points, has
also drawn lots of attentions in recent years. Motivated by the case of
rational points, Colliot-Thélène has formulated the following conjectures on
the local-global principle for zero cycles.

######  Conjecture 1.1.

[CT99, Conjecture 2.2] Let $X$ be a smooth projective variety defined over the
function field $\mathbb{F}_{q}(B)$ of a smooth curve $B$ defined over a finite
field $\mathbb{F}_{q}$. For every place $\nu$ of $\mathbb{F}_{q}(B)$, let
$z_{\nu}\in CH_{0}(X_{v})$. Suppose that for all element $A\in
Br(X)\\{\ell\\}$, we have $\sum_{\nu}Inv(A(z_{\nu}))=0$. Then for all $n>0$,
there is a cycle $z_{n}\in CH_{0}(X)$ such that for all $\nu$ we have that

$cl(z_{n})=cl(z_{\nu})\in
H^{2d}_{\text{\'{e}t}}(X_{\nu},\mu_{\ell^{n}}^{\otimes d}).$

Here $Inv(A(z_{\nu}))$ means the value of $(A,z_{\nu})$ under the pairing

$Br(X_{\nu})\\{\ell\\}\times CH_{0}(X_{\nu})\to\mathbb{Q}/\mathbb{Z}.$

A particular case of the above conjecture is the following.

######  Conjecture 1.2.

Let $X$ be a smooth projective variety defined over the function field
$\mathbb{F}_{q}(B)$ of a smooth curve $B$ defined over a finite field
$\mathbb{F}_{q}$. Suppose that for every place $\nu$ of $\mathbb{F}_{q}(B)$,
there is a zero cycle

$z_{\nu}\in CH_{0}(X_{v})$

of degree prime to $\ell$. Suppose that for all element $A\in
Br(X)\\{\ell\\}$, we have

$\sum_{\nu}Inv(A(z_{\nu}))=0.$

Then there is a cycle $z\in CH_{0}(X)$ of degree prime to $\ell$.

In this paper, for an abelian group $A$, we use
$A\hat{\otimes}\mathbb{Z}_{\ell}$ to denote the inverse limit
$\lim\limits_{\longleftarrow}A/\ell^{n}A$. The following stronger form of the
above conjectures is also well-known.

######  Conjecture 1.3.

Let $X$ be a smooth projective variety defined over a global field $K$. Let
$l$ be a prime number invertible in $K$. There is an exact sequence:

$CH_{0}(X)\hat{\otimes}\mathbb{Z}_{\ell}\to\Pi_{\nu\in\Omega(K)}CH_{0}(X_{\nu})\hat{\otimes}\mathbb{Z}_{\ell}\to
Hom(Br(X)\\{\ell\\},\mathbb{Q}/\mathbb{Z}).$

Conjectures 1.1 and 1.2 are consequences of this via considering the
commutative diagram of various cycle class maps. On the other hand, if the
cycle class map $CH_{0}(X_{\nu})\hat{\otimes}\mathbb{Z}_{\ell}\to
H^{2d}_{\text{\'{e}t}}(X,\mathbb{Z}_{\ell}(d))$ is injective, Conjecture 1.1
and this stronger conjecture 1.3 are equivalent. In general the injectivity
fails. But we will see that in many (and conjecturally all) cases of interest
to us, the injectivity holds.

One of the main theorems of this article is the following.

######  Theorem 1.4.

Let $X$ be a smooth projective geometrically rational surface defined over the
function field $\mathbb{F}_{q}(B)$ of a smooth projective curve $B$. Then
Conjectures 1.3, and hence 1.1 and 1.2 hold true for $X$ .

By a happy coincidence, we deduce a corollary for rational points.

######  Theorem 1.5.

Let $X$ be a del Pezzo surface of degree $4$ defined over a global function
field of odd characteristic. Then Brauer-Manin obstruction is the only
obstruction for Hasse principle for rational points on $X$ to hold.

###### Proof.

If there is a rational point everywhere locally and satisfies the Brauer-Manin
constraint, there is a zero cycle of degree $1$ over the function field by
Theorem 1.4. A del Pezzo surface of degree $4$ is a complete intersection of
$2$ quadrics in $\mathbb{P}^{4}$. Such a complete intersection has a rational
point if and only if it has an odd degree $0$-cycle [Bru78]. Hence we have the
result. ∎

######  Remark 1.6.

One is also interested to study weak approximation for rational points on a
del Pezzo surface of degree $4$. For a del Pezzo surface of degree $4$ over a
number field, assuming that there is a rational point, Salberger and
Skorobogatov[SS91] prove that the Brauer-Manin obstruction is the only
obstruction to weak approximation. As the author has been informed by Colliot-
Thélène, essentially the same argument also proves that over a global function
field of odd characteristic, Brauer-Manin obstruction is the only obstruction
to weak approximation once there is a rational point. In characteristic $2$,
some partial results are contained in the joint work of the author with Letao
Zhang [TZ18].

We finish this section with some previously known results. There is a vast
literature on the local-global principles for zero-cycles/rational points on
geometrically rational surfaces. Let us only mention a few relevant results
and refer the readers to survey articles such as [Wit18] etc. for a more
comprehensive list.

Colliot-Thélène proved Conjecture 1.3 holds for ruled surfaces defined over
number fields [CT00]. The global function field version for ruled surfaces is
proved by Parimala-Suresh [PS16], whose proof depends on the computation of
degree $3$ unramified cohomology and also establishes the integral Tate
conjecture for conic bundle over surfaces defined over finite fields. An
interesting example of cubic surfaces of the form
$(f+tg=0)\subset\mathbb{P}^{3}\times\mathbb{A}^{1}_{t}$ is studied by Colliot-
Thélène-Swinnerton-Dyer [CTSD12]. In addition to proving that Hasse-principle
for zero cycles holds for cubic surfaces of this form, they also prove that
the existence of rational points is equivalent to the existence of a zero
cycle of degree $1$ for such surfaces.

The study of complete intersection of two quadrics has also drawn lots of
attention. It starts with the work of Colliot-Thélène, Sansuc, and Swinnerton-
Dyer [CTSSD87]. Heath-Brown proved that Hasse principle for rational points
holds for smooth complete intersections of two quadrics in $\mathbb{P}^{7}$
over number fields [HB18], see also a different proof by Colliot-Thélène
[CT22]. Under the assumption on finiteness of Tate-Shafarevich groups of
elliptic curves and the validity of Schinzel’s hypothesis, Wittenberg proved
Hasse principle holds for such complete intersections in $\mathbb{P}^{5}$ and
some case in $\mathbb{P}^{4}$ over number fields [Wit07]. The author has shown
in a previous paper [Tia17] that Hasse principle for rational points holds for
smooth complete intersections of two quadrics in $\mathbb{P}^{n},n\geq 5$
defined over a global function field of odd characteristic.

###  1.2. Integral Tate conjecture

Our approach to Theorem 1.4 is based on the close relation between an integral
version of Tate conjecture and Colliot-Thélène’s conjectures, first studied by
Saito [Sai89] and Colliot-Thélène [CT99].

Let $X$ be a smooth projective geometrically irreducible variety of dimension
$d$ defined over a finite field $\mathbb{F}$. We have the cycle class maps:

(1) $CH_{1}(X)\otimes\mathbb{Z}_{\ell}\to H^{2d-2}(X,\mathbb{Z}_{\ell}(d-1)),$
(2) $CH_{1}(X)\otimes\mathbb{Z}_{\ell}\to
H^{2d-2}(X,\mathbb{Z}_{\ell}(d-1))\to
H^{2d-2}(\bar{X},\mathbb{Z}_{\ell}(d-1))^{G},$ (3)
$CH_{1}(\bar{X})\otimes\mathbb{Z}_{\ell}\to\cup_{K/\mathbb{F}}H^{2d-2}(\bar{X},\mathbb{Z}_{\ell}(d-1))^{G_{K}}\subset
H^{2d-2}(\bar{X},\mathbb{Z}_{\ell}(d-1)).$

We also have the corresponding cycle class maps after tensoring with
$\mathbb{Q}_{\ell}$. Tate conjecture predicts that the cycle class map on
codimension $r$ cycles

$CH^{r}(X)\otimes\mathbb{Q}_{\ell}\to
H^{2r}_{\text{\'{e}t}}(X,\mathbb{Q}_{\ell}(r))$

is surjective for any smooth projective varieties defined over a finite field.
While the cycle class map is in general not surjective for $\mathbb{Z}_{\ell}$
coefficients, one is still interested in knowing in which cases surjectivity
still holds. This is usually called the integral Tate conjecture (even though
it is not true in general).

######  Definition 1.7.

Let $X$ be a smooth, proper variety over an algebraically closed field. Given
any $f:\mathbb{P}^{1}\to X$, the pull-back $f^{*}T_{X}$ decomposes as direct
sum of line bundles $\oplus_{i=1}^{\dim X}\mathcal{O}(a_{i})$. We say that $X$
is _separably rationally connected_ , or _SRC_ , if $a_{i}>0$ for every $i$.
We say that $X$ is _separably rationally connected in codimension $1$_ or _SRC
in codimension $1$_ if there is a morphism $f$ for which $a_{i}\geq 0$ for
every $i$, with strict inequality for all but one $a_{i}$.

######  Remark 1.8.

The term SRC is introduced by Kollár-Miyaoka-Mori [KMM92]. The term SRC in
codimension $1$ is introduced in [KT23]. Main examples include SRC varieties
and fibrations over a curve with smooth proper SRC general fibers. In
characteristic $0$, these are all the examples. In positive characteristic
there are more examples. In any case, one can take the quotient by free
rational curves on a variety that is SRC in codimension $1$. The quotient is
either a curve or a point. In particular, the Chow group of $0$-cycles on such
varieties is supported in a curve.

The results of this paper, together with some results proved by the author in
[Tia20], strongly suggest that the following is true.

######  Conjecture 1.9.

Let $X$ be a smooth projective variety defined over a finite field. Assume
that $X$ is separably rationally connected in codimension $1$. Then the cycle
class map

$CH_{1}(X)\otimes\mathbb{Z}_{\ell}\to
H^{2d-2}_{\text{\'{e}t}}(X,\mathbb{Z}_{\ell}(d-1))$

is surjective, where $d=\dim X$.

We refer the readers to Theorem 1.13 and Remark 1.15 for evidences of this
conjecture.

The connection between integral Tate conjecture and Conjecture 1.1, 1.2 is the
following.

######  Theorem 1.10 ([CT99] Proposition 3.2, [Sai89] Corollary (8-6)).

Let $\mathbb{F}$ be a finite field, $C$ a smooth projective geometrically
connected curve over $\mathbb{F}$, and $K$ the function field of $C$. Let
$\mathcal{X}$ be a smooth projective geometrically connected variety of
dimension $d+1$ defined over $\mathbb{F}$, equipped with a morphism
$p:\mathcal{X}\to C$, whose generic fiber is smooth and geometrically
irreducible. Let $l$ be a prime different from the characteristic.

  1. (1)

If the cycle class map

$CH^{d}(\mathcal{X})\otimes\mathbb{Z}_{\ell}\to
H^{2d}_{\text{{\'{e}}t}}(\mathcal{X},\mathbb{Z}_{\ell}(d))$

is surjective, Conjectures 1.1 and 1.2 are true.

  2. (2)

If the cycle class map

$CH^{d}(\mathcal{X})\otimes\mathbb{Z}_{\ell}\to
H^{2d}_{\text{{\'{e}}t}}(\mathcal{X},\mathbb{Z}_{\ell}(d))\to
H^{2d-2}_{\text{{\'{e}}t}}(\overline{\mathcal{X}},\mathbb{Z}_{\ell}(d-1))^{G}$

is surjective, or if

$CH^{d}(\mathcal{X})\otimes\mathbb{Z}_{\ell}\to
H^{2d}_{\text{{\'{e}}t}}(\mathcal{X},\mathbb{Z}_{\ell}(d))$

is surjective modulo torsion, Conjecture 1.2 is true.

######  Remark 1.11.

The cited references only contain a proof of the first statement. But the
second statement follows from the same proof. The general result of Saito
produces a cohomology class $\xi\in H^{2d}(\mathcal{X},\mathbb{Z}_{\ell}(d))$
whose restriction to each local place coincide with the class of
$z_{\mu}$([CT99, Proposition 3.1]). The various types of integral Tate
conjecture are simply used to find a global cycle whose class agrees with
$\xi$ in various cohomology groups. See also Page 19 of the slide of Colliot-
Thélène’s lecture at Cambridge in 2008 (available at
https://www.imo.universite-paris-saclay.fr/~jean-louis.colliot-
thelene/expocambridge240809.pdf).

A result of Schoen [Sch98] says that if the Tate conjecture is true for
divisors on all smooth projective surfaces defined over finite fields, then
for any smooth projective variety $V$ defined over a finite field
$\mathbb{F}$, the cycle class map

$CH_{1}(\bar{V})\otimes\mathbb{Z}_{l}\to\cup_{K/\mathbb{F}}H^{2d-2}(V,\mathbb{Z}_{\ell}(d-1))^{\text{Gal}(\bar{\mathbb{F}}/K)}\subset
H^{2d-2}(\bar{V},\mathbb{Z}_{\ell}(d-1))$

is surjective, where $\bar{V}$ is the base change of $V$ to an algebraic
closure of $\mathbb{F}$.

If furthermore $V$ is SRC in codimnesion $1$, since its Chow group of zero
cycles is supported in a curve, it is easy to see that every class in
$H^{2d-2}(\bar{V},\mathbb{Q}_{\ell}(d-1))$ is algebraic. Thus every class in
$H^{2d-2}(\bar{V},\mathbb{Z}_{\ell}(d-1))$ is fixed by some open subgroup of
the Galois group. So in this case, Schoen’s theorem implies that we always
have a surjection

$CH_{1}(V)\otimes\mathbb{Z}_{\ell}\to H^{2d-2}(V,\mathbb{Z}_{\ell}(d-1)),$

provided that Tate conjecture holds for all surfaces.

The paper [CTS10] discussed the implication of Schoen’s result for varieties
defined over $\bar{\mathbb{F}}(C)$, the function field of a curve defined over
$\bar{\mathbb{F}}$. Colliot-Thélène and Kahn analyzed the surjectivity of the
$\mathbb{Z}_{\ell}$-coeffiecient cycle class map of codimension $2$ cycles and
its relation with degree $3$ unramified cohomology
$H^{3}_{\text{nr}}(X,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}(2))$ in [CTK13] (over
the complex numbers, such a relation is studied in [CTV12]). For the sake of
brevity, and since we will not need these notions for the other parts of this
paper, we will not define this invariant. Instead, we refer the interested
reader to these papers and the references therein for definitions and
properties of unramified cohomology. In particular, if the unramified
cohomology $H^{3}_{\text{nr}}(X,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}(2))$
vanishes, the cokernal of $CH^{2}(X)\otimes\mathbb{Z}_{\ell}\to
H^{4}(X,\mathbb{Z}_{\ell}(2))$ is torsion free [CTK13, Théorème 2.2]. Thus if,
in addition, we know that the cokernal is torsion (for instance, if the Chow
group of zero cycles with rational coefficients is universally supported in a
surface [CTK13, Proposition 3.2]), we know the cycle class map is surjective.
One should also note that by the Tate conjecture, one expects the cokernal to
be torsion. In general, they deduced a short exact sequence relating various
Chow groups of codimension $2$ cycles and degree $3$ unramified cohomology.
Their short exact sequence for varieties over finite fields reads the
following ([CTK13, Théorème 6.8]):

$\displaystyle 0\to$ $\displaystyle\text{Ker}(CH^{2}(X)\to CH^{2}(\bar{X}))\to
H^{1}(\mathbb{F},\oplus_{\ell}H^{3}_{\text{\'{e}t}}(X,\mathbb{Z}_{\ell}(2))_{\text{tors}})$
(4) $\displaystyle\to$
$\displaystyle\text{Ker}(H^{3}_{\text{nr}}(X,\mathbb{Q}/\mathbb{Z}(2))\to
H^{3}_{\text{nr}}(\bar{X},\mathbb{Q}/\mathbb{Z}(2)))$ $\displaystyle\to$
$\displaystyle\text{Coker}(CH^{2}(X)\to CH^{2}(\bar{X})^{G})\to 0$

Of course, one can deduce from this a similar exact sequence for
$\ell$-primary torsions.

In particular, we can apply their results to $3$-folds, which then relates the
integral Tate conjecture to the vanishing of degree $3$ unramified cohomology.
Note that by the Lefschetz hyperplane theorem, if we can prove integral Tate
conjecture for one cycles on all $3$-folds, we prove integral Tate conjecture
for one cycles on all smooth projective varieties.

Several groups of authors proved the vanishing of the degree $3$ unramified
cohomoogy on certain threefolds and deduce the integral Tate conjecture for
one cycles, and thus proving Conjecture 1.1 and 1.2 for some surfaces defined
over a global function field. See, for example, [PS16] for the case of conic
bundles over a surface, [CTS21] and [Sca22] for the case of a product of a
curve with a $CH_{0}$-trivial surface.

We prove Theorem 1.4 as a consequence of the following case of the integral
Tate conjecture for one cycles.

######  Theorem 1.12.

Let $\pi:\mathcal{X}\to B$ be a projective flat family of surfaces over a
smooth projective curve $B$ defined over a finite field $\mathbb{F}_{q}$.
Assume that $\mathcal{X}$ is smooth and that the geometric generic fiber is a
smooth rational surface. Then integral Tate conjecture holds for one cycles.
More concretely, the cycle class map

$CH_{1}(\mathcal{X})\otimes\mathbb{Z}_{\ell}\to
H^{4}_{\text{\'{e}t}}(\mathcal{X},\mathbb{Z}_{\ell}(2))$

is surjective.

In general, one can deduce the following geometric criterion for the validity
of the integral Tate conjecture 1.9 and the local-global principles for
separably rationally connected varieties defined over global function fields.

Given a variety $V$ defined over a field $k$, we denote by $A_{1}(V)$ the
group of one cycles in $V$ modulo algebraic equivalence. We also use
$\overline{V}$ to denote the base change of $V$ to an algebraic closure of
$k$.

######  Theorem 1.13.

Let $\pi:\mathcal{X}\to B$ be a projective flat family of varieties over a
smooth projective curve $B$ defined over a finite field $\mathbb{F}_{q}$.
Assume that $\mathcal{X}$ is smooth and that the generic fiber is smooth,
separably rationally connected, and of dimension $d$. Consider the following
hypothesis:

  * (A)

The cycle class map $A_{1}(\overline{\mathcal{X}})\otimes\mathbb{Z}_{\ell}\to
H^{2d}_{\text{\'{e}t}}(\overline{\mathcal{X}},\mathbb{Z}_{\ell}(d))$ is
surjective.

  * (B)

The cycle class map $A_{1}(\overline{\mathcal{X}})\otimes\mathbb{Z}_{\ell}\to
H^{2d}_{\text{\'{e}t}}(\overline{\mathcal{X}},\mathbb{Z}_{\ell}(d))$ is
injective.

  * (C)

The cycle class map from higher Chow groups

$\lim\limits_{\xleftarrow[n]{}}CH_{1}(\overline{\mathcal{X}},1,\mathbb{Z}/\ell^{n}\mathbb{Z})\to
H^{2d-1}_{\text{\'{e}t}}(\overline{\mathcal{X}},\mathbb{Z}_{\ell}(d))$

is surjective.

  * (D)

The coniveau filtration
$N^{1}H^{2d-1}_{\text{\'{e}t}}(\overline{\mathcal{X}},\mathbb{Z}_{\ell})$ is
the whole cohomology group

$H^{2d-1}_{\text{\'{e}t}}(\overline{\mathcal{X}},\mathbb{Z}_{\ell}).$

If $\overline{\mathcal{X}}$ satisfies hypothesis (A) and (B), then the cycle
maps

$CH_{1}(\mathcal{X})\otimes\mathbb{Z}_{l}\to
H^{2d}_{\text{\'{e}t}}(\mathcal{X},\mathbb{Z}_{l}(d))\to
H^{2d}_{\text{\'{e}t}}(\overline{\mathcal{X}},\mathbb{Z}_{l}(d))^{Gal(\bar{\mathbb{F}}_{q}/\mathbb{F}_{q})}$

is surjective, and Conjecture 1.2 holds for the generic fiber $X$ over
$\mathbb{F}_{q}(B)$.

If $\overline{\mathcal{X}}$ satisfies hypothesis (C) or (D), then the cycle
maps

$CH_{1}(\mathcal{X})_{\text{alg}}\otimes\mathbb{Z}_{l}\to
H^{1}(\mathbb{F}_{q},H^{2d-1}_{\text{\'{e}t}}(\overline{\mathcal{X}},\mathbb{Z}_{l}(d)))$

is surjective.

Thus Conjecture 1.1 holds for the generic fiber $X$ over $\mathbb{F}_{q}(B)$
if either hypothesis (A), (B), (C) or (A), (B), (D) hold.

######  Remark 1.14.

The statements in Hypothesis (A), (B), (C), (D) only depends on the stable
birational class of the generic fiber of $\overline{\mathcal{X}}\to\bar{B}$.
In particular, they only depend on the stable birational class of the generic
fiber $X$ over the field $\mathbb{F}_{q}(B)$ (assuming that there is a smooth
projective model for every stable birational class of $X$). Also note that
Conjectures 1.1, 1.2, 1.3, 1.9 only depend on the stable birational class of
the variety over $\mathbb{F}_{q}(B)$ (or $\mathbb{F}$ for Conjecture 1.9).

######  Remark 1.15.

We make a few simple remarks about the validity of the hypothesis above. First
of all, it is a simple exercise to prove that all these hypothesis hold if we
use $\mathbb{Q}_{\ell}$-coefficient, and that they hold for all but finitely
many $\ell$.

As discussed above, Hypothesis (A) follows from Tate’s conjecture on surfaces.
The author has made conjectures on the Kato homology of rationally connected
fibrations over an algebraically closed field of characteristic $0$ in
[Tia20]. A special case of the conjecture predicts that for a rationally
connected fibration over a curve defined over an algebraically closed field of
characteristic $0$, hypothesis (A), (B), (C), and (D) hold. It is quite
reasonable to believe that the same is true for separably rationally connected
fibrations in characteristic $p>0$. We discuss some examples in Section 6.

Now we explain a corollary of Theorem 1.12, which confirm a conjecture of
Colliot-Thélène and Kahn ([CTK13, Conjecture 5.8]) up to $p$ torsion.

######  Corollary 1.16.

Let $X$ be a smooth projective threefolds defined over a finite field
$\mathbb{F}$. Assume that $X$ admits a fibration structure over a smooth
projective curve with smooth projective geometrically rational generic fiber.
Then we have

$H^{3}_{\text{nr}}(X,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}(2))=0,$

for any $\ell$ invertible in $\mathbb{F}$, and a short exact sequence

$0\to H^{1}(\mathbb{F},H^{3}(\bar{X},\mathbb{Z}_{\ell}(2))\\{\ell\\}\to
CH_{1}(X)\otimes\mathbb{Z}_{\ell}\to
CH_{1}(\bar{X})^{G}\otimes\mathbb{Z}_{\ell}\to 0.$

###### Proof.

The vanishing of $H^{3}_{\text{nr}}(X,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}(2))$
follows from Theorem 1.12, [CTK13, Theorem 2.2, Proposition 3.2], and the fact
that $CH_{0}(\bar{X})$ is supported in a curve.

It then follows from the exact sequence (1.2) that we have the above
descritption of the Chow gruops of $X$ and $\bar{X}$. ∎

######  Remark 1.17.

Theorem 1.13 holds for smooth projective separably rationally connected
varieties. We have made the proof works in both cases. A cheaper way to get
this result is to note that the validity of the integral Tate conjecture is a
stable birational invariant and apply the above theorems to the product
$\mathbb{P}^{1}\times X$ ($X$ separably rationally connected) as a fibration
over $\mathbb{P}^{1}$. Unfortunately, for a separably rationally connected
threefold $V$ defined over ${\mathbb{F}}_{q}$, we do not know if the cycle
class map

$CH_{1}(\bar{V})\otimes\mathbb{Z}_{\ell}\to
H^{4}(\bar{V},\mathbb{Z}_{\ell}(2))$

is surjective. Once we know this (e.g. if we are willing to assume the Tate
conjecture for surfaces), the same argument as above shows that

$CH_{1}({V})\otimes\mathbb{Z}_{\ell}\to H^{4}({V},\mathbb{Z}_{\ell}(2))$

is surjective. One can also deduce the vanishing of degree $3$ unramified
cohomology and the short exact sequence of Chow groups as above.

###  1.3. Coniveau and strong coniveau

This section is a digression into several a priori different notions of
conveau filtraions on the cohomology of a variety introduced by Benoist-Ottem
and Voisin in [BO21, Voi22]. These notions will play an important role when we
return to discuss the integral Tate conjecture in the next section.

Let us first review the definitions.

######  Definition 1.18.

Let $X$ be a smooth projective variety of dimension $n$ defined over an
algebraically closed field. Given an abelian group $A$ that is one of
$\mathbb{Z}/n\mathbb{Z},\mathbb{Z}_{\ell},\mathbb{Z},\mathbb{Q}$, or
$\mathbb{Q}_{\ell}$, we simply write $H^{k}(X,A)$ as either the étale
cohomology with coefficient $A$ or the singular cohomology with coefficient
$A$ (if $X$ is a complex variety), we have the following closely related
filtrations on the cohomolgy $H^{k}(X,A)$.

  1. (1)

The coniveau filtration:

$N^{c}H^{k}(X,A):=\sum_{f:Y\to X}f_{*}(H_{2n-k}(Y,A))\subset
H_{2n-k}(X,A)\cong H^{k}(X,A),$

where the sum is taken over all morphisms from projective algebraic sets
$f:Y\to X,\dim Y\leq n-c$;

  2. (2)

The strong coniveau filtration:

$\tilde{N}^{c}H^{k}(X,A):=\sum_{f:Y\to X}f_{*}(H_{2n-k}(Y,A))\subset
H_{2n-k}(X,A)\cong H^{k}(X,A),$

where the sum is taken over all morphisms from smooth projective varieties
$f:Y\to X,\dim Y\leq n-c$.

  3. (3)

The strong cylindrical filtration:

$\tilde{N}_{c,\text{cyl}}H^{k}(X,A):=\sum\Gamma_{*}(H_{2n-k-2c}(Z,A))\subset
H_{2n-k}(X,A)\cong H^{k}(X,A)$

where the sum is taken over all _smooth_ projective varieties $Z$ and
correspondences $\Gamma\subset Z\times X$ of relative dimension $c$ over $Z$.

  4. (4)

The strong equidimensional cylindrical filtration:

$\tilde{N}_{c,\text{cyl},\text{eq}}H^{k}(X,A):=\sum\Gamma_{*}(H_{2n-k-2c}(Z,A))\subset
H_{2n-k}(X,A)\cong H^{k}(X,A)$

where the sum is taken over all smooth projective varieties $Z$ and
correspondences $\Gamma\subset Z\times X$ that is equidimensional of relative
dimension $c$ over $Z$.

  5. (5)

The semi-stable filtration: $N_{c,\text{st},\text{cyl}}H^{k}(X,\mathbb{Z})$ as
the group generated by the cylinder homomorphisms

$f_{*}\circ p^{*}:H_{2n-k-2c}(Z,\mathbb{Z})\to H_{2n-k}(X,\mathbb{Z})\cong
H^{k}(X,\mathbb{Z}),$

for all morphisms $f:Y\to X$, and flat projective morphisms $p:Y\to Z$ of
relative dimension $c$ with simple normal crossing fibers, where $\dim Z\leq
2n-k-2c$.

  6. (6)

We use the notations $N^{c}H_{k}$ etc. to denote the filtrations on Borel-
Moore or singular homology $H_{k}$. Since $X$ is smooth, this is the same as
the filtrations $N^{c}H^{2d-k}$.

A general relation between these filtrations is the following.

######  Lemma 1.19.

[Voi22, Proposition 1.3] We have the following inclusions:

$\tilde{N}_{n-c,\text{cyl},\text{eq}}H^{2c-1}(X,A)\subset\tilde{N}_{n-c,\text{cyl}}H^{2c-1}(X,A)=\tilde{N}^{c}H^{2c-1}(X,A)\subset
N^{c}H^{2c-1}(X,A).$

The only non-obvious part, the equality in the middle, is proved by Voisin
[Voi22, Proposition 1.3].

A natural question is whether or not these filtrations agree with each other.
If we use $\mathbb{Q}$ or $\mathbb{Q}_{\ell}$ coefficients, the theory of
weights shows that the strong coniveau and coniveau filtrations are
equivalent. Since the difference between some of these filtrations also gives
stable birational invariants, one wonders if this could be used to prove non-
stable-rationality for some rationally connected varieties.

Examples with $\mathbb{Z}$-coefficients where the strong coniveau filtration
and coniveau filtration differ are constructed in [BO21]. More precisely, they
prove the following.

######  Theorem 1.20.

[BO21, Theorem 1.1] For all $c\geq 1$ and $k\geq 2c+1$, there is a smooth
projective complex variety $X$ such that the inclusion

$\tilde{N^{c}}H^{k}(X,\mathbb{Z})\subset N^{c}H^{k}(X,\mathbb{Z})$

is strict. One may choose $X$ to have torsion canonical bundle. If $c\geq 2$,
one may choose $X$ to be rational.

The examples as above usually have large dimension especially when $c$ is
large. For lower dimensional examples, Benoist-Ottem proved the following.

######  Theorem 1.21.

[BO21, Theorem 1.2] For $k\in\\{3,4\\}$, there is a smooth projective complex
variety $X$ of dimension $k+1$ with torsion canonical bundle such that the
inclusion

$\tilde{N}^{1}H^{k}(X,\mathbb{Z})\subset N^{1}H^{k}(X,\mathbb{Z})$

is strict.

These examples leave the case of $c=1$ open for threefolds and for rationally
connected varieties. Voisin studied the strong coniveau filtrations on
$H^{2d-3}$ [Voi22].

######  Theorem 1.22.

[Voi22, Theorem 2.6, Corollary 2.7, Theorem 2.17] Let $X$ be a smooth
projective variety of dimension $d$ defined over $\mathbb{C}$.

  1. (1)

Assume the Walker Abel-Jacobi map ([Wal07])

$\phi:CH_{1}(X)_{\text{alg}}\to J(N^{1}H^{2d-3}(X,\mathbb{Z}))$

is injective on torsions. Then we have

$N_{1,\text{st},\text{cyl}}H^{2d-3}(X,\mathbb{Z})/\text{Tor}=N^{1}H^{2d-3}(X,\mathbb{Z})/\text{Tor}.$

  2. (2)

If $\dim X$ is $3$, we have

$N_{1,\text{cyl},\text{st}}H^{3}(X,\mathbb{Z})/\text{Tor}=N^{1}H^{3}(X,\mathbb{Z})/\text{Tor}.$

  3. (3)

If $X$ is rationally connected, we have

$N_{1,\text{cyl},\text{st}}H^{2d-3}(X,\mathbb{Z})=\tilde{N}_{1}H^{2d-3}(X,\mathbb{Z}).$

As a consequence,

$N_{1,\text{cyl},\text{st}}H^{2d-3}(X,\mathbb{Z})/\text{Tor}=\tilde{N}^{1}H^{2d-3}(X,\mathbb{Z})/\text{Tor}=N^{1}H^{2d-3}(X,\mathbb{Z})/\text{Tor}.$

We prove an improvement of Voisin’s results.

######  Theorem 1.23.

Let $X$ be a complex smooth projective variety of dimension $d$. Then the
following two filtrations agree with each other:

$N_{1,\text{st},\text{cyl}}H_{3}(X,\mathbb{Z})=N^{1}H^{2d-3}(X,\mathbb{Z}).$

Assume furthermore that $X$ is SRC in codimension $1$. There is a smooth
projective curve $C$ with a family of $1$-dimensional cycles $\Gamma\subset
C\times X$ such that

$\Gamma_{*}:H_{1}(C,\mathbb{Z})\to H_{3}(X,\mathbb{Z})$

has the same image as the s-map $s:L_{1}H_{3}(X)\to H_{3}(X)$, which is the
same as the $N^{1}H^{3}(X,\mathbb{Z})$. In particular, the following
filtrations on $H^{2d-3}(X,\mathbb{Z})$ introduced in Definition 1.18 are the
same:

$\tilde{N}_{1,\text{cyl},\text{eq}}H^{2d-3}(X,\mathbb{Z})=\tilde{N}_{1,\text{cyl}}H^{2d-3}(X,\mathbb{Z})=\tilde{N}^{d-1}H^{2d-3}(X,\mathbb{Z})=N^{d-1}H^{2d-3}(X,\mathbb{Z}).$

For the definition of Lawson homology and s-map, see Definition 3.5 and 3.10
in Section 3.

An immediate corollary is the following.

######  Theorem 1.24.

Let $X$ be a complex smooth projective variety, that is SRC in codimension
$1$. Then the following filtrations on $H^{3}(X,\mathbb{Z})$ introduced in
Definition 1.18 coincide with the whole cohomology group:

$\tilde{N}_{1,\text{cyl},\text{eq}}H^{3}(X,\mathbb{Z})=\tilde{N}_{1,\text{cyl}}H^{3}(X,\mathbb{Z})=\tilde{N}^{1}H^{3}(X,\mathbb{Z})=N^{1}H^{3}(X,\mathbb{Z})=H^{3}(X,\mathbb{Z}).$

######  Remark 1.25.

Using the decomposition of the diagonal argument, one can show that when $X$
is SRC in codimension $1$, for each $i$, there is a smooth projective variety
$Y_{i}$ and a family of cycles $\Gamma_{i}\subset Y_{i}\times X$ such that the
cokernal of $\Gamma_{*}:H_{i}(Y_{i})\to H_{i+2}(X)$ is $N$-torsion for a fixed
$N$. So we may consider the s-map from $\mathbb{Z}/N$ Lawson homology (defined
as the homotopy group of the topological group $Z_{r}(X)\otimes\mathbb{Z}/N$)
to $H_{3}(X,\mathbb{Z}/N)$. We have long exact sequences

$\begin{CD}L_{1}H_{i}(X,\mathbb{Z})@>{\cdot
N}>{}>L_{1}H_{i}(X,\mathbb{Z})@>{}>{}>L_{1}H_{i}(X,\mathbb{Z}/N)@>{}>{}>L_{1}H_{i-1}(X,\mathbb{Z})\\\
@V{}V{}V@V{}V{}V@V{}V{}V@V{}V{}V\\\ H_{i}(X,\mathbb{Z})@>{\cdot
N}>{}>H_{i}(X,\mathbb{Z})@>{}>{}>H_{i}(X,\mathbb{Z}/N)@>{}>{}>H_{i-1}(X,\mathbb{Z}).\\\
\end{CD}$

By results of Suslin-Voevodsky [SV96] and the Bloch-Kato conjecture proved by
Voevodsky, there is an isomorphism

$L_{1}H_{2+i}(X,\mathbb{Z}/N)\cong
CH_{1}(X,i,\mathbb{Z}/N)\cong\mathbb{H}^{i}(X,\tau^{\leq\dim
X-1}R\pi_{*}(\mathbb{Z}/N))$

between torsion Lawson homology, Bloch’s higher Chow group, and certain
Zariski cohomology group, where $\pi:X_{cl}\to X_{zar}$ is the continuous map
from $X(\mathbb{C})$ with the analytic topology to $X$ with the Zariski
topology.

We also have a long exact sequence:

$\ldots\to L_{1}H_{k}(X,\mathbb{Z}/N)\to H_{k}(X,\mathbb{Z}/N)\to
KH_{k}(X,\mathbb{Z}/N)\to L_{1}H_{k-1}(X,\mathbb{Z}/N)\to\ldots$

where $KH_{k}(X,\mathbb{Z}/N)=H^{\dim X-k}(X,R^{\dim X}\pi_{*}\mathbb{Z}/N)$
is the so-called Kato homology. The author has made a number of conjectures
about Kato homologies of a rationally connected fibration in [Tia20]. Special
cases of these conjectures imply that there is an isomorphism
$L_{1}H_{i}(X,\mathbb{Z}/N)\cong H_{i}(X,\mathbb{Z}/N)$ for all $k$ and all
rationally connected varieties and rationally connected fibrations over a
curve defined over the complex numbers. This in turn would imply the s-maps
$L_{1}H_{i}(X,\mathbb{Z})\to H_{i}(X,\mathbb{Z})$ are isomorphisms.

We have a similar result that applies to fields of positive characteristic.

######  Theorem 1.26 (=Theorem 4.14).

Let $X$ be a $d$-dimensional smooth projective variety defined over an
algebraically closed field, which is separably rationally connected in
codimension $1$. There is a smooth projective curve $C$ with a family of
$1$-dimensional cycles $\Gamma\subset C\times X$ such that

$\Gamma_{*}:H_{1}^{\text{BM}}(C,\mathbb{Z}_{\ell})\to
H_{3}^{\text{BM}}(X,\mathbb{Z}_{\ell})\cong
H^{2d-3}_{\text{\'{e}t}}(X,\mathbb{Z}_{\ell})$

surjects onto $N^{1}H^{2d-3}(X,\mathbb{Z}_{\ell})$.

######  Theorem 1.27 (=Theorem 4.16).

Let $X$ be a smooth projective defined over an algebraically closed field,
which is separably rationally connected in codimension $1$. Assume $X$ is a
$3$-fold. Then the following filtrations on $H^{3}(X,\mathbb{Z}_{\ell})$
introduced in Definition 1.18 equal the whole cohomology group:

$\tilde{N}_{1,\text{cyl},\text{eq}}H^{3}(X,\mathbb{Z}_{\ell})=\tilde{N}_{1,\text{cyl}}H^{3}(X,\mathbb{Z}_{\ell})=\tilde{N}^{1}H^{3}(X,\mathbb{Z}_{\ell})=N^{1}H^{3}(X,\mathbb{Z}_{\ell})=H^{3}(X,\mathbb{Z}_{\ell}).$

###  1.4. Integral Tate conjecture for one cycles: arithmetic part

We continue the discussion on integral Tate conjecture in this section.

The Serre-Hochschild spectral sequence gives an exact sequence:

$0\to
H^{1}(\mathbb{F}_{q},H^{2d-2r-1}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(d-r)))\to
H^{2d-2r}_{\text{\'{e}t}}(X,\mathbb{Z}_{\ell}(d-r))\to
H^{2d-2r}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(d-r))^{G}\to 0.$

Thus the integral Tate conjecture consists of a geometric part, i.e.
surjectivity of

$CH_{r}(X)\otimes\mathbb{Z}_{\ell}\to
H^{2d-2r}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(d-r))^{G},$

and an arithmetic part, i.e. surjectivity of

$CH_{r}(X)_{\text{hom}}\otimes\mathbb{Z}_{\ell}\to
H^{1}(\mathbb{F}_{q},H^{2d-2r-1}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(d-r))),$

where $CH_{r}(X)_{\text{hom}}\otimes\mathbb{Z}_{\ell}$ is the “geometrically
homologically trivial” part, i.e. the kernal of

$CH_{r}(X)\otimes\mathbb{Z}_{\ell}\to
H^{2d-2r}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(d-r))^{G}.$

In a recent preprint [SS22], Scavia and Suzuki systematically investigated the
question of the surjectivity in the arithmetic part and relate them to the
strong coniveau filtration. For codimension $2$ cycles, they obtain the
following results.

######  Theorem 1.28.

[SS22, Theorem 1.3] Let $\mathbb{F}$ be a finte field and $\ell$ be a prime
number invertible in $\mathbb{F}$,and suppose that $\mathbb{F}$ contains a
primitive $\ell^{2}$-th root of unity. There exists a smooth projective
geometrically connected $\mathbb{F}$-variety X of dimension $2\ell+2$ such
that the map

$CH^{2}(X)\otimes\mathbb{Z}_{\ell}\to
H^{4}_{\text{\'{e}t}}(X,\mathbb{Z}_{\ell}(4))^{G}$

is surjective whereas the map

$CH^{2}(X)_{\text{hom}}\otimes\mathbb{Z}_{\ell}\to
H^{1}(\mathbb{F}_{q},H^{3}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(2)))$

is not.

######  Theorem 1.29.

[SS22, Theorem 1.4] Let $p$ be an odd prime. There exist a finite field
$\mathbb{F}$ of characteristic $p$ and a smooth projective geometrically
connected fourfold $X$ over $\mathbb{F}$ for which the image of the
composition

$H^{1}(\mathbb{F},H^{3}_{\text{\'{e}t}}(X,\mathbb{Z}_{2}(2))_{\text{tors}})\to
H^{1}(\mathbb{F},H^{3}_{\text{\'{e}t}}(X,\mathbb{Z}_{2}(2)))\to
H^{4}(X,\mathbb{Z}_{2}(2))$

contains a non-algebraic torsion class.

Shortly after Scavia-Suzuki’s preprint appeared, Benoist studied steenrod
operations on Chow groups and cohomologies [Ben22]. He obtained new examples
of non-algebraic cohomology classes over many fields
($\mathbb{C},\mathbb{R},\bar{\mathbb{F}}_{q},\mathbb{F}_{q}$) and for
cohomology classes on algebraizable smooth manifolds. In the case of finite
fields, his results removed the assumptions on $\ell^{2}$-th roots of unity in
Scavia-Suzuki’s results.

######  Theorem 1.30.

[Ben22, Theorem 4.12] Let $p\neq\ell$ be prime numbers, and let $\mathbb{F}$
be a finite subfield of $\bar{\mathbb{F}}_{p}/\mathbb{F}_{p}$. There exist a
smooth projective geometrically connected variety $X$ of dimension $2\ell+3$
over $\mathbb{F}$ and a non-algebraic class
$x\in\text{Ker}(H^{4}_{\text{\'{e}t}}(X,\mathbb{Z}_{\ell}(2))\to
H^{4}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(2)))$.

The failure of the surjectivity is related to the discrepancy of the strong
coniveau and coniveau filtration by the following.

######  Theorem 1.31.

[SS22, Theorem 1.5, Proposition 7.6] Let $X$ be a smooth projective
geometrically connected variety over a finite field $\mathbb{F}$ and $\ell$ be
a prime number invertible in $\mathbb{F}$. Suppose that the coniveau and
strong coniveau on $H^{3}_{\text{\'{e}t}}(X,\mathbb{Z}_{\ell}(2))$ coincide:
$N^{1}H^{3}(X,\mathbb{Z}_{\ell}(2))=\tilde{N}^{1}H^{3}_{\text{\'{e}t}}(X,\mathbb{Z}_{\ell}(2))$.
Then the $\ell$-adic algebraic Abel-Jacobi map is an isomorphism:

$CH^{2}(X)_{\text{alg}}\otimes\mathbb{Z}_{\ell}\to
H^{1}(\mathbb{F},N^{1}H^{3}(X,\mathbb{Z}_{\ell}(2))).$

In general, we have a surjection

$CH^{r}(X)_{\text{F-alg}}\otimes\mathbb{Z}_{\ell}\to\text{Image}(H^{1}(\mathbb{F},\tilde{N}^{r-1}H^{2r-1}(X,\mathbb{Z}_{\ell}(r)))\to
H^{1}(\mathbb{F},N^{r-1}H^{2r-1}(X,\mathbb{Z}_{\ell}(r)))).$

In the paper of Scavia-Suzuki [SS22], they use $CH^{r}(X)_{\text{F-alg}}$ to
denote cycles algebraically equivalent to zero over $\mathbb{F}$, and
$CH^{r}(X)_{\text{alg}}$ to denote cycles defined over $\mathbb{F}$ that are
algebraically equivalent to zero over $\bar{\mathbb{F}}$. However, for
codimension $2$ cycles on varieties defined over $\mathbb{F}$, there is no
known example where $CH^{2}(X)_{\text{F-alg}}$ and $CH^{2}(X)_{\text{alg}}$
differ (Question 8.2, [SS22]), and
$CH^{2}(X)_{\text{F-alg}}\otimes\mathbb{Z}_{\ell}$ and
$CH^{2}(X)_{\text{alg}}\otimes\mathbb{Z}_{\ell}$ are isomorphic if the strong
coniveau coincides with the coniveau filtration on $H^{3}$ ([SS22, Propostion
7.10, 7.11]). Moreover, for one cycles on a separably rationally connected
variety or a separably rationally connected fibration over a curve defined
over a finite field, we know $CH_{1}(X)_{\text{F-alg}}$ and
$CH_{1}(X)_{\text{alg}}$ are the same [KT23, Theorem 6].

While the surjectivity in the arithmetic part is not true in general, we do
expect this to be true for one cycles on varieties that is separably
rationally connected in codimension $1$.

As a corollary of Theorem 4.7 and the work of Scavia-Suzuki, we get the
following results regarding the arithemtic part of the cycle class map.

######  Corollary 1.32 (=Corollary 4.17).

Let $X$ be a smooth projective variety of dimension $d$ defined over a finite
field $\mathbb{F}_{q}$, which is separably rationally connected in codimension
$1$. Then we have a surjection

$CH_{1}(X)_{\text{alg}}\to
H^{1}(\mathbb{F}_{q},N^{1}H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}_{\ell}(d-1)).$

Furthermore, assume one of the followings

  1. (1)

$N^{1}H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}_{\ell}(d-1))=H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}_{\ell}(d-1))$.

  2. (2)

The cycle class map

$cl:\lim\limits_{\xleftarrow[n]{}}CH_{1}(\bar{X},1,\mathbb{Z}/\ell^{n})\to
H^{2d-3}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(d-1))$

is surjective.

Then every class in
$H^{1}(\mathbb{F}_{q},H^{3}(\bar{X},\mathbb{Z}_{\ell}(d-1)))$ is the class of
an algebraic cycle. In particular, this holds if $X$ has dimension $3$.

###  1.5. Algebraic equivalence

The key technical result in proving the main theorems is the joint work of the
author with János Kollár [KT23] studying algebraic equivalences of one cycles
on smooth projective varieties.

Algebraically equivalence between two cycles means that one has to add
complicated cycles to both of them and then get a family of cycles over a
curve. Given two stable maps, supposing that they give algebraically
equivalent cycles, it is not clear that this algebraic equivalence of cycles
can be realized as a deformation of stable maps. In the joint work with
Kollár, we prove that this is always possible for curves on smooth projective
varieties.

######  Theorem 1.33.

[KT23, Theorem 1] Let $X$ be a smooth, projective variety over an
algebraically closed field $K$. Let $\pi_{i}:C_{i}\to X$ (for $i\in I$) be
finitely many morphisms of nodal curves to $X$ such that the
$(\pi_{i})_{*}[C_{i}]$ are algebraically equivalent to each other. Then there
is a morphism from a single nodal curve $\pi_{R}:R\to X$ and a family of
connected nodal curves over a connected curve $B$ with a morphism to $X$:
$B\xleftarrow{}S\xrightarrow{\pi}X$ such that for any $i\in I$, there is a
point $b_{i}\in B$ with fiber $S_{i}\cong C_{i}\cup R$ and
$(\pi|_{S_{i}}:C_{i}\to X)\cong(\pi_{i}:C_{i}\to X),(\pi|_{S_{i}}:R\to
X)\cong(\pi_{R}:R\to X)$.

That is, the deformation of cycles is visible as deformation of maps. This
result yields many interesting applications. For example, the study leads to
the following theorem.

######  Theorem 1.34.

[KT23, Theorem 6] Let $X_{k}$ be a smooth, projective variety over a perfect
field $k$, which is separably rationally connected in codimension $1$. Then
the kernel of the natural map

$A_{1}(X_{k})\to A_{1}({X}_{\bar{k}})$

is either trivial or $\mathbb{Z}/2\mathbb{Z}$. More precisely,

  1. (1)

the kernel is trivial if $X_{k}$ contains an odd degree $0$-cycle, and

  2. (2)

if $Z=\sum_{i}d_{i}C_{i}$ and $Z_{\bar{k}}$ is algebraically equivalent to $0$
over $\bar{k}$, then $Z$ is algebraically equivalent to $0$ over $k$ if and
only if the index of $X$ divides
$\chi(Z):=\sum_{i}d_{i}\chi(C_{i},\mathcal{O}_{C_{i}})$.

Recall that a _pseudo algebraically closed field_ (or a _PAC field_ for short)
is a field where every geometrically integral variety has a rational point.

######  Theorem 1.35.

[KT23, Theorem 7] Let $X_{k}$ be a smooth projective variety defined over a
perfect field $k$. Assume that every geometrically irreducible $k$-variety has
a $0$-cycle of degree $1$ (e.g. $k$ is a finite field or a PAC field). Assume
that $X$ is separably rationally connected in codimension $1$. We have an
isomorphism

$A_{1}(X_{k})\cong A_{1}(X_{\bar{k}})^{G},$

where $G$ is the absolute Galois group of $k$.

If $k$ is not perfect (and every geometrically irreducible $k$-variety has a
$0$-cycle of degree $1$), then we have an isomorphism after inverting the
characteristic $p$.

###  1.6. Structure of the paper

This paper consists of applications of the results in [KT23].

The first application is in Section 2, where we describe some structural
results of the space of one cycles. These structural results are then used in
Sections 3 4 to study the coniveau filtration on $H_{3}$ of a separably
rationally connected fibration over a curve. The case of complex varieties
uses Lawson homology and is topological. Moreover, it gives “integral”
results. We present this first to give the readers some flavor of the
argument. The general case has to use the Chow sheaves introduced by Suslin
and Voevodsky, hence more abstract. Unfortunately, in this case we only have
results for torsion coefficients and have to pass to the inverse limit from
time to time. The criterion for the surjection of the cycle class map onto the
arithmetic part is proved in Corollary 4.17.

Finally, the applications to local-global principles are discussed in Section
5. In Section 6, we give some examples where the criterion in Theorem 1.13 can
be effectively checked.

Acknowledgment: I would like to thank Jean-Louis Colliot-Thélène and Olivier
Wittenberg for many helpful and constructive comments. I am grateful to János
Kollár for generously sharing his ideas and for co-authoring the article
[KT23] which produces much stronger results than those proved in the first
version of this paper, and which provides the technical results needed for
this paper. This work is partially supported by NSFC grants No. 11890660 and
No.11890662.

##  2\. Space of one cycles

In this section, we fix an algebraically closed field $k$ of any
characteristic. We remind the readers that a variety is always assumed to be
irreducible through out the paper, and thus connected. Sometimes we add the
word irreducible just to emphasis this.

######  Definition 2.1.

Let $X,Y$ be finite type reduced separated $k$-schemes. A family of relative
cycles of equi-dimension $r$ over $Y$ is a formal linear combination of
integral subschemes $\mathcal{Z}=\sum m_{i}Z_{i},Z_{i}\subset Y\times
X,m_{i}\in\mathbb{Z}$ such that

  1. (1)

Each $Z_{i}$ dominates one irreducible component of $Y$

  2. (2)

Each fiber of $Z_{i}\to Y$ has dimension $r$.

  3. (3)

A fat point condition is satisfied (see [Kol96, Chapter I, 3.10.4] or [SV00,
Definition 3.1.3]).

  4. (4)

A field of definition condition is satisfied. Namely, for any point $y\in Y$,
there are integral subschemes $\gamma_{i}$ of $X$ defined over the residue
field $\kappa(y)$ such that the cycle theoretic fiber ([Kol96, Chapter I,
3.10.4]) of $\mathcal{Z}$ over $y$ is $\sum m_{i}\gamma_{i}$.

We write $\mathcal{Z}^{+}$ and $\mathcal{Z}^{-}$ as the positive and negative
parts, i.e. the sum of $Z_{i}$’s with positive and negative coefficients.

We say this family has proper support if $Z_{i}\to Y$ are proper for every
$i$.

It is often convenient to allow some redundancy in the linear combination,
especially when we consider pull-back and restrictions. For example, we allow
an expression of the form $(Z_{1}+Z_{3})-(Z_{2}+Z_{3})$. When this happens, we
think of $(Z_{1}+Z_{3})$ (resp. $(Z_{2}+Z_{3})$) as the positive (resp.
negative) part.

We refer the interested readers to [Kol96] Section I.3, I.4 and [SV00] Section
3 for details about the last two conditions and the subtle points about these
definitions. We only remark that with this definition, one can pull-back
families of cycles.

We adopt Kollár’s [Kol96] convention of only considering families of cycles
over a reduced base. Suslin and Voevodsky [SV00] consider more general base.

For our purpose, it is perfectly fine to always work over a reduced base. We
call a separated, finite type, reduced $k$-scheme an _algebraic set_.

######  Remark 2.2.

We would also like to mention that for a normal variety, condition $3$ is
automatic. Condition $4$ is always satisfied in characteristic $0$, or if the
base is regular, or if all the $m_{i}$’s are invertible in the field $k$. It
is introduced to make sure that pulling-back of families of cycles is always
defined and one has a presheaf of relative cycles.

######  Definition 2.3.

Given a family of projective curves $S\to B$ and a $B$-morphisms $F:S\to
B\times X$, we can associate to a family of cycles $\Gamma_{S}\subset B\times
X$. So given a family of cycles $\Gamma\subset B\times X$ over $B$, we say it
is _nodal_ , if it is induced from a family of nodal curves as above.

######  Proposition 2.4.

Let $S_{1},S_{2}$ be two connected algebraic sets and $\Gamma_{1}\subset
S_{1}\times X,\Gamma_{2}\subset S_{2}\times X$ be two equidimensional families
of $r$-cycles in $X$. Assume that the cycles in the two families are
algebraically equivalent. There is a connected algebraic set $S$ and an
equidimensional family of $r$-cycles $\Gamma\subset S\times X$, and morphisms
$f_{1}:S_{1}\to S,f_{2}:S_{2}\to S$ such that
$\Gamma_{1}=f_{1}^{*}\Gamma,\Gamma_{2}=f_{2}^{*}\Gamma$. Moreover, if both
$S_{1}$ and $S_{2}$ are normal/smooth/projective, we may choose $S$ to be
normal/smooth/projective.

###### Proof.

Take two points $s_{1}\in S_{2},s_{2}\in S_{2}$. Denote by
$\gamma_{1},\gamma_{2}$ the cycle over $s_{1},s_{2}$. By assumption,
$\gamma_{1},\gamma_{2}$ are algebraically equivalent. Thus there is a smooth
projective curve $S_{3}$, two points $a,b\in S_{3}$ and a family of $r$-cycles
$\Gamma_{3}\subset S_{3}\times X$ such that the cycle over $a,b$ are
$\gamma_{1},\gamma_{2}$. This follows from the definition of algebraic
equivalence. Indeed, by definition of algebraic equivalence, we may find a
cycle $\Delta$ and a family of cycles $\Gamma_{T}$ over a smooth projective
curve $T$ and two points $t_{1},t_{2}\in T$ such that the cycle over $t_{1}$
(resp. $t_{2}$) is $\gamma_{1}+\Delta$ (resp. $\gamma_{2}+\Delta$). Then we
take $S_{3}$ to be $T$ and the family of cycles $\Gamma_{3}$ to be
$\Gamma_{T}-T\times\Delta$.

Define $S=S_{1}\times S_{2}\times S_{3}$, with $p_{i}:S\to S_{i},i=1,2,3,$ the
projections. Define
$\Gamma=p_{1}^{*}\Gamma_{1}+p_{2}^{*}\Gamma_{2}-p_{3}^{*}\Gamma_{3}$. Finally,
define

$f_{1}:S_{1}\to S,x\mapsto(x,s_{2},b)$

and

$f_{2}:S_{2}\to S,y\mapsto(s_{1},y,a).$

One easily checks these satisfy the conditions.

Since $S_{3}$ is a smooth projective curve, if both $S_{1},S_{2}$ are normal,
or smooth, or projective, so is $S$. ∎

Now we can state the main technical result of this section.

######  Theorem 2.5.

Let $X$ be a smooth projective variety defined over an algebraically closed
field $k$. Let $(U,\Gamma_{U})$ be an equi-dimensional family of one
dimensional cycles over an irreducible variety $U$ and let $u_{0},u_{1}\in U$
be two points in $U$ such that

$\gamma_{0}:=\Gamma_{U}|_{u_{0}}=\gamma_{1}:=\Gamma_{U}|_{u_{1}}=\gamma$

as cycles. Then there is a family of one-dimensional cycles $(V,\Gamma_{V})$
over a smooth quasi-projective variety $V$ with a morphism $f:V\to U$ such
that $f^{*}\Gamma_{U}=\Gamma_{V}$, and a lifting $v_{0},v_{1}$ of
$u_{0},u_{1}$, such that

  1. (1)

The morphism $f:V\to U$ is projective and surjective.

  2. (2)

We may take the family over $V$ to be nodal as in Definition 2.3. We still use
$\Gamma_{V}$ to denote the family of nodal curves in the following.

  3. (3)

There is a nodal deformation equivalence $T\leftarrow\Gamma_{T}\to X$ over a
two pointed connected nodal curve $(T,t_{0},t_{1})$ between
$\Gamma_{V}|_{v_{0}}$ and $\Gamma_{V}|_{v_{1}}$.

  4. (4)

For each $k$-point $t$ of $T$, the cycle over $t$ is $\gamma$.

  5. (5)

For a general point $u$ in $U$, and for any pair of points $v,v^{\prime}$ in
the inverse image of $u$ in $V$, there is a nodal deformation equivalence

$(C_{v,v^{\prime}},c,c^{\prime})\leftarrow\Gamma_{C}\to
X,\Gamma_{C}|_{c}\cong\Gamma_{V}|_{v},\Gamma_{C}|_{c^{\prime}}\cong\Gamma_{V}|_{v^{\prime}}$

over a connected two pointed nodal curve $(C_{v,v^{\prime}},c,c^{\prime})$.

  6. (6)

For any point $c\in C$, the cycle of $\Gamma_{C}$ at $c$ is the same as that
of $\Gamma_{U}$ at the point $u$.

######  Theorem 2.6.

Keep the same assumptions as in Theorem 2.5. Assume furthermore that $X$ is
separably rationally connected in codimension $1$. Then $V,T,C_{v,v^{\prime}}$
admit a morphism to $(W,\Gamma_{W})$, where $W$ is a normal projective variety
and $\Gamma_{W}$ is a family cycles over $W$. In characteristic $0$, we may
choose $W$ to be irreducible, smooth and projective. In general, we may take
$W$ to be irreducible, normal, projective and smooth near the nodal points of
$T,C$, and $v_{0},v_{1}$.

######  Remark 2.7.

This theorem is special to one cycles on varieties that is SRC in codimension
$1$.

Indeed, if the statements were true for a variety $X$ and families of
$r$-dimensional cycles, then the same argument as in Sections 3 and 4 would
prove that the strong coniveau filtration $\tilde{N}^{r}H^{2r+1}(X)$ coincide
with the coniveau filtration $N^{r}H^{2r+1}$. But the examples of Benoist-
Ottem [BO21], Scavia-Suzuki [SS22] (cited in Section 1.3 and 1.4 as Theorems
1.20, 1.21, 1.28, 1.29) shows that this is not true in general.

######  Remark 2.8.

We remark that the positive and negative part of the families of cycles may
vary along $T$. The theorem only states that the difference remains constant.

######  Remark 2.9.

Even if we start with a family of effective cycles, for the statements to be
true, we have to use non-effective cycles.

Moreover, the statement is not a simple corollary of the existence of the
universal family over the Chow variety (which is true only in characteristic
$0$). This is because we require that the family is parameterized by a normal
variety, while the Chow variety is only semi-normal in [Kol96] by definition
or satisfies no such normality condition at all in some other references such
as [Fri91].

It is possible that the morphism from the normalization of the Chow variety to
the Chow variety maps two points to the same point. If this happens, we take
$U$ to be the normalization of the Chow variety, $u,u^{\prime}$ to be the two
points mapping to the same point in the Chow variety, the existence of $V,T,W$
in this case cannot be deduced from the existence of the universal family over
the Chow variety.

######  Remark 2.10.

Finally we remark that $U$ being irreducible is not essential in the proof.
But it simplifies the argument. If $U$ is reducible and connected, one can use
similar argument as in [KT23, Section 8] to find a connected algebraic set
$V$. But in this case, we cannot choose $V$ to admit a morphism to $U$. The
best one can have is that for each irreducible component of $U$, there is an
irreducible component of $V$ with a projective, surjetive morphism to this
component.

###### Proof of Theorem 2.5.

First, using Nagata’s compactification and Chow lemma, we can make a base
change that is a birational projective morphism and replace $U$ with a quasi-
projective variety. So in the following, we assume $U$ is quasi-projective.

Up to a purely inseparable base change, we may assume that the generic fiber
of the family is comes from nodal curves.

We write $\Gamma_{U}=\Gamma^{+}_{U}-\Gamma^{-}_{U}$ as its positive and
negative part. We write

$\gamma_{0}^{+}=\Gamma^{+}_{U}|_{u_{0}},\gamma_{0}^{-}=\Gamma^{-}_{U}|_{u_{0}},\gamma_{1}^{+}=\Gamma^{+}_{U}|_{u_{1}},\gamma_{1}^{-}=\Gamma^{-}_{U}|_{u_{1}}.$

By assumption,

$\gamma_{0}^{+}-\gamma_{0}^{-}=\gamma_{1}^{+}-\gamma_{1}^{-}.$

We take a general complete intersection $V^{\prime}$ (of the same dimension as
$U$) containing $v_{0}=(u_{0},u_{1})$ and $v_{1}=(u_{1},u_{0})$ in the product
$U\times U$. There are two families of nodal curves
$\Gamma_{p}=\Gamma_{p}^{+}-\Gamma_{p}^{-},\Gamma_{q}=\Gamma_{q}^{+}-\Gamma_{q}^{-}$
over $V^{\prime}$ induced by the two projections $p,q:V^{\prime}\to U$. We
have the family of cycles over $V^{\prime}$

${\Gamma}_{V^{\prime}}=(\Gamma_{p}^{+}+\Gamma_{q}^{-})-(\Gamma_{p}^{-}+\Gamma_{q}^{-}).$

Then the positive part over $v_{0}$ is $\gamma_{0}^{+}+\gamma_{1}^{-}$, and
the positive part over $v_{1}$ is $\gamma_{1}^{+}+\gamma_{0}^{-}$. Thus they
are the same as cycles. Moreover, the restriction of the negative parts of
$\Gamma_{V^{\prime}}$ to $v_{0},v_{1}$ are the same.

So now we have two families of cycles, and the restriction of each family to
$v_{0},v_{1}$ gives the same cycle. We first prove the statement for each
family. Then we take base changes for both families such that they are over
the same base and subtract them. Here we use the fact that the fiber of the
family over the singular points of $T,C$ and $v_{0},v_{1},v,v^{\prime}$ are
all nodal curves, and thus the base change will change nothing in their
neighborhood.

From now on we assume there is a nodal family of effective cycles.

The existence of a generically finite base change $V^{\prime\prime}\to U$, the
lifting $v_{0},v_{1}$ and the curve $T$ follows from [KT23, Theorem 58].
Unwrapping the content of this theorem, one gets the following:

  1. (1)

There is a nodal curve $Z$ and a family of r-tuple pf complete intersection
curves $r|H^{\text{ci}}|\to V^{\prime\prime}$.

  2. (2)

One can glue them together to get a family of nodal curves
$C_{V^{\prime\prime}}\to V^{\prime\prime}$.

  3. (3)

There is a family of nodal curves over a two pointed curve $X\leftarrow
C_{T}\to(T,t_{0},t_{1})$, such that the restriction of the family $C_{T}$ to
$t_{0},t_{1}$ coincide with the restriction of of $C_{V^{\prime\prime}}$ to
$v_{0},v_{1}$.

  4. (4)

For each $t\in T$, the cycle over $t$ is $Z+\sum_{i=1}^{r}L_{i}(t)$, where
$\cup L_{i}\to T$ is a family of $r$-tuple complete intersection curves.

  5. (5)

The restriction of $\cup L_{i}|$ to ${t_{0}},t_{1}$ coincide with the
restriction of $r|H^{\text{ci}}|$ to $v_{0},v_{1}$.

  6. (6)

The families of curves induces a morphism $V^{\prime\prime}\cup
T\to\text{Hilb}_{1}(X\times\mathbb{P}^{3})$.

This is almost what we want, except that the cycles over $V^{\prime\prime}$
and $T$ changes by a constant cycle $Z$ and a family of r-tuples of complete
intersection curves in $r|H^{\text{ci}}|$. So we subtract the corresponding
family.

To get a finite base change $V\to U$, one can take the graph closure of
$V^{\prime}$ with respect to the morphism to
$\text{Hilb}_{1}(X\times\mathbb{P}^{3})$ and then apply semi-stable reduction.

We may assume that $V$ is smooth using de Jong’s alteration (or resolution of
singularities in characteristic zero).

We note that the base change $V\to U$ consists of two steps: first a purely
inseparable base change $V^{\prime}\to U$ such that a general fiber becomes
nodal, then a further base change $V\to V^{1}\to U$ such that all fibers
becomes semi-stable. The second step does not change general fibers.
Therefore, for a general point $u\in U$, denote by $v^{1}\in V^{1}$ its
inverse image in $V^{\prime}$, and two of its inverse image by
$v,v^{\prime}\in V$. The fiber of the family of stable maps over
$v,v^{\prime}$ consists of complete intersection curves and the nodal curve
over $v^{1}$. They only differ by the complete intersection curves. So we can
construct a deformation over a curve $C$ from the fiber over $v$ to the fiber
over $v^{\prime}$ by deforming the complete intersection curves. ∎

###### Proof of Theorem 2.6.

This is essentially [KT23, Corollary 59]. As in the proof of Theorem 2.5, we
reduces the theorem to the case of an effective family. The point is, we can
attach families of curves constructed in [KT23, Theorem 43] to the family
(after a base change), so that the fiber (of the new family of curves) over
the nodes of $T,C$ and $v_{0},v_{1}$ has unobstructed deformation. Thus the
nodes in $T,C$ and $v_{0},v_{1}$ map to smooth points in the Hilbert scheme of
$X\times\mathbb{P}^{3}$. We take $W$ to be the normalization of the unique
geometrically irreducible component containing the image of $V,T,C$. In
characteristic $0$, we may even take a resolution of singularities of $W$ that
is isomorphic over the smooth locus. ∎

##  3\. Lawson homology

Let $X$ be a complex projective variety and we fix a very ample line bundle
$\mathcal{O}(1)$.

All the degree’s are taken with respect to this line bundle. Let
$\text{Chow}_{r,d}(X)$ be the Chow variety parameterizing degree $d$,
$r$-dimensional cycles of $X$ and

$\text{Chow}_{r}(X)=\coprod_{d\geq 0}\text{Chow}_{r,d}(X),$

where $\text{Chow}_{r,0}(X)$ is defined to be a single point corrsponding to
the zero cycle. We give the set $\text{Chow}_{r}(X)(\mathbb{C})$ the structure
of a topological monoid, where the topological structure comes from the
analytic topology on $\text{Chow}_{r,d}(X)(\mathbb{C})$ and the monoid
structure is the sum of cycles. Define $Z_{r}(X)$ to be the group completion
of $\text{Chow}_{r}(X)(\mathbb{C})$. It has a topological group structure. The
topology can be defined in several equivalent ways. These are studied by Lima-
Filho [LF94].

######  Definition 3.1.

We first define the category $I^{\text{eq}}$. The objects are pairs
$(S,\Gamma)$ consisting of a normal variety $S$ and a family of equi-
dimensional $r$-dimensional cycles $\Gamma$, and whose morphisms between
$(S,\Gamma)$ and $(S^{\prime},\Gamma^{\prime})$ are all the morphisms $f:S\to
S^{\prime}$ such that $\Gamma=f^{*}\Gamma^{\prime}$.

Define the topological space $Z_{r}(X)^{\text{eq}}$ as the colimit of all the
topological spaces $S(\mathbb{C})$ over the category $I^{\text{eq}}$.

More precisely, each $(S,\Gamma)$ in $I^{\text{eq}}$ gives a map of sets
$\phi_{S}:S(\mathbb{C})\to Z_{r}(X)$. The topology of $Z_{r}(X)^{\text{eq}}$
is defined in such a way that a subset $T\subset Z_{r}(X)$ is closed if and
only if $\phi_{S}^{-1}(T)$ is closed for all $(S,\Gamma)$.

######  Lemma 3.2.

In the definition of $Z_{r}(X)^{\text{eq}}$, we may take a subset consisting
of family of equidimensional cycles over normal projective varieties (or
smooth projective varieties).

###### Proof.

Given any family of equidimensional cycles $\Gamma\to S$, we may find a normal
projective variety (resp. smooth projective variety) $T$, a family
$\Gamma_{T}\to T$, and an open subset $T^{0}$ of $T$ such that there is a
surjective proper map $p:T^{0}\to S$ and $\Gamma_{T}|_{T^{0}}$ is
$\Gamma\times_{S}T^{0}$.

Note that we have a factorization $T^{0}(\mathbb{C})\to S(\mathbb{C})\to
Z_{r}(X)$. A set in $S(\mathbb{C})$ is closed if and only if its inverse image
under $p^{-1}$ in $T^{0}(\mathbb{C})$ is closed. That is, the topology of
$S(\mathbb{C})$ is the quotient topology coming from $T^{0}(\mathbb{C})\to
S(\mathbb{C})$.

Thus the topology on $Z_{r}(X)^{\text{eq}}$ is determined by families over
normal varieties (resp. smooth varieties) such that the family has an
extension over a normal (resp. smooth) projective compactification.

Therefore, when defining $Z_{r}(X)^{\text{eq}}$ as a colimit, we may take only
normal (resp. smooth) projective varieties. ∎

######  Definition 3.3.

Define the topological space $Z_{r}(X)^{\text{Chow}}$ as the quotient of

$\text{Chow}_{r}(X)(\mathbb{C})\times\text{Chow}_{r}(X)(\mathbb{C})$

by $\text{Chow}_{r}(X)(\mathbb{C})$, where the action is
$(a,b)\mapsto(a+c,b+c)$ for $c\in\text{Chow}_{r}(X)(\mathbb{C})$.

######  Theorem 3.4 ([LF94], Theorem 3.1, Theorem 5.2, Corollary 5.4).

The identity map induces homeomorphisms

$Z_{r}(X)^{\text{eq}}\cong Z_{r}(X)^{\text{Chow}}.$

Here is the definition of Lawson homology, first studied in [Law89].

######  Definition 3.5.

Let $X$ be a complex projecitve variety. Define the Lawson homology
$L_{r}H_{n+2r}(X)$ as the homotopy group $\pi_{n}(Z_{r}(X))$.

######  Example 3.6 (Dold-Thom isomorphism).

Consider $Z_{0}(X)$, the group of zero cycles on $X$. The classical Dold-Thom
theorem implies that there is an isomorphism

$L_{0}H_{n}(X)\cong H_{n}(X,\mathbb{Z}),$

######  Example 3.7 (Hurewitz map).

The Hurewitz map is induced by the inclusion $X\to Z_{0}(X)$:

$\pi_{k}(X)\to\pi_{k}(Z_{0}(X))\cong H_{k}(X,\mathbb{Z}).$

######  Theorem 3.8.

Let $X$ be a complex smooth projective variety. Then for any loop $L$ in
$Z_{1}(X)$, there is a projective algebraic set $Y$ with a family of nodal
curves $\Gamma\to Y\times X$ over $Y$ such that the map

$\Phi:L_{0}H_{1}(Y)=\pi_{1}(Z_{0}(Y))\to L_{1}H_{3}(X)=\pi_{1}(Z_{1}(X))$

induced by the family $\Gamma$ contains the class $[L]$ in $L_{1}H_{3}(X)$.
Assume furthermore that either $X$ is rationally connected or $X$ is a
rationally connected fibration over a curve, we may take $Y$ to be smooth.

We first introduce some notations. Given a projective algebraic set $S$
parameterizing a family of one dimensional cycles of $X$, there is an induced
continuous map between topological groups:

$Z_{0}(S)\to Z_{1}(X).$

We denote by $I(S)$ the image of this map, i.e. the closed subgroup of
$Z_{1}(X)$ generated by the cycles over $S$, and $K(S)$ the kernal of this
map.

The first observation in the proof of Theorem 3.8 is the following.

######  Lemma 3.9.

Let $X$ be a complex smooth projective variety. For any class $[L]$ in
$L_{1}H_{3}(X)=\pi_{1}(Z_{1}(X))$, there is a normal projective variety $U$
and a family of equidimensional one cycles $\gamma_{U}$ over $U$ such that
$[L]$ is represented by a continuous map

$I=[0,1]\to U\to Z_{1}(X).$

Note that $I\to U$ is not a loop in general.

###### Proof.

Denote by $Z_{1}(X)^{0}$ the neutral component of the topological group
$Z_{1}(X)$, i.e., the connected component containing the identity. We may
assume $L$ lies in $Z_{1}(X)^{0}$. Cycles in $Z_{1}(X)^{0}$ are precisely the
cycles algebraically equivalent to $0$. By Proposition 2.4, the topological
group $Z_{1}(X)^{0}$ is a filtered colimit over closed subgroups generated by
one-dimensional cycles parameterized by normal projective varieties.

Homotopy groups commutes with filtered colimits. Thus there is an irreducible
normal projective variety $S$ with a family of one dimensional cycles over $S$
such that the induced map

$\pi_{1}(I(S))\to\pi_{1}(Z_{1}(X))\cong L_{1}H_{3}(X)$

contains the class $[L]$ in $\pi_{1}(Z_{1}(X))$.

The fibration $K(S)\to Z_{0}(S)\to I(S)$ gives a long exact sequence of
homotopy groups:

$\ldots\to\pi_{1}(Z_{0}(S))\to\pi_{1}(I(S))\to\pi_{0}(K(S))\to\ldots.$

A loop in $I(S)$ lifts to a continuous map from the unit interval $I=[0,1]$ to
$Z_{0}(S)$, such that $0,1$ map to two points in $Z_{0}(S)$ that parameterize
the same cycle in $X$.

We may assume that the family over $I$ is the difference of two families of
effective $0$-cycles of degree $d+/d-$ in $S$. That is, it corresponds to the
difference of two continuous maps $f^{+}:I\to S^{(d+)},f^{-}:I\to S^{(d-)}$,
which is the same as a continuous map $f:I\to S^{(d+)}\times S^{(d-)}$ with
$0$ mapping to a point $x=(x^{+},x^{-})$ and $1$ mapping to a point
$y=(y^{+},y^{-})$.

A family of one cycles over $S$ induces a family of cycles over $S^{(d+)}$ and
$S^{(d-)}$. Let us denote them by $\Gamma_{d+},\Gamma_{d-}$.

The loop is the composition $I\to S^{(d+)}\times S^{(d-)}\to Z_{0}(S)\to
Z_{1}(X)$, where the middle map is taking the difference.

Let us use a different family of cycles
$\pi_{+}^{*}\Gamma_{d+}-\pi_{d-}^{*}\Gamma_{d-}$ on the product
$S^{(d+)}\times S^{(d-)}$, where $\pi_{+/-}$ is the projection to
$S^{(d+)/(d-)}$. This family of cycles induces a continuous map
$S^{(d+)}\times S^{(d-)}\to Z_{1}(X)$ such that the composition $I\to
S^{(d+)}\times S^{(d-)}\to Z_{1}(X)$ is the loop $L$.

We take $U$ to be $S^{(d+)}\times S^{(d-)}$ and $\gamma_{U}$ to be
$\pi_{+}^{*}\Gamma_{d+}-\pi_{d-}^{*}\Gamma_{d-}$. ∎

###### Proof of Theorem 3.8.

By Lemma 3.9, there is a normal projective variety $U$ and a family of
equidimensional one cycles $\gamma_{U}$ over $U$ such that $[L]$ is
respresented by a continuous map

$f:I=[0,1]\to U\to Z_{1}(X).$

Denote by $x,y\in U$ the image of $0,1$ by $f$. The cycle over $x,y$ are the
same by assumption. Now we are in the set-up of Theorem 2.5. Thus there is a
smooth projective variety $V$ with a generically finite surjective morphism
$V\to U$, a lifting $x_{V},y_{V}$ of $x,y$ to $V$, and families of constant
cycles parameterized by curves $T,C$ connecting $x_{V},y_{V}$ and respectively
inverse images of a general point in $S$. We take $Y$ to be $V\cup T\cup C$ in
this case.

The morphism $V\to U$ induces a continuous map between topological groups
$Z_{0}(V)\to Z_{0}(U)$. Denote by $K$ the kernal topological group. As a
group, $K$ is generated by cycles of the form $a-b$, where $a,b$ are points in
a fiber of $V\to U$. Note that $I(V)=I(U)$. Thus we have a fibration sequence
of topological groups:

$0\to K\to K(V)\to K(U)\to 0.$

We have commutative diagrams:

$\begin{CD}\pi_{1}(Z_{0}(Y))@>{}>{}>\pi_{1}(I(Y))@>{}>{}>\pi_{0}(K(Y))\\\
@A{}A{}A@A{}A{}A@A{}A{}A\\\
\pi_{1}(Z_{0}(V))@>{}>{}>\pi_{1}(I(V))@>{}>{}>\pi_{0}(K(V))@<{}<{}<\pi_{0}(K)\\\
@V{}V{}V@V{}V{=}V@V{}V{}V\\\
\pi_{1}(Z_{0}(U))@>{}>{}>\pi_{1}(I(U))@>{}>{}>\pi_{0}(K(U))\\\ \end{CD}$

The obstruction of lifting the class $[L]$ in $\pi_{1}(I(V))$ is in
$\pi_{0}(K(V))$ and maps to $[x-y]$ in $\pi_{1}(K(U))$. The class
$[x_{V}-y_{V}]$ differs from the obstruction class by an element in
$\pi_{0}(K)$.

We take the stein factorization $V\to V^{\prime}\to U$, where $V\to
V^{\prime}$ has connected fibers (hence birational) and $V^{\prime}\to U$ is
finite. Therefore $\pi_{0}(K)$ is finitely generated by classes of the form
$[a-b]$, where $a,b$ are points in the fiber over a general point in $U$.

The class $[L]$ maps to $\pi_{1}(I(Y))$, with obstruction class the push-
forward of $[x_{V}-y_{V}]$ modulo classes in $\pi_{0}(K)$.

By the existence of the families of constant cycles over the curves $T,C$ in
Theorem 2.6, we have

  1. (1)

The composition

$\pi_{0}(K)\to\pi_{0}(K(V))\to\pi_{0}(K(Y))$

is the zero map.

  2. (2)

The push-forward of the class $[x_{V}-y_{V}]$ vanishes in $\pi_{0}(K(Y))$.

Thus the class of the loop $L$ in $\pi_{1}(I(Y))$ is contained in
$\pi_{1}(Z_{0}(Y))$.

Finally, if $X$ is rationally connected in codimension $1$, by Theorem 2.6,
all these families over $V,T,C$ come from pulling back from a family of cycles
over a smooth projective variety. In this case, we take $Y$ to be this smooth
projective variety. ∎

Now we introduce another ingredient.

######  Lemma 3.10.

[FM94, Page 709, 1.2.1] There is a continuous map, the _s-map_ :
$Z_{r}(X)\wedge\mathbb{P}^{1}\to Z_{r-1}(X)$ inducing the s-map on Lawson
homologies $s:L_{r}H_{k}(X)\to L_{r-1}H_{k}(X)$.

######  Remark 3.11.

The construction of the s-map depends on a deep result: Lawson’s algebraic
suspension theorem. A geometric way of describing the s-map is the following.
Given a cycle $\Gamma$, take a general pencil of divisors
$D_{t}(t\in\mathbb{P}^{1})$ that intersect $\Gamma$ properly, and the s-map
sends $([\Gamma],t)$ to the cycle $\Gamma\cdot D_{t}-\Gamma\cdot D_{0}$.

######  Definition 3.12.

Let $Y$ be a semi-normal variety. Let $Z\subset Y\times X$ be a family of
$r$-dimensional cycle over $Y$ corresponding to a morphism $f:Y\to Z_{r}(X)$.
We define the _correspondence homomorphism_

$\Phi_{f}:H_{k}(Y,\mathbb{Z})\to H_{k+2r}(X,\mathbb{Z})$

as the composition

$H_{k}(Y,\mathbb{Z})\cong\pi_{k}(Z_{0}(Y))\to\pi_{k}(Z_{r}(X))\xrightarrow{s^{\circ
k}}\pi_{k+2r}(Z_{0}(X))\cong H_{k+2r}(X,\mathbb{Z}),$

where the map $\pi_{k}(Z_{r}(X))\to\pi_{k+2r}(Z_{0}(X))$ is induced by $k$-th
iterations of the $s$-map.

######  Theorem 3.13 ([FM94] Theorem 3.4).

Let $Y$ be a smooth projective variety and $\Gamma\subset Y\times X$ be a
family of $r$-dimensional cycle over $Y$ corresponding to a morphism $f:Y\to
Z_{r}(X)$. We have

$\Phi_{f}=\Gamma_{*}:H_{k}(Y,\mathbb{Z})\to H_{k+2r}(X,\mathbb{Z}),$

where $\Gamma_{*}$ is the map defined using $Z$ as a correspondence.

With this result, we can prove the main results over complex numbers.

######  Theorem 3.14.

Let $X$ be a smooth projective variety. There is a projective curve $C$ with a
nodal family of cycles $\Gamma\subset C\times X$ inducing a map $f:C\to
Z_{1}(X)$ such that

$\Phi_{f}:H_{1}(C,\mathbb{Z})\to H_{3}(X,\mathbb{Z})$

has the same image as the s-map $s:L_{1}H_{3}(X)\to H_{3}(X)$, which is the
same as the coniveau filtration $N^{1}H_{3}(X,\mathbb{Z})$.

Furthermore, if $X$ is rationally connected in codimension $1$, we may take
$C$ to be a smooth projective curve, and $\Phi_{f}=\Gamma_{*}$.

###### Proof.

Recall that there is an isomorphism $L_{0}H_{k}(S)\cong H_{k}(S)$ for any
projective algebraic set $S$ by the Dold-Thom theorem. We have a commutative
diagram:

$\begin{CD}L_{0}H_{1}(Y)@>{f_{*}}>{}>L_{1}H_{3}(X)\\\
@V{\cong}V{}V@V{s}V{}V\\\ H_{1}(Y)@>{\Phi_{f}}>{}>H_{3}(X)\end{CD}$

The image of the s-map is finitely generated. Thus we may find finitely many
projective algebraic sets $Y_{i}$ and families of semistable curves
$\Gamma_{i}\to Y_{i}\times X$ such that the generators of the image of s-map
are contained in the image of the correspondence homomophisms
$\Phi_{i*}:H_{1}(Y_{i})\to H_{3}(X)$. Then we take $Y$ to be the product
$\Pi_{i}Y_{i}$ and $\Gamma=\sum_{i}\pi_{i}^{*}\Gamma_{i}$. Clearly $\Phi_{*}$
contains the image of all the $\Phi_{i*}$.

By taking general hyperplane sections containing all the singularities and all
the one dimensional irreducible components, we may find a projective curve
$C\subset Y$ such that $\pi_{1}(C_{1})\to\pi_{1}(Y)$ is surjective. Then we
simply restrict the family of cycles to $C$.

If $X$ is rationally connected in codimension $1$, we may take all the
$Y_{i}$’s to be smooth by Theorem 3.8, and thus $C$ to be a general complete
intersection of very ample divisors, which is a smooth projective curve.

Finally, we note that the image of the s-map

$s:L_{1}H_{3}(X)\to H_{3}(X)$

is $N^{1}H_{3}(X,\mathbb{Z})$ by [Wal07, Proposition 2.8]. ∎

The immediate consequence is the following.

######  Theorem 3.15.

Let $X$ be a smooth projective rationally connected variety or a rationally
connected fibration over a curve. Assume $X$ is a $3$-fold. Then all the
filtrations on $H^{3}(X,\mathbb{Z})$ introduced in Definition 1.18 equal the
whole cohomology group:

$\tilde{N}_{1,\text{cyl},\text{eq}}H^{3}(X,\mathbb{Z})=\tilde{N}_{1,\text{cyl}}H^{3}(X,\mathbb{Z})=\tilde{N}^{1}H^{3}(X,\mathbb{Z})=N^{1}H^{3}(X,\mathbb{Z})=H^{3}(X,\mathbb{Z}).$

###### Proof.

By the decomposition of the diagonal argument,

$L_{1}H_{k}(X)\otimes\mathbb{Q}\cong H_{k}(X,\mathbb{Q})\cong
H^{k}(X,\mathbb{Q}).$

Thus we know that

$s:L_{1}H_{3}(X)\to H_{3}(X,\mathbb{Z})\cong H^{3}(X,\mathbb{Z})$

is surjective by [Voi08, Corollary 3.1]. ∎

##  4\. Chow sheaves

In this section we discuss the general case over an algebraically closed field
$k$.

Sometimes we may “invert $p$”, by taking the tensor product of a sheaf with
$\mathbb{Z}[\frac{1}{p}]$. In this scenario, we understand that $p$ is $1$ if
the base field has characteristic $0$ and equal the characteristic otherwise.
We use $\mathbb{Z}_{l}$ coefficient étale cohomology for $\ell$ a prime
number, non-zero in the field $k$.

One can also define Lawson homology with $\mathbb{Z}_{\ell}$ coefficients in
this context [Fri91]. But the construction of the analogue of $Z_{r}(X)$ is
more complicated . Also Lawson homology in this context is much less studied.
Many of the known results for complex varieties have not been explicitly
stated to hold, even though one can imagine that they are still true. For
example, the author do not know a reference for the construction of the s-map,
Neither could the author find the analogue of Friedlander-Mazur’s result
(Theorem 3.13 ) explicitly stated. If one had developed all the necessary
results in this general context, presumably the argument in last section works
the same way.

So we decided to use another approach for the general case.

######  Definition 4.1.

A finite correspondence from $Y$ to $X$ is a family of relative cycles of
dimension $0$ with proper support over $Y$.

######  Definition 4.2.

Let $\text{Sch}_{k}$ be the category of finite type separated $k$-schemes. Let
$\text{Cor}_{k}$ be category whose objects are separated finite type
$k$-schemes and morphisms finite correspondences. Let $\text{SmCor}_{k}$ be
the full subcategory whose objects are smooth $k$-varieties. In this
subcategory a finite correspondence between from $X$ to $Y$ is a linear
combination of closed subvarieties of $X\to Y$ that are finite surjective onto
one of the irreducible components of $X$.

Recall that the h-topology is generated by covers that are universal
topological epimorphisms. Since we will only deal with noetherian schemes,
this is the same as the topology generated by Zariski covers and covers that
are proper surjective morphisms.

The qfh-topology is generated by covers in the h-topology that are also quasi-
finite.

Later we will only use the fact that a surjective proper morphism is an
h-cover.

######  Definition 4.3.

We define the presheaf $Z_{\text{fl}}(X,r)$ on the category $\text{Sch}_{k}$
whose value on a scheme $S$ is a formal linear combination of integral
subschemes $Z\subset S\times X$ that is flat, of equidimension $r$ over $S$.

We also define $Z_{\text{eq}}(X,r)$ on the category $\text{Sch}_{k}$ whose
value on a scheme $S$ is the group of families of cycles in $X$ of
equidimension $r$ over $S$.

We also define $Z(X,r)$ on the category $\text{Sch}_{k}$ whose value on a
scheme $S$ is the group of families of cycles of dimension $r$ in $X$ over
$S$.

We define $Z_{\text{fl}}^{\text{eff}}(X,r)$ (resp.
$Z_{\text{eq}}^{\text{eff}}(X,r),Z^{\text{eff}}(X,r)$) on the category
$\text{Sch}_{k}$ whose value on a scheme $S$ is the monoid of families of
effective cycles in $X$ of equidimension $r$ over $S$.

Similarly, we define
$C_{\text{fl}}(X,r),C_{\text{eq}}(X,r),C(X,r),C_{\text{fl}}^{\text{eff}}(X,r),C_{\text{eq}}^{\text{eff}}(X,r),C^{\text{eff}}(X,r)$
as the counterpart of the above presheaves for families of cycles with proper
support over $S$.

The sheaf $C_{\text{fl}}(X,r)$ is denoted by
$\mathbb{\mathbb{Z}}\text{PropHilb}$ in [SV00].

Since later we will consider cycles on proper schemes, with the purpose of
keeping the names consistent with previous section, we will use
$Z_{\text{fl}}(X,r)$ etc. notations.

Note that we do not require the subschemes to be equidimensional over $S$ in
the definition of $Z(X,r)$ and $C(X,r)$. It is possible to have higher
dimensional fibers ([SV00, Example 3.1.9]). However, $Z^{\text{eff}}(X,r)$ is
the same as $Z^{\text{eff}}_{\text{eq}}(X,r)$. Similarly for the properly
supported version.

We have the following.

######  Proposition 4.4.

[SV00, Proposition 4.2.7, 4.2.6, Lemma 4.2.13] The presheaf
$Z_{\text{eq}}(X,r)\otimes\mathbb{Z}[\frac{1}{p}]$ is a qfh-sheaf and the
presheaf $Z(X,r)\otimes\mathbb{Z}[\frac{1}{p}]$ is an h-sheaf. Moreover, the
sheafification in the h topology of $Z_{\text{eq}}(X,r)$ is the same as that
of $Z(X,r)$.

In the following, we write $Z_{r}^{h}(X)$ as the h-sheaf associated to
$Z_{\text{fl}}(X,r)$ (which is the same as that of $Z_{\text{eq}}(X,r)$ or
$Z(X,r)$).

In the following, given a presheaf $\mathcal{F}$, we use $C^{*}(\mathcal{F})$
to denote the Suslin complex of presheaves (with non-positive degrees). That
is, $C^{-i}(\mathcal{F})(S)=\mathcal{F}(S\times\Delta^{i})$, where
$\Delta^{i}=\text{Spec }k[t_{0},\ldots,t_{i}]/\langle\sum_{j}t_{j}=1\rangle$
is the algebraic $i$-simplex.

If $\mathcal{F}$ is a torsion, homotopy invariant étale sheaf with transfers,
or a qfh, or h sheaf on the category of schemes over $X$, with torsion order
prime to $p$, the Suslin rigidity theorem [SV96, Theorem 4.5] implies that
$\mathcal{F}$ is a locally constant sheaf. Since we work over an algebraically
closed field, locally constant is the same as constant. Thus for any torsion
sheaf $\mathcal{G}$ with torsion order prime to $p$, its Suslin complex
$C^{*}(\mathcal{G})$ is isomorphic to the pull-back of a complex of locally
constant sheaves. Moreover if $\mathcal{F}$ is a constant étale sheaf, we have
isomorphisms ([SV96, Theorem 10.2, 10.7])

$H^{i}_{\text{\'{e}t}}(X,\mathcal{F})\cong
H^{i}_{\text{qfh}}(X,\mathcal{F^{\text{qfh}}})\cong
H^{i}_{\text{h}}(X,\mathcal{F^{\text{h}}}).$

Since we assume that $k$ is algebraically closed, $\text{Spec }k$ has no
higher cohomology for any sheaf in any of these three topologies. Therefore
for any complex of constant sheaves, we also have the isomorphism of
cohomologies for $X=\text{Spec }k$. In particular, above discussions apply to
the complex $C^{*}(Z_{\text{eq}}(X,r))\otimes\mathbb{Z}/N\mathbb{Z}$. We may
identify the cohomology of this complex.

######  Theorem 4.5.

Let $X$ be a quasi-projective variety defined over an algebraically closed
field $k$. Let $N$ be an integer, non-zero in the field. We have the following
isomorphisms.

$\displaystyle H^{i}_{\text{h}}(\text{Spec
}k,C^{*}(Z_{\text{eq}}^{\text{h}}(X,r)\otimes\mathbb{Z}/N\mathbb{Z}))\cong
H^{i}_{\text{qfh}}(\text{Spec
}k,C^{*}(Z_{\text{eq}}^{\text{qfh}}(X,r)\otimes\mathbb{Z}/N\mathbb{Z}))$
$\displaystyle\cong$ $\displaystyle H^{i}_{\text{\'{e}t}}(\text{Spec
}k,C^{*}(Z_{\text{eq}}(X,r)\otimes\mathbb{Z}/N\mathbb{Z}))\cong
H^{i}_{\text{Ab}}(C^{*}(Z_{\text{eq}}(X,r)\otimes\mathbb{Z}/N\mathbb{Z})(\text{Spec
}k))$ $\displaystyle\cong$ $\displaystyle
CH_{r}(X,-i,\mathbb{Z}/N\mathbb{Z}).$

###### Proof.

The first three cohomology groups are isomorphic as discussed above. They are
all equal to the cohomology of the complex
$C^{*}(Z_{\text{eq}}(X,r)\otimes\mathbb{Z}/N\mathbb{Z})(\text{Spec }k))$,
since this complex computes the cohomology group in the qfh and étale
topology.

Finally, under the hypothesis that resolution of singularities holds, Suslin
[Sus00, Theorem 3.2] proves that for any quasi-projective variety $X$, we have
an isomorphism

$H^{i}_{\text{Ab}}(C^{*}(Z_{\text{eq}}(X,r)\otimes\mathbb{Z}/N\mathbb{Z})(\text{Spec
}k))\cong CH_{r}(X,-i,\mathbb{Z}/N\mathbb{Z}).$

Using Gabber’s refinement of de Jong’s alteration theorem, Kelly [Kel17,
Theorem 5.6.4] removed the resolution of singularities hypothesis. ∎

######  Corollary 4.6.

Let $X$ be a quasi-projective variety defined over an algebraically closed
field $k$. Let $N$ be an integer, non-zero in the field. We have the following
isomorphisms.

$\displaystyle H^{i}_{\text{\'{e}t}}(\text{Spec
}k,C^{*}(Z_{\text{eq}}(X,0)\otimes\mathbb{Z}/N\mathbb{Z}))\cong
H^{i}_{\text{qfh}}(\text{Spec
}k,C^{*}(Z_{\text{eq}}(X,0)\otimes\mathbb{Z}/N\mathbb{Z}))$
$\displaystyle\cong$ $\displaystyle H^{i}_{\text{h}}(\text{Spec
}k,C^{*}(Z_{\text{eq}}(X,0)\otimes\mathbb{Z}/N\mathbb{Z}))\cong
H^{i}_{\text{Ab}}(C^{*}(Z_{\text{eq}}(X,0)\otimes\mathbb{Z}/N\mathbb{Z})(\text{Spec
}k))$ $\displaystyle\cong$ $\displaystyle
CH_{0}(X,-i,\mathbb{Z}/N\mathbb{Z})\cong
H^{2d-i}_{\text{\'{e}t}}(X,\mathbb{Z}/N\mathbb{Z}),$

where $d$ is the dimension of $X$. In particular, all the groups are finite.

###### Proof.

The last equality follows from [SV96, Corollary 7.8] and [Gei00, Theorem 3.5],
[Kel17, Theorem 5.6.1]. Clearly the étale cohomology group is finite. ∎

We use $A_{1}(X)$ to denote the group of one cycles in $X$ modulo algebraic
equivalence. For any abelian group $A$ and any integer $m$, we use $A[m]$ to
denote the group of $m$-torsions in $A$, and $A/m$ to denote the quotient
$A/mA$.

For any integer $N$ invertible in the field $k$, we have a homomorphism

$CH_{1}(X,1,\mathbb{Z}/N\mathbb{Z})\to CH_{1}(X,0,\mathbb{Z})[N]$

that comes from the long exact sequence of higher Chow groups with
$\mathbb{Z}$ and $\mathbb{Z}/N$ coeffiecients. Composing with the surjective
map

$CH_{1}(X,0,\mathbb{Z})[N]\to A_{1}(X)[N],$

we have a homomorphism

$CH_{1}(X,1,\mathbb{Z}/N\mathbb{Z})\to A_{1}(X)[N].$

Now we can state the counterpart of Theorem 3.8.

######  Theorem 4.7.

Let $X$ be a smooth projective variety defined over an algebraically closed
field $k$ of characteristic $p$. Fix an integer $N$ that is invertible in $k$.
For any class $[L]$ in the kernal of the map

$H_{h}^{-1}(\text{Spec
}k,C^{*}(Z_{1}^{h}(X))\otimes\mathbb{Z}/N\mathbb{Z})\cong
CH_{1}(X,1,\mathbb{Z}/N\mathbb{Z})\to A_{1}(X)[N],$

there is a projective algebraic set $Z$ and a family of one-dimensional nodal
curves over $Z$ such that the class $[L]$ is in the image of

$H_{h}^{-1}(\text{Spec }k,C^{*}(Z_{0}^{h}(Z))\otimes\mathbb{Z}/N\mathbb{Z})\to
H_{h}^{-1}(\text{Spec }k,C^{*}(Z_{1}^{h}(X))\otimes\mathbb{Z}/N\mathbb{Z})$

induced by this family of cycles. Assume furthermore that $X$ is separably
rationally connected in codimension $1$. We may take $Z$ to be normal.

######  Remark 4.8.

This result is a priori weaker than Theorem 3.8 over complex numbers. We have
a short exact sequence

$0\to L_{1}H_{3}(X)/N\to L_{1}H_{3}(X,\mathbb{Z}/N)\to L_{1}H_{2}(X)[N]\to 0.$

Since $L_{1}H_{3}(X,\mathbb{Z}/N)\cong CH_{1}(X,1,\mathbb{Z}/N)$ and
$L_{1}H_{2}(X)[N]\cong A_{1}(X)[N]$, Theorem 4.7 only says that classes in
$L_{1}H_{3}(X)/N$ comes from a smooth projective variety.

But if we know that $L_{1}H_{3}(X)$ is finitely generated, then we can find
the lift. Conjecturally, this group is isomorphism to $H_{3}(X,\mathbb{Z})$,
thus finitely generated.

The proof of Theorem 4.7 is analogous to that of Theorem 3.8. We first prove
the analogue of Lemma 3.9.

######  Lemma 4.9.

Let $X$ be a smooth projective variety defined over an algebraically closed
field $k$ of characteristic $p$. Assume that $X$ is either a separablly
rationally connected variety or a separably rationally connected fibration
over a curve. Fix an integer $N$ that is invertible in $k$. For any class
$[L]$ in

$H_{h}^{-1}(\text{Spec
}k,C^{*}(Z_{1}^{h}(X))\otimes\mathbb{Z}/N\mathbb{Z})\cong
CH_{1}(X,1,\mathbb{Z}/N\mathbb{Z}),$

there is a normal projective variety $U$, a family of equidimensional one
cycles $\gamma_{U}$ over $U$ and a morphism $f:\Delta^{1}\to U$ such that
$[L]$ is represented by $f^{*}\gamma_{U}$ over $\Delta^{1}$.

###### Proof.

We could translate the proof of Theorem 3.8 in the context of h-sheaves. But
here is an easier argument using Hilbert schemes.

The class $[L]$ is represented by a family of cycles
$\sum_{i}m_{i}\Gamma_{i},m_{i}\in\mathbb{Z}/N$ over $\Delta^{1}$, where
$\Gamma_{i}\subset\Delta^{1}\times X$ is an integral subvariety. Since
$\Delta^{1}$ is one dimensional, the projection $\Gamma_{i}\to\Delta^{1}$ is
flat. Thus we get a morphism $f_{i}$ from $\Delta^{1}$ to the Hilbert scheme.
The universal subscheme over the Hilbert scheme gives a family of cycles over
the Hilbert scheme. Therefore, we may take $U$ to be the normalization of of
products of irreducible components of the Hilbert scheme and $\gamma_{U}$ to
be the family of cycles (with appropriate multiplicity) coming from universal
subschemes. ∎

We will need the following observation later in the proof.

######  Lemma 4.10.

Let $T$ be a connected projective algebraic set over an algebraically closed
field $k$, and let $x,y$ be two points in $T$. Let $\mathcal{F}$ be a sheaf of
abelian groups in the qfh or h topology, or an étale sheaf with transfers. Fix
an integer $N$ invertible in $k$. Write $F_{x}$ (resp. $F_{y}$) the
restriction of $F\in\mathcal{F}\otimes\mathbb{Z}/N\mathbb{Z}(T)$ to $x$ (resp.
$y$). Then $F_{x}=F_{y}$ in $H^{0}(\text{Spec
}k,C^{*}(\mathcal{F})\otimes\mathbb{Z}/N\mathbb{Z})$, where the cohomology in
taken in the étale topology, qfh topology or h topology.

###### Proof.

Elements in $\mathcal{F}\otimes\mathbb{Z}/N\mathbb{Z}(T)$ induces a unique
morphism

$Z_{0}(T)\otimes\mathbb{Z}/N\to\mathcal{F}\otimes\mathbb{Z}/N\mathbb{Z}.$

If $\mathcal{F}$ is a sheaf with transfers, this is the Yoneda Lemma. If
$\mathcal{F}$ is a qfh sheaf or h sheaf, this follows from the fact that the
qfh sheafification of $Z_{0}(T)[\frac{1}{p}]$ is the free sheaf
$\mathbb{Z}[\frac{1}{p}][T]$ generated by the presheaf of sets
$\text{Hom}(\cdot,T)$ ([SV96, Theorem 6.7]). Thus the class
$[\mathcal{F}_{x}]$ (resp. $[\mathcal{F}_{y}]$) is the image of $[x]$ (resp.
$[y]$) under the map

$H^{0}(\text{Spec }k,C^{*}(Z_{0}(T))\otimes\mathbb{Z}/N\mathbb{Z})\to
H^{0}(\text{Spec }k,C^{*}(\mathcal{F})\otimes\mathbb{Z}/N\mathbb{Z}).$

So it suffices to show that $[x]=[y]$ in $H^{0}(\text{Spec
}k,Z_{0}(T)\otimes\mathbb{Z}/N\mathbb{Z})$. But the latter cohomology group is
$CH_{0}(T)\otimes\mathbb{Z}/N\mathbb{Z}\cong\mathbb{Z}/N\mathbb{Z}$ by Lemma
4.6 and the isomorphism is given by the degree map. Any two points $x,y$ give
the same class in $H^{0}(\text{Spec
}k,Z_{0}(T)\otimes\mathbb{Z}/N\mathbb{Z})$. ∎

Now we begin the proof Theorem 4.7.

###### Proof of Theorem 4.7.

Given a normal projective variety $S$ parameterizing a family of one
dimensional cycles of $X$, there is an induced morphism of h-sheaves:

$Z_{0}^{h}(S)\to Z_{1}^{h}(X).$

We denote by $I(S)$ (resp. $K(S)$) the image h-sheaf (resp. kernal h-sheaf) of
this map.

By Lemma 4.9, there is a normal projective variety $U$, a family of
equidimensional one cycles $\gamma_{U}$ over $U$ and a morphism
$f:\Delta^{1}\to U$ such that $[L]$ is represented by $f^{*}\gamma_{U}$ over
$\Delta^{1}$.

Denote by $\gamma_{0},\gamma_{1}$ the restriction of the family of cycles
$\gamma_{U}$ over $U$ to $0,1\in\Delta^{1}$. Then
$\gamma_{0}-\gamma_{1}=N(\gamma_{0,1})$ for some cycle $\gamma_{0,1}$. The
image of $[L]$ in $CH_{1}(X,0,\mathbb{Z})[N]$ and $A_{1}(X)[N]$ is the class
of $\gamma_{0,1}$.

If $\gamma_{0,1}$ is zero in $A_{1}(X)[N]$, that is, if $\gamma_{0,1}$ is
algebraically equivalent to $0$, then by Proposition 2.4, there is a smooth
projective curve $D$ with a family of cycles $\gamma_{D}$ and two points
$d,d^{\prime}$ such that $\Gamma_{d}$ is $0$ and $\Gamma_{d^{\prime}}$ is
$\gamma_{0,1}$.

Consider the product $U\times D$. We have a family of cycles
$\gamma=\pi_{U}^{*}\gamma_{U}+N\pi_{D}^{*}\gamma_{D}$.

There are three points in $S=U\times D$,

$x=(f(0),d),y=(f(1),d),z=(f(1),d^{\prime})$

such that

  1. (1)

$\gamma_{x}=\gamma_{z}=\gamma_{0}$.

  2. (2)

There is a curve $C$ containing $y,z$ such that for every point $c\in C$, the
cycle $\gamma_{c}$ equals $\gamma_{1}$ in
$Z_{1}(X)\otimes\mathbb{Z}/N(\text{Spec }k)$.

As in the proof of Theorem 3.8, we apply the second part of Theorem 2.6 to
find a normal projective variety $V$, a projective algebraic set $Y$ with a
surjective projective morphism $V\to S$ and an embedding $V\to Y$, and
liftings $x_{V},y_{V},z_{V}$ of the points $x,y,z$ such that

  1. (1)

The two points $x_{V}$ and $z_{V}$ are connected by a chain of curves in $Y$
parameterizing constant cycles.

  2. (2)

The two points $y_{V}$ and $z_{V}$ are connected by a curve $D_{V}$ in $V$
parameterizing cycles divisible by $N$.

Denote by $K$ the kernal of the morphism between h sheaves

$Z_{0}(V)\to Z_{0}(S).$

Here $V\to S$ is proper and surjective. So the above morphism of h sheaves is
surjective. Then we have an isomorphism of h sub-sheaves of $Z_{1}(X)$

$I(V)\cong I(S).$

It follows that we have a short exact sequence of h sheaves:

$0\to K\to K(V)\to K(S)\to 0.$

We have commutative diagrams:

$\begin{CD}H^{-1}_{h}(C^{*}(Z_{0}(Y))/N)@>{}>{}>H^{-1}_{h}(C^{*}(I(Y)/N))@>{}>{}>H^{0}_{h}(C^{*}(K(Y))/N)\\\
@A{}A{}A@A{}A{}A@A{}A{}A\\\
H^{-1}_{h}(C^{*}(Z_{0}(V))/N)@>{}>{}>H^{-1}_{h}(C^{*}(I(V))/N)@>{}>{}>H^{0}_{h}(C^{*}(K(V))/N)\\\
@V{}V{}V@V{}V{=}V@V{}V{}V\\\
H^{-1}_{h}(C^{*}(Z_{0}(S))/N)@>{}>{}>H^{-1}_{h}(C^{*}(I(S))/N)@>{}>{}>H^{0}_{h}(C^{*}(K(S))/N)\\\
\end{CD}$

The obstruction of lifting the class $[L]$ in $H^{-1}_{h}(\text{Spec
}k,C^{*}(I(V))\otimes\mathbb{Z}/N\mathbb{Z})$ is in $H^{0}_{h}(\text{Spec
}k,C^{*}(K(V))$ and maps to $[x-y]$ in $H^{0}_{h}(\text{Spec
}k,C^{*}(K(S))\otimes\mathbb{Z}/N\mathbb{Z})$. The class $[x_{T}-y_{T}]$
differs from the obstruction class by an element in

$H^{0}_{h}(\text{Spec }k,C^{*}(K)\otimes\mathbb{Z}/N\mathbb{Z}).$

Given the morphism $V\to S$, we have a long exact sequence

$\displaystyle\ldots\to$ $\displaystyle H^{-1}_{h}(\text{Spec
}k,C^{*}(Z_{0}(V))\otimes\mathbb{Z}/N\mathbb{Z})\to H^{-1}_{h}(\text{Spec
}k,C^{*}(Z_{0}(S))\otimes\mathbb{Z}/N\mathbb{Z})$ $\displaystyle\to$
$\displaystyle H^{0}_{h}(\text{Spec
}k,C^{*}(K)\otimes\mathbb{Z}/N\mathbb{Z})\to H^{0}_{h}(\text{Spec
}k,C^{*}(Z_{0}(T))\otimes\mathbb{Z}/N\mathbb{Z})\to\ldots$

Therefore $H^{0}_{h}(\text{Spec }k,C^{*}(K)\otimes\mathbb{Z}/N\mathbb{Z})$ is
finitely generated by Corollary 4.6. By Lemma 4.12, any class in
$H^{0}_{h}(\text{Spec }k,C^{*}(K)\otimes\mathbb{Z}/N\mathbb{Z})$ is equivalent
to the class of the form $[a-b]$, where $a,b$ are points in the fiber over a
general point in $S$.

The class $[L]$ maps to $H^{-1}_{h}(\text{Spec
}k,C^{*}(I(Y))\otimes\mathbb{Z}/N\mathbb{Z})$, with obstruction class the
push-forward of $[x_{V}-y_{V}]$ modulo classes in $H^{0}_{h}(\text{Spec
}k,C^{*}(K)\otimes\mathbb{Z}/N\mathbb{Z})$.

By the existence of the families of constant cycles in the second part of
Theorem 2.6, and by Lemma 4.10, we have

  1. (1)

The composition

$H^{0}_{h}(\text{Spec }k,C^{*}(K)/N)\to H^{0}_{h}(\text{Spec
}k,C^{*}(K(V))/N)\to H^{0}_{h}(\text{Spec }k,C^{*}(K(Y))/N)$

is the zero map.

  2. (2)

The push-forward of the class $[x_{V}-y_{V}]$ vanishes in
$H^{0}_{h}(\text{Spec }k,C^{*}(K(Y))\otimes\mathbb{Z}/N\mathbb{Z})$ (by
applying Lemma 4.10 to $[x_{V}-y_{V}]$ and $[x_{V}-x_{V}]=0$).

Thus the class $[L]$ in $H^{-1}_{h}(\text{Spec }k,C^{*}(I(Y))/N)$ comes from
$H^{-1}_{h}(\text{Spec }k,C^{*}(Z_{0}(Y))/N)$.

Finally, if $X$ is separably rationally connected in codimension $1$, we may
take $Y$ to be normal by Theorem 2.6. We use Gabber’s refinement of de Jong’s
alteration to find a smooth projective variety $Z$ and a projective alteration
$Z\to Y$ whose degree is relatively prime to $N$. Then

$CH_{0}(Z,1,\mathbb{Z}/N\mathbb{Z})\to CH_{0}(Y,1,\mathbb{Z}/N\mathbb{Z})$

is surjective by Lemma 4.13. Pulling back the families of cycles over $Y$
gives a family of cycles over $Z$. Then the theorem follows from the following
commutative diagram

$\begin{CD}CH_{0}(Z,1,\mathbb{Z}/N)@>{}>{}>CH_{0}(Y,1,\mathbb{Z}/N)@>{\Gamma_{*}}>{}>CH_{1}(X,1,\mathbb{Z}/N)\\\
@V{\cong}V{}V@V{\cong}V{}V@V{}V{}V\\\
H_{1}(Z,\mathbb{Z}/N)@>{}>{}>H_{1}(Y,\mathbb{Z}/N)@>{\Gamma_{*}}>{}>H_{3}(X,\mathbb{Z}/N).\end{CD}$

∎

The lemmas used in the proof are the following.

######  Lemma 4.11.

Let $X\to Y$ be a flat and finite morphism defined over an algebraically
closed field $k$, where $Y$ is a normal variety (but $X$ is not necessarily
normal). Let $N$ be an integer invertible over $k$. Denote by $K$ the kernal h
sheaf of $Z_{0}(X)\to Z_{0}(Y)$. Then for any chosen general point in $Y$,
$H_{h}^{0}(\text{Spec }k,C^{*}(K)\otimes\mathbb{Z}/N\mathbb{Z})$ is generated
by class of the form $[t_{1}-t_{2}]$, for $t_{1},t_{2}$ in the fiber of this
chosen general point in $Y$.

###### Proof.

Let $x_{1},x_{2}$ be two points in the fiber over $y\in Y$. Clearly
$H_{h}^{0}(\text{Spec }k,C^{*}(K)\otimes\mathbb{Z}/N\mathbb{Z})$ is generated
by class of the form $[x_{1}-x_{2}]$ for all the pairs of points with the same
image in $Y$. We will show that for any chosen general point $t\in Y$, the
class $[x_{1}-x_{2}]$ is equivalent to a class $[t_{1}-t_{2}]$ for some points
$t_{1},t_{2}$ in the fiber over $t$. Consider the correspondence
$X\times_{Y}X\subset X\times X$. Since $X\to Y$ is assumed to be flat,
$X\times_{Y}X\to X$ is flat. We take an irreducible component $D$ containing
$(x_{1},x_{2})$, which dominates (and thus surjects onto) $X$. There are two
points $x_{D},t_{D}$ in $D$ such that the following conditions are satisfied.

  1. (1)

There is a surjective morphism $f:D\to Y$ that maps $x_{D}$ (resp. $t_{D}$) to
$y$ (resp. $t$).

  2. (2)

There are two morphisms $f_{1},f_{2}:D\to X$ such that
$f_{1}(x_{D})=x_{1},f_{2}(x_{D})=x_{2}$.

  3. (3)

The composition of $f_{1},f_{2}$ with the morphism $q:X\to Y$ gives the
morphism $f:D\to Y$.

Then by Lemma 4.10, the class $[x_{1}-x_{2}]$ is the same as
$[f_{1}(t_{D})-f_{2}(t_{D})]$. ∎

######  Lemma 4.12.

Let $p:X\to Y$ be a generically finite surjective morphism between normal
projective varieties over an algebraically closed field $k$. Let $N$ be an
integer invertible over $k$. Denote by $K$ the kernal h sheaf of $Z_{0}(X)\to
Z_{0}(Y)$. Then the cohomology group $H_{h}^{0}(\text{Spec
}k,C^{*}(K)\otimes\mathbb{Z}/N\mathbb{Z})$ is generated by class of the form
$[t_{1}-t_{2}]$, for $t_{1},t_{2}$ in the fiber of any chosen general point in
$Y$.

###### Proof.

There is a birational projective variety morphism $Y^{\prime}\xrightarrow{q}Y$
such that the strict transform $X^{\prime}$ of $X$ is flat over $Y^{\prime}$.
That is, we have a commutative diagram:

$\begin{CD}X^{\prime}@>{q^{\prime}}>{}>X\\\ @V{p^{\prime}}V{}V@V{p}V{}V\\\
Y^{\prime}@>{q}>{}>Y\\\ \end{CD}$

We denote by $K(p)$ etc. to denote the kernal h sheaf of $Z_{0}(X)\to
Z_{0}(Y)$ etc.. There is a commuative diagram of short exact sequences of h
sheaves

$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>K(q^{\prime})@>{}>{}>Z_{0}(X^{\prime})@>{q_{*}^{\prime}}>{}>Z_{0}(X)@>{}>{}>0\\\
@V{}V{}V@V{}V{}V@V{p_{*}^{\prime}}V{}V@V{p_{*}}V{}V@V{}V{}V\\\
0@>{}>{}>K(q)@>{}>{}>Z_{0}(Y^{\prime})@>{q_{*}}>{}>Z_{0}(Y)@>{}>{}>0,\end{CD}$

which also gives commuatative diagrams after tensoring with
$\mathbb{Z}/N\mathbb{Z}$. Then we have long exact sequences:

$CH_{1}(X,1,\mathbb{Z}/N\mathbb{Z})\to CH_{1}(Y,1,\mathbb{Z}/N\mathbb{Z})\to
H_{h}^{0}(\text{Spec }k,C^{*}(K(p))\otimes\mathbb{Z}/N\mathbb{Z})\ldots,$
$CH_{1}(X^{\prime},1,\mathbb{Z}/N\mathbb{Z})\to
CH_{1}(Y^{\prime},1,\mathbb{Z}/N\mathbb{Z})\to H_{h}^{0}(\text{Spec
}k,C^{*}(K(p^{\prime}))\otimes\mathbb{Z}/N\mathbb{Z})\ldots.$

The cohomology group $H_{h}^{0}(\text{Spec
}k,C^{*}(K(p))\otimes\mathbb{Z}/N\mathbb{Z})$ (resp. $H_{h}^{0}(\text{Spec
}k,C^{*}(K(p^{\prime}))\otimes\mathbb{Z}/N\mathbb{Z})$ ) is generated by
cycles of the form $y_{1}-y_{2}$ for $y_{1},y_{2}$ in the same fiber. So it
suffices to show that such cycles are zero.

We first show that

$CH_{0}(Y^{\prime},1,\mathbb{Z}/N\mathbb{Z})\to
CH_{0}(Y,1,\mathbb{Z}/N\mathbb{Z})$

is surjective. This is because $Y^{\prime}\to Y$ has connected fibers. So for
any two points in the same fiber, by Lemma 4.10, the class of the difference
is zero. Since

$CH_{0}(Y^{\prime},0,\mathbb{Z}/N\mathbb{Z})\to
CH_{0}(Y,0,\mathbb{Z}/N\mathbb{Z})$

is an isomorphism, we know that $H_{h}^{0}(\text{Spec
}k,C^{*}(K(q))\otimes\mathbb{Z}/N\mathbb{Z})$ vanishes.

By the same argument, $H_{h}^{0}(\text{Spec
}k,C^{*}(K(q^{\prime}))\otimes\mathbb{Z}/N\mathbb{Z})$ vanishes. Then a simple
diagram chasing shows that

$H_{h}^{0}(\text{Spec }k,C^{*}(K(p^{\prime}))\otimes\mathbb{Z}/N\mathbb{Z})\to
H_{h}^{0}(\text{Spec }k,C^{*}(K(p))\otimes\mathbb{Z}/N\mathbb{Z})$

is surjective.

Thus the statements follow form Lemma 4.11. ∎

######  Lemma 4.13.

Let $p:X\to Y$ be a generically finite morphism between normal projective
varieties over an algebraically closed field $k$. Let $N$ be an integer
invertible over $k$. Assume that $\deg p$ is relatively prime to $N$. Then we
have a surjection

$CH_{0}(X,1,\mathbb{Z}/N\mathbb{Z})\to CH_{0}(Y,1,\mathbb{Z}/N\mathbb{Z}).$

###### Proof.

By Lemma 4.12, and the long exact sequence

$\displaystyle CH_{0}(X,1,\mathbb{Z}/N\mathbb{Z})\to
CH_{0}(Y,1,\mathbb{Z}/N\mathbb{Z})\to H^{0}_{h}(\text{Spec }k,C^{*}(K)/N)$
$\displaystyle\to$ $\displaystyle
CH_{0}(X,0,\mathbb{Z}/N)\xrightarrow{\cong}CH_{0}(Y,0,\mathbb{Z}/N),$

it suffices to show that for a general point $y\in Y$ and any two points
$x_{1},x_{2}$ in the fiber of $y$, the class $[x_{1}-x_{2}]$ is zero in
$H^{0}_{h}(\text{Spec }k,C^{*}(K)/N)$.

By the Bertini theorem for étale fundamental groups, there is a general
complete intersection curve $H$ such that the inverse image $H^{\prime}$ in
$Y^{\prime}$ is irreducible. For $H$ general, the morphism $H^{\prime}\to H$
is flat and finite of degree prime to $N$. Thus

$\displaystyle CH_{0}(H^{\prime},1,\mathbb{Z}/N\mathbb{Z})\to
CH_{0}(H,1,\mathbb{Z}/N\mathbb{Z})\to H_{h}^{0}(\text{Spec
}k,C^{*}(K_{H})\otimes\mathbb{Z}/N\mathbb{Z})$ $\displaystyle\to
CH_{0}(H^{\prime},\mathbb{Z}/N\mathbb{Z})\to
CH_{0}(H,\mathbb{Z}/N\mathbb{Z}),$

where $K_{H}$ is the kernal h sheaf of $Z_{0}(H^{\prime})\to Z_{0}(H)$. The
map

$CH_{0}(H^{\prime},\mathbb{Z}/N\mathbb{Z})\to
CH_{0}(H,\mathbb{Z}/N\mathbb{Z})$

is an isomorphism. On the other hand, since $p:H^{\prime}\to H$ is flat and
finite, we have pull-back and push-forward on all the higher Chow groups. The
composition of pull-back and push-forward

$CH_{0}(H,1,\mathbb{Z}/N\mathbb{Z})\xrightarrow{p^{*}}CH_{0}(H^{\prime},1,\mathbb{Z}/N\mathbb{Z})\xrightarrow{p_{*}}CH_{0}(H,1,\mathbb{Z}/N\mathbb{Z})$

is multiplcation by $\deg p$. Since the degree of the map is relatively prime
to $N$,

$CH_{0}(H^{\prime},1,\mathbb{Z}/N\mathbb{Z})\xrightarrow{q_{*}}CH_{0}(H,1,\mathbb{Z}/N\mathbb{Z})$

is surjective. Thus for any two points $t_{1},t_{2}$ over a general point
$t\in Y$, the class $[t_{1}-t_{2}]$ vanishes in $H_{h}^{0}(\text{Spec
}k,C^{*}(K_{H})\otimes\mathbb{Z}/N\mathbb{Z})$. So does its push-forward in
$H_{h}^{0}(\text{Spec }k,C^{*}(K)\otimes\mathbb{Z}/N\mathbb{Z})$. ∎

Fix a prime number $\ell$ different from the characteristic of $k$. In the
following theorem, we omit all the Tate twists for simplicity of notations.

######  Theorem 4.14.

Let $X$ be a smooth projective variety defined over an algebraically closed
field, which is separably rationally connected in codimension $1$. There is a
smooth projective curve $C$ with a family of $1$-dimensional cycles
$\Gamma\subset C\times X$ such that

$\Gamma_{*}:H_{1}^{\text{BM}}(C,\mathbb{Z}_{\ell})\to
H_{3}^{\text{BM}}(X,\mathbb{Z}_{\ell})$

surjects onto $N^{1}H_{3}(X,\mathbb{Z}_{\ell})$.

###### Proof.

In the following, we use Borel-Moore homology. For simplicity of notations, we
only write them as $H_{1},H_{3}$. Let $NH_{3}(X,\mathbb{Z}/\ell^{n})$ be the
coniveau filtration on the homology $N^{1}H_{3}(X,\mathbb{Z}/\ell^{n})$.
Denote by $\tilde{N}H_{3}(X,\mathbb{Z}/\ell^{n})$ the strong coniveau
filtration $\tilde{N}^{1}H_{3}(X,\mathbb{Z}/\ell^{n})$.

For a smooth projective variety $Y$, we have
$H_{1}(Y,\mathbb{Z}_{\ell})/\ell^{n}\cong H_{1}(Y,\mathbb{Z}/\ell^{n})$, since
$H_{0}(Y,\mathbb{Z}_{\ell})\cong\mathbb{Z}_{\ell}$ is torsion free. Therefore,
$\tilde{N}H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n}\to\tilde{N}H_{3}(X,\mathbb{Z}/\ell^{n})$
is surjective.

We have a commutative diagram

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
73.91812pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\oplus_{(S,\Gamma_{S})}CH_{0}(S,1,\mathbb{Z}/\ell^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
57.49832pt\raise 6.075pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.70833pt\hbox{$\scriptstyle{\oplus\Gamma_{S*}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 97.91812pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
97.91812pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{CH_{1}(X,1,\mathbb{Z}/\ell^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
197.08075pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
135.49944pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
197.08075pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{A_{1}(X)[\ell^{n}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\to
0}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
226.58502pt\raise-30.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0\to
H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n}\cap
NH_{3}(X,\mathbb{Z}/\ell^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
73.91814pt\raise-41.15pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
102.0091pt\raise-41.15pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
102.0091pt\raise-41.15pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{NH_{3}(X,\mathbb{Z}/\ell^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
197.32379pt\raise-41.15pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
197.32379pt\raise-41.15pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H_{2}(X,\mathbb{Z}_{\ell})[\ell^{n}]}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where the direct sum is taken over families of equidimensional one cycles over
smooth projective varieties.

By Theorem 4.7, the upper row is exact. The lower row is also exact, since it
comes from

$0\to H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n}\to H_{3}(X,\mathbb{Z}/\ell^{n})\to
H_{2}(X,\mathbb{Z}_{\ell})[\ell^{n}]\to 0.$

The vertical maps are cycle class maps.

The middle vertical map

$CH_{1}(X,1,\mathbb{Z}/\ell^{n})\to NH_{3}(X,\mathbb{Z}/\ell^{n})$

is surjective, since for any surface $\Sigma$, not necessarily smooth, we have
a surjection

$CH_{1}(\Sigma,1,\mathbb{Z}/\ell^{n})\to H_{3}(\Sigma,\mathbb{Z}/\ell^{n}).$

When this surface is smooth, it is a consequence of the Bloch-Kato conjecture.
The general case can be proven using the localization sequence for higher Chow
groups and Borel-Moore homology.

The left vertical arrow is the direct sum of the composition

$CH_{0}(S,1,\mathbb{Z}/\ell^{n})\to H_{1}(S,\mathbb{Z}/\ell^{n})\cong
H_{1}(S,\mathbb{Z}_{\ell})/\ell^{n}\xrightarrow{\Gamma_{S*}}H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n},$

Since the cycle class map induces an isomorphism
$CH_{0}(S,1,\mathbb{Z}/\ell^{n})\cong H_{1}(S,\mathbb{Z}/\ell^{n})\cong
H_{1}(S,\mathbb{Z}_{\ell})/\ell^{n}$, the left vertical arrow has the same
cokernal as

$\tilde{N}H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n}\to
H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n}\cap NH_{3}(X,\mathbb{Z}/\ell^{n}).$

By the snake lemma, this cokernal is isomorphic to the cokernal $C_{n}$ of

$\text{Ker}(CH_{1}(X,1,\mathbb{Z}/\ell^{n})\to
H_{3}(X,\mathbb{Z}/\ell^{n}))\to\text{Ker}(A_{1}[\ell^{n}]\to
H_{2}(X,\mathbb{Z}_{\ell})[\ell^{n}]).$

The connecting maps $C_{n+m}\to C_{n}$ are multiplication by $\ell^{m}$.
Therefore the inverse limit $\lim\limits_{\xleftarrow{}}C_{n}$ is torsion
free.

We have a factorization

$\tilde{N}H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n}\to{N}H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n}\to
H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n}\cap NH_{3}(X,\mathbb{Z}/\ell^{n})\subset
H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n}.$

Taking inverse limit, we get

$\tilde{N}H_{3}(X,\mathbb{Z}_{\ell})\to{N}H_{3}(X,\mathbb{Z}_{\ell})\to\lim\limits_{\xleftarrow[n]{}}H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n}\cap
NH_{3}(X,\mathbb{Z}/\ell^{n})\subset H_{3}(X,\mathbb{Z}_{\ell}).$

Therefore the map

${N}H_{3}(X,\mathbb{Z}_{\ell})/\to\lim\limits_{\xleftarrow[n]{}}H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n}\cap
NH_{3}(X,\mathbb{Z}/\ell^{n})$

is injective. Since the cokernal of
${N}H_{3}(X,\mathbb{Z}_{\ell})/\to\lim\limits_{\xleftarrow[n]{}}H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n}\cap
NH_{3}(X,\mathbb{Z}/\ell^{n})$ is torsion free, so is the cokernal of
$\tilde{N}H_{3}\to NH_{3}$. On the other hand, we know that the cokernal is
torsion. So it has to be zero. That is, the strong coniveau filtration
coincide with the coniveau filtration.

Therefore there is a smooth projective variety $Y$ and a family of cycles
$\Gamma_{Y}$ such that the induced map

$\Gamma_{Y*}:H_{1}(Y,\mathbb{Z}_{\ell})\to{N}^{1}H_{3}(X,\mathbb{Z}_{\ell})$

is surjective. By taking hyperplane sections in $Y$, we may find a smooth
projective curve $C$ with a family of cycles $\Gamma$ such that

$\Gamma_{*}:H_{1}(C,\mathbb{Z}_{\ell})\to{N}^{1}H_{3}(X,\mathbb{Z}_{\ell})$

is surjective.

Finally, for later use, we note that in the proof, we also prove that for $X$
SRC in codimension $1$, the maps

(5) $\tilde{N}^{1}H_{3}(X,\mathbb{Z}_{\ell})\to
N^{1}H_{3}(X,\mathbb{Z}_{\ell})\to\lim\limits_{\xleftarrow[n]{}}H_{3}(X,\mathbb{Z}_{\ell})/\ell^{n}\cap
NH_{3}(X,\mathbb{Z}/\ell^{n})$

are isomorphisms. The first isomorphism is already shown above. We have
already shown that the second map is injective. The cokernal is torsion since
the cokernal of $NH_{3}\to H_{3}$ is torsion. By the first isomorphism and the
fact that the composition has torsion free cokernal, the cokernal of the
second map is also torsion free, and thus zero. ∎

######  Remark 4.15.

When $X$ is only smooth projective, one can prove that the filtration
$N_{1,\text{cyl},\text{st}}H_{3}(X,\mathbb{Z}_{\ell})$ is the same as
$N^{1}H_{3}(X,\mathbb{Z}_{\ell})$ by the same argument.

######  Theorem 4.16.

Let $X$ be a smooth projective $3$-fold over an algebraically closed field.
Assume that $X$ is separably rationally connected in codimension $1$. Then all
the filtrations on $H^{3}(X,\mathbb{Z}_{\ell})$ introduced in Definition 1.18
equal the whole cohomology group:

$\tilde{N}_{1,\text{cyl},\text{eq}}H^{3}(X,\mathbb{Z}_{\ell})=\tilde{N}_{1,\text{cyl}}H^{3}(X,\mathbb{Z}_{\ell})=\tilde{N}^{1}H^{3}(X,\mathbb{Z}_{\ell})=N^{1}H^{3}(X,\mathbb{Z}_{\ell})=H^{3}(X,\mathbb{Z}_{\ell}).$

######  Corollary 4.17.

Let $X$ be a smooth projective variety of dimension $d$ defined over a finite
field $\mathbb{F}_{q}$, that is separably rationally connected in codimension
$1$. Assume one of the followings

  1. (1)

$N^{1}H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}_{\ell}(d-1))=H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}_{\ell}(d-1))$.

  2. (2)

The cycle class map

$cl:\lim\limits_{\xleftarrow[n]{}}CH_{1}(\bar{X},1,\mathbb{Z}/\ell^{n})\to
H^{2d-3}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(d-1))$

is surjective.

Then every class in
$H^{1}(\mathbb{F}_{q},H^{3}(\bar{X},\mathbb{Z}_{\ell}(d-1)))$ is the class of
an algebraic cycle defined over $\mathbb{F}_{q}$. In particular, this holds if
$X$ has dimension $3$.

###### Proof.

We first show that the surjectivity of the cycle class map

$cl:\lim\limits_{\xleftarrow[n]{}}CH_{1}(\bar{X},1,\mathbb{Z}/\ell^{n})\to
H^{2d-3}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(d-1))$

implies that
$N^{1}H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}_{\ell}(d-1))=H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}_{\ell}(d-1))$.
In fact, we have

$\displaystyle\lim\limits_{\xleftarrow[n]{}}CH_{1}(\bar{X},1,\mathbb{Z}/\ell^{n})\to\lim\limits_{\xleftarrow[n]{}}N^{1}H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}/\ell^{n}(d-1))$
$\displaystyle\to$
$\displaystyle\lim\limits_{\xleftarrow[n]{}}H^{2d-3}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}/\ell^{n}(d-1))=H^{2d-3}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(d-1)).$

Therefore

$\lim\limits_{\xleftarrow[n]{}}N^{1}H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}/\ell^{n}(d-1))\to\lim\limits_{\xleftarrow[n]{}}H^{2d-3}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}/\ell^{n}(d-1))=H^{2d-3}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(d-1))$

is surjective. On the other hand, since $N^{1}H_{3}(X,\mathbb{Z}/\ell^{n})$ is
a subgroup of $H_{3}(X,\mathbb{Z}/\ell^{n})$, the inverse limit is injective,
hence an isomorphism. We have an exact sequence

$\displaystyle
0\to\lim\limits_{\xleftarrow[n]{}}H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}_{\ell}(d-1))/\ell^{n}\cap
N^{1}H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}/\ell^{n}(d-1))$
$\displaystyle\xrightarrow{\phi}$
$\displaystyle\lim\limits_{\xleftarrow[n]{}}N^{1}H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}/\ell^{n}(d-1))\to\lim\limits_{\xleftarrow[n]{}}H^{2d-2}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(d-1))[\ell^{n}],$

where the first inverse limit is
$N^{1}H^{2d-3}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(d-1))$ by (5), and
the last inverse limit is torsion free. Since the quotient of
$H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}_{\ell}(d-1))/N^{1}H_{\text{\'{e}t}}^{2d-3}(\bar{X},\mathbb{Z}_{\ell}(d-1))$
is torsion for separably rationally connected varieties or separably
rationally connected fibrations over a curve, we know that $\phi$ is an
isomorphism and thus $N^{1}H^{2d-3}_{\text{\'{e}t}}(X,\mathbb{Z}_{\ell})\to
H^{2d-3}_{\text{\'{e}t}}(X,\mathbb{Z}_{\ell})$ is an isomorphism.

Therefore, by Theorem 4.14, there is a smooth projective curve $C$ defined
over $\bar{\mathbb{F}}_{q}$ with a family of one-dimensional cycles
$\Gamma\subset C\times\bar{X}$ such that

$\Gamma_{*}:H^{1}_{\text{\'{e}t}}(C,\mathbb{Z}_{\ell}(1))\to
H^{2d-3}_{\text{\'{e}t}}(\bar{X},\mathbb{Z}_{\ell}(d-1))$

is surjective. Then this corollary follows from [SS22, Proposition 7.6], the
statement of which is recalled in Theorem 1.31. ∎

##  5\. Integral Tate conjecture and local-global principle for zero cycles

Let $X$ be a smooth projective geometrically irreducible variety of dimension
$d$ defined over a finite field $\mathbb{F}$. We have the cycle class maps:

(6) $CH_{1}(X)\otimes\mathbb{Z}_{\ell}\to H^{2d-2}(X,\mathbb{Z}_{\ell}(d-1)),$

Recall the integral Tate conjecture asks the following question.

######  Question 5.1.

For which smooth projective variety $X$ defined over $\mathbb{F}$, and which
$r$, is the cycle class map (6) surjective?

We mention another closely related question.

######  Question 5.2.

Let $X$ be a smooth projecitve variety defined over a henselian local field
with finite residue field. Is the cycle class map

$CH_{0}(X)\hat{\otimes}\mathbb{Z}_{\ell}\to H^{2d}(X,\mathbb{Z}_{\ell}(d))$

injective? Here $\ell$ is invertible in the residue field.

######  Remark 5.3.

Question 5.2 has a positive answer if $X$ is a geometrically rational surface,
and has a regular model with SNC central fiber ([EW16, Theorem 3.1] in general
and [Sai91, Theorem A] for the case of $p$-adic fields). In this case, the
proof in [EW16] also shows that the closed fiber also satisfies a version of
the integral Tate conjecture. For $X$ defined over a Laurant field
$\mathbb{F}_{q}\left(\leavevmode\hbox{\set@color$\left(\leavevmode\hbox{\set@color$t$}\right)$}\right)$,
a regular model with SNC central fiber always exists since we have resolution
of singularities for $3$-folds.

If Question 5.2 has a positivie answer for the generic fiber $X$, then
Conjectures 1.1 and 1.3 are equivalent for $X$.

######  Remark 5.4.

We also note that the results in [Tia20] suggest that Question (5.2) should be
true for separably rationally connected varieties, provided that the
characteristic $p$ analogues of the conjecture $\textbf{R}(n,3)$ about Kato
homology in loc. cit. is true, and that we have the minimal model program
established in positive and mixed characteristic.

As discussed in Theorem 1.10 and the remark that follows this theorem in the
introduction, various types of integral Tate conjectures would imply various
versions of Colliot-Thélène’s conjectures 1.1, 1.2.

We can deduce Theorem 1.13 from Theorem 1.35 and Corollary 4.17.

###### Proof of Theorem 1.13.

Recall that $G=\text{Gal}(\bar{\mathbb{F}}_{q}/\mathbb{F}_{q})$ is the
absolute Galois group. By Theorem 1.35, we have an isomorphism

$A_{1}(\mathcal{X})\cong A_{1}(\overline{\mathcal{X}})^{G}.$

Under the assumptions (A), (B) of Theorem 1.13, we know that there is an
isomorphism of $\text{Gal}(\bar{\mathbb{F}}_{q}/\mathbb{F}_{q})$-modules:

$A_{1}(\overline{\mathcal{X}})\otimes\mathbb{Z}_{\ell}\cong
H^{2d}(\overline{\mathcal{X}},\mathbb{Z}_{\ell}(d)).$

Note that $G$ is generated by the Frobenius $F$ and
$A_{1}(\overline{\mathcal{X}})^{G}$ is the kernal of $F^{*}-\text{id}$. Since
$\mathbb{Z}_{\ell}$ is a flat $\mathbb{Z}$-module, we have
$A_{1}(\overline{\mathcal{X}})^{G}\otimes\mathbb{Z}_{\ell}$ is the kernal of

$(F^{*}-\text{id})\otimes\text{id}_{\mathbb{Z}_{\ell}}:A_{1}(\overline{\mathcal{X}})\otimes\mathbb{Z}_{\ell}\to
A_{1}(\overline{\mathcal{X}})\otimes\mathbb{Z}_{\ell}.$

That is,

$A_{1}(\overline{\mathcal{X}})^{G}\otimes\mathbb{Z}_{\ell}\cong(A_{1}(\overline{\mathcal{X}})\otimes\mathbb{Z}_{\ell})^{G}\cong
H^{2d}(\overline{\mathcal{X}},\mathbb{Z}_{\ell})^{G},$

where $\mathbb{Z}_{\ell}$ is equipped with the trivial action of $G$.

By part (2) of Theorem 1.10, this proves the first part of the theorem. The
second part of the theorem is just Corollary 4.17. ∎

###### Proof of Theorem 1.12.

First note that the surjectivity of the cycle class map is a birational
invariant. So using resolution of singularities for $3$-folds [CP08, CP09,
Abh98], we may assume that the singular fibers are SNC divisors. The result of
Bloch-Srinivas [BS83] shows that the Griffiths group of $1$ cycles on
$\overline{\mathcal{X}}$ is $p$-torsion. Thus the hypothesis (B) in Theorem
1.13 is satisfied. As for hypothesis (A), we have a commutative diagram of
localization exact sequences:

$\begin{CD}\oplus
CH_{1}(\overline{\mathcal{X}}_{i})\otimes\mathbb{Z}_{\ell}@>{}>{}>CH_{1}(\overline{\mathcal{X}})\otimes\mathbb{Z}_{\ell}@>{}>{}>CH_{1}(\overline{\mathcal{X}}^{0})\otimes\mathbb{Z}_{\ell}@>{}>{}>0\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ \oplus
H^{4}_{\overline{\mathcal{X}}_{i}}(\overline{\mathcal{X}},\mathbb{Z}_{\ell}(2))@>{}>{}>H^{4}(\overline{\mathcal{X}},\mathbb{Z}_{\ell}(2))@>{}>{}>H^{4}(\overline{\mathcal{X}}^{0},\mathbb{Z}_{\ell}(2))\\\
\end{CD}$

Here $\mathcal{X}_{i}$ is a singular fiber of the fibration $\mathcal{X}\to B$
and $\mathcal{X}^{0}$ is the complement of all the singular fibers
$\mathcal{X}_{i}$. By Section 4.3 in [EW16], the first vertical map is
surjective. We may assume that $\overline{\mathcal{X}}^{0}$ is over an affine
curve (i.e. the direct sum on the left is non-trivial). A simple calculation
then shows that $H^{4}(\overline{\mathcal{X}}^{0})$ is one dimensional and
spanned by the class of a section. Thus the third cycle class map is also
surjective by [dJS03]. So the middle one is surjective.

Hypothesis (C) and (D) are also satisfied in this case. For simplicity, we
only explain how to prove hypothesis (D). On the one hand, the cokernal is
torsion for separably rationally connected fibrations by a decomposition of
diagonal argument. On the other hand, Bloch-Kato conjecture proved by
Voevodsky (in dimension $3$, we can also use the Merkurjev-Suslin theorem)
implies that it is torsion free for all smooth projective $3$-fold. Hence the
cokernal has to vanish.

Thus Theorem 1.13 implies that $CH_{1}(\mathcal{X})\otimes\mathbb{Z}_{\ell}\to
H^{4}(\mathcal{X},\mathbb{Z}_{l}(2))$ is surjective. ∎

###### Proof of Theorem 1.4.

This follows from combining Theorem 1.10, 1.12, and Remark 5.3. ∎

##  6\. Examples

We conclude this article with some examples where one can check the conditions
in Theorem 1.13.

######  Proposition 6.1.

Let $\mathcal{X}\subset\mathbb{P}_{B}(E)$ be a family of complete
intersections of degree $d_{1},\ldots,d_{c}$ in $\mathbb{P}^{n}$ over a smooth
projective curve $B$ over $\mathbb{F}_{q}$. Assume that the generic fiber $X$
is smooth separably rationally connected of dimension $d,d\geq 5$ and that
$\sum d_{i}^{2}\leq n$. Also assume that $\mathcal{X}$ is smooth. Then the
cycle class map

$CH^{d}(\mathcal{X})\otimes\mathbb{Z}_{\ell}\to
H^{2d}_{\text{{\'{e}}t}}(\mathcal{X},\mathbb{Z}_{\ell}(d))$

is surjective and Conjectures 1.1 and 1.2 hold for the generic fiber $X$ over
$\mathbb{F}_{q}(B)$.

###### Proof.

By the dimension assumption and the affine vanishing for étale cohomology,
there is an isomorphism

$H^{2d}_{\text{{\'{e}}t}}(\mathcal{X},\mathbb{Z}_{\ell}(d))\cong
H^{2d-2}_{\text{{\'{e}}t}}(\overline{\mathcal{X}},\mathbb{Z}_{\ell}(d-1))^{G},$

which is spanned by the class of a section and a line in a fiber. So it
suffices to show that over the algebraic closure $\overline{\mathbb{F}}_{q}$,
every multisection is rationally equivalent to a multiple of any fixed section
modulo lines in fibers, and that every line in a fiber is algebraically
equivalent to any line in any smooth fiber.

Both statements follows from some well-known argument. More precisely, the
space of a chain of two lines passing through two points in a complete
intersection of degree $d_{1},\ldots,d_{c}$ is defined by equations of degree
$1,1,2,2,\ldots,d_{1}-1,d_{1}-1,d_{1},1,1,\ldots,d_{2}-1,d_{2}-1,d_{2},\ldots,1,1,\ldots,d_{c}-1,d_{c}-1,d_{c}$
in $\mathbb{P}^{n}$ (see, for example, Lemma 3.4 in [Pan18]). Thus by the
classical Tsen-Lang theorem, for any family of complete intersections of
degree $(d_{1},\ldots,d_{c})$ over a smooth curve $T/\bar{\mathbb{F}}_{q}$,
and for any two sections of this family, there is a family of chain of two
lines in the complete intersections over $T$ such that the two sections lie in
this family of chain of two lines. Any two sections in a
$\mathbb{P}^{1}$-bundle over a curve are rationally equivalent modulo general
fibers. Thus any two sections are rationally equivalent up to lines in general
fibers. This in turn implies that any two multi-sections of the same degree
are rationally equivalent up to lines in general fibers. Since any curve in a
fiber is rationally equivalent to the difference of two multi-sections of the
same degree, it is also rationally equivalent to lines in general fibers.
Finally, since the Fano scheme of lines of a complete intersection is
connected as long as it has positive dimension [DM98, Théroème 2.1], all lines
in a fiber are algebraically equivalent. ∎

######  Remark 6.2.

The dimension assumption is not restrictive. The only low dimensional examples
satisfying the numerical conditions are quadrics and linear spaces. One can
check by hands that integral Tate conjecture holds for them.

######  Remark 6.3.

In general it is still an open question if a smooth Fano complete intersection
is separably rationally connected. However, one can show that if the
characteristic $p$ is larger than all the $d_{i}$, then every smooth Fano
complete intersection of degree $d_{1},\ldots,d_{c}$ is separably rationally
connected [STZ22].

######  Proposition 6.4.

Let $X$ be a smooth proper variety that is also a homogeneous variety under an
integral linear algebraic group $G$ over $\mathbb{F}_{q}(B)$. Assume that $X$
admits a regular projective model $\mathcal{X}\to B$. Then the cycle class map

$CH^{d}(\mathcal{X})\otimes\mathbb{Z}_{\ell}\to
H^{2d}_{\text{{\'{e}}t}}(\mathcal{X},\mathbb{Z}_{\ell}(d))$

is surjective and Conjecture 1.2 holds for $X$.

###### Proof.

It is well-known that $\bar{G}$ over $\bar{\mathbb{F}}_{q}(B)$ is rational.
Thus $\overline{\mathcal{X}}\to\bar{B}$ is birational to
$\bar{B}\times_{\bar{\mathbb{F}}_{q}}\mathbb{P}^{n}\to\bar{B}$. So the
conditions in Theorem 1.13 are satisfied. ∎

######  Remark 6.5.

Liang [Lia13] proved that if the Brauer-Manin obstruction is the only
obstruction to weak approximation of rational points in a rationally connected
variety over a number field $K$ and all of its finite field extensions, the
number field analogue of Conjecture 1.3 is true. As a corollary, he proved
Conjecture 1.3 for all smooth proper varieties birational to a homogeneous
space under a linear algebraic groups with connected stablizer. Harpaz and
Wittenberg [HW20] proved Conjecture 1.3 for all smooth proper varieties
birational to a homogeneous space under a linear algebraic group. One could
expect that this also holds in the global function field case by essentially
the same proof (modulo some characteristic p issues).

######  Theorem 6.6.

Let $C$ be a smooth projective geometrically integral curve defined over a
global function field $\mathbb{F}(B)$. Assume that $C$ has a zero cycle of
degree $1$ over $\bar{\mathbb{F}}(B)$. Let $X(r,L)$ be the moduli space of
stable vector bundles of rank $r$ and fixed determinant
$L\in\text{Pic}^{d}(X)$, with $r,d$ relatively prime to each other. Assume
that $X(r,d)$ has a smooth projective model over $B$. Then the local-global
principle for zero cycles (i.e. Conjecture 1.1) holds for $X(r,L)$.

###### Proof.

It is well-knonw that $X(r,L)$ is geometrically rational [Hof07, New75, New80,
KS99] and has geometric Picard group isomorphic to $\mathbb{Z}$. In fact, as
long as the curve, defined over any field $k$, has a $k$-rational point,
$X(r,L)$ is rational over $k$ [Hof07]. Using a norm argument, one can prove
that the hypothesis (A)-(D) in Theorem 1.13 holds for
$X(r,L)\otimes\bar{\mathbb{F}}$ under our assumptions. Hence Theorem 1.13
implies the statements. ∎

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