# Integral Fourier transforms and the integral Hodge conjecture for one-cycles
on abelian varieties

Thorsten Beckmann⋆ $\quad\&\quad$ Olivier de Gaay Fortman⋆⋆

###### Abstract

We prove the integral Hodge conjecture for one-cycles on a principally
polarized complex abelian variety whose minimal class is algebraic. In
particular, the Jacobian of a smooth proper complex curve satisfies the
integral Hodge conjecture for one-cycles. The main ingredient is a lift of the
Fourier transform to integral Chow groups. Similarly, we prove the integral
Tate conjecture for one-cycles on Jacobians of smooth proper curves over the
separable closure of a finitely generated field. Furthermore, abelian
varieties over the complex numbers (respectively an algebraically closed field
of positive characteristic) satisfying the integral Hodge (respectively
integral Tate) conjecture for one-cycles are dense in their moduli space.

##  1 Introduction

Let $g$ be a positive integer and let $A$ be an abelian variety of dimension
$g$ over a field $k$ with dual abelian variety ${\widehat{A}}$. The
correspondence attached to the Poincaré bundle ${\mathcal{P}}_{A}$ on
$A\times{\widehat{A}}$ defines a powerful duality between the derived
categories, rational Chow groups and cohomology of $A$ and ${\widehat{A}}$
[Muk81, Bea83, Huy06]. We shall refer to such morphisms as Fourier transforms.

On the level of cohomology, the Fourier transform preserves integral
$\ell$-adic étale cohomology when $k=k_{s}$ and integral Betti cohomology when
$k=\mathbb{C}$. It is thus natural to ask whether the Fourier transform on
rational Chow groups preserves integral cycles modulo torsion or, more
generally, lifts to a homomorphism between integral Chow groups. This question
was raised by Moonen–Polishchuk [MP10] and Totaro [Tot21]. More precisely,
Moonen and Polishchuk gave a counterexample for abelian varieties over non-
closed fields and asked about the case of algebraically closed fields.

In this paper we further investigate this question with a view towards
applications concerning the integral Hodge conjecture for one-cycles when $A$
is defined over $\mathbb{C}$. To state our main result, we recall that
whenever $\iota\colon C\hookrightarrow A$ is a smooth curve, the image of the
fundamental class under the pushforward map
$\iota_{\ast}\colon{\mathrm{H}}_{2}(C,\mathbb{Z})\to{\mathrm{H}}_{2}(A,\mathbb{Z})\cong{\mathrm{H}}^{2g-2}(A,\mathbb{Z})$
defines a cohomology class $[C]\in{\mathrm{H}}^{2g-2}(A,\mathbb{Z})$. This
construction extends to one-cycles and factors modulo rational equivalence. As
such, it induces a canonical homomorphism, called the cycle class map,

$cl\colon\textnormal{CH}_{1}(A)\to\textnormal{Hdg}^{2g-2}(A,\mathbb{Z}),$

which is a direct summand of a natural graded ring homomorphism
$cl\colon\textnormal{CH}(A)\to{\mathrm{H}}^{\bullet}(A,\mathbb{Z})$.

The liftability of the Fourier transform turns out to have important
consequences for the image of the cycle class map. Recall that an element
$\alpha\in{\mathrm{H}}^{\bullet}(A,\mathbb{Z})$ is called algebraic if it is
in the image of $cl$, and that $A$ satisfies the integral Hodge conjecture for
$k$-cycles if all Hodge classes in ${\mathrm{H}}^{2g-2k}(A,\mathbb{Z})$ are
algebraic. Although the integral Hodge conjecture fails in general [AH62, Tre,
Tot97], it is an open question for abelian varieties. Our main result is as
follows.

††⋆Max–Planck Institute for Mathematics, Bonn<EMAIL_ADDRESS>††⋆⋆École normale supérieure de Paris<EMAIL_ADDRESS>
######  Theorem 1.1.

Let $A$ be a complex abelian variety of dimension $g$ with Poincaré bundle
${\mathcal{P}}_{A}$. The following three statements are equivalent:

  1. 1.

The cohomology class
$c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic.

  2. 2.

The Chern character
$\textnormal{ch}({\mathcal{P}}_{A})=\exp(c_{1}({\mathcal{P}}_{A}))\in{\mathrm{H}}^{\bullet}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic.

  3. 3.

The integral Hodge conjecture for one-cycles holds for $A\times{\widehat{A}}$.

Any of these statements implies that

  1. 4.

The integral Hodge conjecture for one-cycles holds for $A$ and
${\widehat{A}}$.

Suppose that $A$ is principally polarized by
$\theta\in\textnormal{Hdg}^{2}(A,\mathbb{Z})$ and consider the following
statements:

  1. 5.

The minimal cohomology class
$\gamma_{\theta}\coloneqq\theta^{g-1}/(g-1)!\in{\mathrm{H}}^{2g-2}(A,\mathbb{Z})$
is algebraic.

  2. 6.

The cohomology class
$c_{1}({\mathcal{P}}_{A})^{2g-2}/(2g-2)!\in{\mathrm{H}}^{4g-4}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic.

Then statements
$\ref{introitem:minimalpoincare}-\ref{introitem:minimalpoincare2}$ are
equivalent. If they hold, then
$\theta^{i}/i!\in{\mathrm{H}}^{2i}(A,\mathbb{Z})$ is algebraic for $i\geq 1$.

Remark that Condition 5 is stable under products, so a product of principally
polarized abelian varieties satisfies the integral Hodge conjecture for one-
cycles if and only if each of the factors does. More importantly, if $J(C)$ is
the Jacobian of a smooth projective curve $C$ of genus $g$, then every
integral Hodge class of degree $2g-2$ on $J(C)$ is a $\mathbb{Z}$-linear
combination of curves classes:

######  Theorem 1.2.

Let $C_{1},\dotsc,C_{n}$ be smooth projective curves over $\mathbb{C}$. Then
the integral Hodge conjecture for one-cycles holds for the product of
Jacobians $J(C_{1})\times\cdots\times J(C_{n})$.

See Remark 4.2.1 for another approach towards Theorem 1.2 in the case $n=1$. A
second consequence of Theorem 1.1 is that the integral Hodge conjecture for
one-cycles on principally polarized complex abelian varieties is stable under
specialization, see Corollary 4.3. An application of somewhat different nature
is the following density result, proven in Section 4.2:

######  Theorem 1.3.

Let $\delta=(\delta_{1},\dotsc,\delta_{g})$ be positive integers such that
$\delta_{i}|\delta_{i+1}$ and let ${\mathsf{A}}_{g,\delta}(\mathbb{C})$ be the
coarse moduli space of polarized abelian varieties over $\mathbb{C}$ with
polarization type $\delta$. There is a countable union
$X\subset{\mathsf{A}}_{g,\delta}(\mathbb{C})$ of closed algebraic subvarieties
of dimension at least $g$, that satisfies the following property: $X$ is dense
in the analytic topology and the integral Hodge conjecture for one-cycles
holds for those polarized abelian varieties whose isomorphism class lies in
$X$.

######  Remark 1.4.

The lower bound that we obtain on the dimension of the components of $X$
actually depends on $\delta$ and is often greater than $g$. For instance, when
$\delta=1$ and $g\geq 2$, there is a set $X$ as in the theorem, whose elements
are prime-power isogenous to products of Jacobians of curves. Therefore, the
components of $X$ have dimension $3g-3$ in this case, c.f. Remark 4.7.

One could compare Theorem 1.1 with the following statement, proven by
Grabowski [Gra04]: if $g$ is a positive integer such that the minimal
cohomology class $\gamma_{\theta}=\theta^{g-1}/(g-1)!$ of every principally
polarized abelian variety of dimension $g$ is algebraic, then every abelian
variety of dimension $g$ satisfies the integral Hodge conjecture for one-
cycles. In this way, he proved the integral Hodge conjecture for abelian
threefolds, a result which has been extended to smooth projective threefolds
$X$ with $K_{X}=0$ by Voisin and Totaro [Voi06, Tot21]. For abelian varieties
of dimension greater than three, not many unconditional statements seem to
have been known.

The idea behind the proof of Theorem 1.1 is the following. Let $A$ be a
complex abelian variety of dimension $g$ and let $i\geq 0$ be an integer. Then
Poincaré duality induces a canonical isomorphism
$\varphi\colon{\mathrm{H}}^{2i}(A,\mathbb{Z})\cong{\mathrm{H}}^{2g-2i}(A,\mathbb{Z})^{\vee}\cong{\mathrm{H}}^{2g-2i}({\widehat{A}},\mathbb{Z})$.
The map $\varphi$ respects the Hodge structures and thus induces an
isomorphism
$\textnormal{Hdg}^{2i}(A,\mathbb{Z})\cong\textnormal{Hdg}^{2g-2i}({\widehat{A}},\mathbb{Z})$.
However, it is unclear a priori whether $\varphi$ sends algebraic classes to
algebraic classes. We prove that the algebraicity of
$c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!$ forces $\varphi$ to be algebraic,
i.e. to be induced by a correspondence
$\Gamma\in\textnormal{CH}(A\times{\widehat{A}})$. In particular, one then has
${\mathrm{Z}}^{2i}(A)\coloneqq\textnormal{Hdg}^{2i}(A,\mathbb{Z})/{\mathrm{H}}^{2i}(A,\mathbb{Z})_{\textnormal{alg}}\cong{\mathrm{Z}}^{2g-2i}({\widehat{A}})$.
To prove this, we lift the cohomological Fourier transform to a homomorphism
between integral Chow groups whenever
$c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!$ is algebraic. For this we use a
theorem of Moonen–Polishchuk saying that the ideal of positive dimensional
cycles in the Chow ring with Pontryagin product of an abelian variety admits a
divided power structure [MP10, Theorem 1.6].

In Section 5, we consider an abelian variety $A_{/\mathbb{C}}$ of dimension
$g$ and ask: if $n\in\mathbb{Z}_{\geq 1}$ is such that $n\cdot
c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})_{\textnormal{alg}}$,
can we bound the order of ${\mathrm{Z}}^{2g-2}(A)$ in terms of $g$ and $n$?
For a smooth complex projective $d$-dimensional variety $X$,
${\mathrm{Z}}^{2d-2}(X)$ is called the degree $2d-2$ Voisin group of $X$
[Per20], is a stably birational invariant [Voi16, Lemma 2.20], and related to
the unramified cohomology groups by Colliot-Thélène–Voisin and Schreieder
[CTV12, Sch21]. We prove that if $n\cdot
c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!$ is algebraic, then
$\gcd(n^{2},(2g-2)!)\cdot{\mathrm{Z}}^{2g-2}(A)=(0)$. In particular,
$(2g-2)!\cdot{\mathrm{Z}}^{2g-2}(A)=(0)$ for any $g$-dimensional complex
abelian variety $A$. Moreover, if $A$ is principally polarized by
$\theta\in\textnormal{NS}(A)$ and if
$n\cdot\gamma_{\theta}\in{\mathrm{H}}^{2g-2}(A,\mathbb{Z})$ is algebraic, then
$n\cdot c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!$ is algebraic. Since it is
well known that for Prym varieties, the Hodge class $2\cdot\gamma_{\theta}$ is
algebraic, these observations lead to the following result (see also Theorem
5.3).

######  Theorem 1.5.

Let $A$ be a $g$-dimensional Prym variety over $\mathbb{C}$. Then
$4\cdot{\mathrm{Z}}^{2g-2}(A)=(0)$.

For the study of the liftability of the Fourier transform, which was initiated
by Moonen and Polishchuk in [MP10], it is more natural to consider abelian
varieties defined over arbitrary fields. For this reason we define and study
integral Fourier transforms in this generality, see Section 3. We provide, for
an abelian variety principally polarized by a symmetric ample line bundle,
necessary and sufficient conditions for an integral Fourier transform to
exist, see Theorem 3.8.

This generality also allows to obtain the analogue of Theorem 1.1 over the
separable closure $k$ of a finitely generated field. Recall that a smooth
projective variety $X$ of dimension $d$ over $k$ satisfies the integral Tate
conjecture for one-cycles over $k$ if, for every prime number $\ell$ different
from $\textnormal{char}(k)$ and for some finitely generated field of
definition $k_{0}\subset k$ of $X$, the cycle class map

$cl\colon\textnormal{CH}_{1}(X)_{\mathbb{Z}_{\ell}}=\textnormal{CH}_{1}(X)\otimes_{\mathbb{Z}}{\mathbb{Z}_{\ell}}\to\bigcup_{U}{\mathrm{H}}_{\textnormal{\'{e}t}}^{2d-2}(X,\mathbb{Z}_{\ell}(d-1))^{U}$
(1)

is surjective, where $U$ ranges over the open subgroups of
$\textnormal{Gal}(k/k_{0})$.

######  Theorem 1.6.

Let $A$ be an abelian variety of dimension $g$ over the separable closure $k$
of a finitely generated field.

  * •

The abelian variety $A$ satisfies the integral Tate conjecture for one-cycles
over $k$ if the cohomology class
$c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}_{\textnormal{\'{e}t}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z}_{\ell}(2g-1))$
is the class of a one-cycle with $\mathbb{Z}_{\ell}$-coefficients for every
prime number $\ell<(2g-1)!$ unequal to $\textnormal{char}(k)$.

  * •

Suppose that $A$ is principally polarized and let
$\theta_{\ell}\in{\mathrm{H}}^{2}_{\textnormal{\'{e}t}}(A,\mathbb{Z}_{\ell}(1))$
be the class of the polarization. The map (1) is surjective if
$\gamma_{\theta_{\ell}}\coloneqq\theta_{\ell}^{g-1}/(g-1)!\in{\mathrm{H}}_{\textnormal{\'{e}t}}^{2g-2}(A,\mathbb{Z}_{\ell}(g-1))$
is in its image. In particular, if $\ell>(g-1)!$, then this always holds. Thus
$A$ satisfies the integral Tate conjecture for one-cycles if
$\gamma_{\theta_{\ell}}$ is in the image of (1) for every prime number
$\ell<(g-1)!$ unequal to $\textnormal{char}(k)$.

Theorem 1.6 implies in particular that products of Jacobians of smooth
projective curves over $k$ satisfy the integral Tate conjecture for one-cycles
over $k$. Moreover, for an abelian variety $A_{K}$ over a number field
$K\subset\mathbb{C}$, the integral Hodge conjecture for one-cycles on
$A_{\mathbb{C}}$ is equivalent to the integral Tate conjecture for one-cycles
on $A_{\bar{K}}$ (Corollary 6.2), which in turn implies the integral Tate
conjecture for one-cycles on the geometric special fiber
$A_{\overline{k}({\mathfrak{p}})}$ of the Néron model of $A_{K}$ over
$\mathcal{O}_{K}$ for any prime ${\mathfrak{p}}\subset\mathcal{O}_{K}$ at
which $A_{K}$ has good reduction (Corollary 6.3).

Finally, Theorem 1.3 has an analogue in positive characteristic. The
definition for a smooth projective variety over the algebraic closure $k$ of a
finitely generated field to satisfy the integral Tate conjecture for one-
cycles over $k$ is analogous to the definition above (see e.g. [CP15]).

######  Theorem 1.7.

Let $k$ be the algebraic closure of a finitely generated field of
characteristic $p>0$. Let ${\mathsf{A}}_{g}$ be the coarse moduli space over
$k$ of principally polarized abelian varieties of dimension $g$ over $k$. The
subset of ${\mathsf{A}}_{g}(k)$ of isomorphism classes of principally
polarized abelian varieties over $k$ that satisfy the integral Tate conjecture
for one-cycles over $k$ is Zariski dense in ${\mathsf{A}}_{g}$.

Acknowledgements. We are grateful for the encouragement and support of our
respective PhD advisors Daniel Huybrechts and Olivier Benoist. We would like
to thank them, as well as Fabrizio Catanese and Frans Oort, for stimulating
conversations. We thank Giuseppe Ancona, Olivier Benoist, Daniel Huybrechts
and Burt Totaro for useful comments on an earlier draft of this paper. The
content of this article has been presented for the Research Seminar Algebraic
Geometry in Hanover, the Algebraic Geometry Seminar in Berlin, and the
seminars Autour des cycles algébriques and Séminaire de géométrie algébrique
in Paris during the months January and February 2022. We would like to thank
the participants for ample feedback.

We stress that work of Moonen and Polishchuk [MP10] has been essential for our
results.

The first author is funded by the IMPRS program of the Max–Planck Society, the
second by the European Union’s Horizon 2020 research and innovation programme
under the Marie Skłodowska-Curie grant agreement No 754362 . The authors thank
the Hausdorff Center for Mathematics as well as the École normale supérieure
for their hospitality and pleasant working conditions during the respective
stays of the authors which made this project possible.

##  2 Notation

  * •

We let $k$ be a field with separable closure $k_{s}$ and $\ell$ a prime number
different from the characteristic of $k$. For a smooth projective variety $X$
over $k$, we let $\textnormal{CH}(X)$ be the Chow group of $X$ and define
$\textnormal{CH}(X)_{\mathbb{Q}}=\textnormal{CH}(X)\otimes\mathbb{Q}$,
$\textnormal{CH}(X)_{\mathbb{Q}_{\ell}}=\textnormal{CH}(X)\otimes{\mathbb{Q}_{\ell}}$
and
$\textnormal{CH}(X)_{\mathbb{Z}_{\ell}}=\textnormal{CH}(X)\otimes{\mathbb{Z}_{\ell}}$.
We let
${\mathrm{H}}_{\textnormal{\'{e}t}}^{i}(X_{k_{s}},\mathbb{Z}_{\ell}(a))$ be
the $i$-th degree étale cohomology group with coeffients in
$\mathbb{Z}_{\ell}(a)$, $a\in\mathbb{Z}$.

  * •

Often, $A$ will denote an abelian variety of dimension $g$ over $k$, with dual
abelian variety ${\widehat{A}}$ and (normalized) Poincaré bundle on
${\mathcal{P}}_{A}$. The abelian group $\textnormal{CH}(A)$ will in that case
be equipped with two ring structures: the usual intersection product $\cdotp$
as well as the Pontryagin product $\star$. Recall that the latter is defined
as follows:

$\star\colon\textnormal{CH}(A)\times\textnormal{CH}(A)\to\textnormal{CH}(A),\quad
x\star y=m_{\ast}(\pi_{1}^{\ast}(x)\cdot\pi_{2}^{\ast}(y)).$

Here, as well as in the rest of the paper, $\pi_{i}$ denotes the projection
onto the $i$-th factor, $\Delta\colon A\to A\times A$ the diagonal morphism,
and $m\colon A\times A\to A$ the group law morphism of $A$.

  * •

For any abelian group $M$ and any element $x\in M$, we will denote by
$x_{\mathbb{Q}}\in M\otimes_{\mathbb{Z}}\mathbb{Q}$ the image of $x$ in
$M\otimes_{\mathbb{Z}}\mathbb{Q}$ under the canonical homomorphism $M\to
M\otimes_{\mathbb{Z}}\mathbb{Q}$.

##  3 Integral Fourier transforms and one-cycles on abelian varieties

Our goal in this section is to provide necessary and sufficient conditions for
the Fourier transform on rational Chow groups or cohomology to lift to a
motivic homomorphism between integral Chow groups. We will relate such lifts
to the integral Hodge conjecture when $k=\mathbb{C}$. In subsequent Section 4
we will use the theory developed in this section to prove Theorem 1.1.

###  3.1 Integral Fourier transforms and integral Hodge classes

For abelian varieties $A$ over $k=k_{s}$, it is unknown whether the Fourier
transform
${\mathcal{F}}_{A}\colon\textnormal{CH}(A)_{\mathbb{Q}}\to\textnormal{CH}({\widehat{A}})_{\mathbb{Q}}$
preserves the subgroups of integral cycles modulo torsion. A sufficient
condition for this to hold is that ${\mathcal{F}}_{A}$ lifts to a homomorphism
$\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$. In this section we
outline a second consequence of such a lift
$\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$ when $A$ is defined over
the complex numbers: the existence of an integral lift of ${\mathcal{F}}_{A}$
implies the integral Hodge conjecture for one-cycles on ${\widehat{A}}$.

Let $A$ be an abelian variety over $k$. The Fourier transform on the level of
Chow groups is the group homomorphism

${\mathcal{F}}_{A}\colon\textnormal{CH}(A)_{\mathbb{Q}}\to\textnormal{CH}({\widehat{A}})_{\mathbb{Q}}$

induced by the correspondence
$\textnormal{ch}({\mathcal{P}}_{A})\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$,
where $\textnormal{ch}({\mathcal{P}}_{A})$ is the Chern character of
${\mathcal{P}}_{A}$. Similarly, one defines the Fourier transform on the level
of étale cohomology:

${\mathscr{F}}_{A}\colon{\mathrm{H}}_{\textnormal{\'{e}t}}^{\bullet}(A_{k_{s}},\mathbb{Q}_{\ell}(\bullet))\to{\mathrm{H}}_{\textnormal{\'{e}t}}^{\bullet}({\widehat{A}}_{k_{s}},\mathbb{Q}_{\ell}(\bullet)).$

In fact, ${\mathscr{F}}_{A}$ preserves the integral cohomology classes and
induces, for each integer $j$ with $0\leq j\leq 2g$, an isomorphism [Bea83,
Proposition 1], [Tot21, page 18]:

${\mathscr{F}}_{A}\colon{\mathrm{H}}_{\textnormal{\'{e}t}}^{j}(A_{k_{s}},\mathbb{Z}_{\ell}(a))\to{\mathrm{H}}_{\textnormal{\'{e}t}}^{2g-j}({\widehat{A}}_{k_{s}},\mathbb{Z}_{\ell}(a+g-j)).$

Similarly, if $k=\mathbb{C}$, then $\textnormal{ch}({\mathcal{P}}_{A})$
induces, for each integer $i$ with $0\leq i\leq 2g$, an isomorphism of Hodge
structures

${\mathscr{F}}_{A}\colon{\mathrm{H}}^{i}(A,\mathbb{Z})\to{\mathrm{H}}^{2g-i}({\widehat{A}},\mathbb{Z})(g-i).$
(2)

In [MP10], Moonen and Polishchuk consider an isomorphism $\phi\colon
A\xrightarrow{\sim}{\widehat{A}}$, a positive integer $d$, and define the
notion of motivic integral Fourier transform of $(A,\phi)$ up to factor $d$.
The definition goes as follows. Let ${\mathcal{M}}(k)$ be the category of
effective Chow motives over $k$ with respect to ungraded correspondences, and
let $h(A)$ be the motive of $A$. Then a morphism ${\mathcal{F}}\colon h(A)\to
h(A)$ in ${\mathcal{M}}(k)$ is a motivic integral Fourier transform of
$(A,\phi)$ up to factor $d$ if the following three conditions are satisfied:
(i) the induced morphism $h(A)_{\mathbb{Q}}\to h(A)_{\mathbb{Q}}$ is the
composition of the usual Fourier transform with the isomorphism
$\phi^{\ast}\colon
h({\widehat{A}})_{\mathbb{Q}}\xrightarrow{\sim}h(A)_{\mathbb{Q}}$, (ii) one
has $d\cdot{\mathcal{F}}\circ{\mathcal{F}}=d\cdot(-1)^{g}\cdot[-1]_{\ast}$ as
morphisms from $h(A)$ to $h(A)$, and (iii) $d\cdot{\mathcal{F}}\circ
m_{\ast}=d\cdot\Delta^{\ast}\circ{\mathcal{F}}\otimes{\mathcal{F}}\colon
h(A)\otimes h(A)\to h(A)$.

For our purposes, we will consider similar homomorphisms
$\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$. However, to make their
existence easier to verify (c.f. Theorem 3.8) we relax some of the above
conditions:

######  Definition 3.1.

Let $A_{/k}$ be an abelian variety and let
${\mathcal{F}}\colon\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$ be a
group homomorphism. We call ${\mathcal{F}}$ a weak integral Fourier transform
if the following diagram commutes:

$\textstyle{\textnormal{CH}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{F}}}$$\textstyle{\textnormal{CH}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textnormal{CH}(A)_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{F}}_{A}}$$\textstyle{\textnormal{CH}({\widehat{A}})_{\mathbb{Q}}.}$
(3)

We call a weak integral Fourier transform ${\mathcal{F}}$ motivic if it is
induced by a cycle $\Gamma\in\textnormal{CH}(A\times{\widehat{A}})$ that
satisfies
$\Gamma_{\mathbb{Q}}=\textnormal{ch}({\mathcal{P}}_{A})\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$.
A group homomorphism
${\mathcal{F}}\colon\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$ is an
integral Fourier transform up to homology if the following diagram commutes:

$\textstyle{\textnormal{CH}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{F}}}$$\textstyle{\textnormal{CH}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\oplus_{r\geq
0}{\mathrm{H}}_{\textnormal{\'{e}t}}^{2r}(A_{k_{s}},\mathbb{Z}_{\ell}(r))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathscr{F}}_{A}}$$\textstyle{\oplus_{r\geq
0}{\mathrm{H}}_{\textnormal{\'{e}t}}^{2r}({\widehat{A}}_{k_{s}},\mathbb{Z}_{\ell}(r)).}$
(4)

Similarly, a $\mathbb{Z}_{\ell}$-module homomorphism
${\mathcal{F}}_{\ell}\colon\textnormal{CH}(A)_{\mathbb{Z}_{\ell}}\to\textnormal{CH}({\widehat{A}})_{\mathbb{Z}_{\ell}}$
is called an $\ell$-adic integral Fourier transform up to homology if
${\mathcal{F}}_{\ell}$ is compatible with ${\mathscr{F}}_{A}$ and the
$\ell$-adic cycle class maps. Finally, an integral Fourier transform up to
homology ${\mathcal{F}}$ (resp. an $\ell$-adic integral Fourier transform up
to homology ${\mathcal{F}}_{\ell}$) is called motivic if it is induced by a
cycle $\Gamma\in\textnormal{CH}(A\times{\widehat{A}})$ (resp.
$\Gamma_{\ell}\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Z}_{\ell}})$
such that $cl(\Gamma)$ (resp. $cl(\Gamma_{\ell})$) equals
$\textnormal{ch}({\mathcal{P}}_{A})\in\oplus_{r\geq
0}{\mathrm{H}}_{\textnormal{\'{e}t}}^{2r}((A\times{\widehat{A}})_{k_{s}},\mathbb{Z}_{\ell}(r))$.

######  Remark 3.2.

If ${\mathcal{F}}\colon\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$ is
a weak integral Fourier transform, then ${\mathcal{F}}$ is an integral Fourier
transform up to homology, the $\mathbb{Z}_{\ell}$-module $\oplus_{r\geq
0}{\mathrm{H}}_{\textnormal{\'{e}t}}^{2r}({\widehat{A}}_{k_{s}},\mathbb{Z}_{\ell}(r))$
being torsion-free. If $k=\mathbb{C}$, then
${\mathcal{F}}\colon\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$ is an
integral Fourier transform up to homology if and only if ${\mathcal{F}}$ is
compatible with the Fourier transform
${\mathscr{F}}_{A}\colon{\mathrm{H}}^{\bullet}(A,\mathbb{Z})\to{\mathrm{H}}^{\bullet}({\widehat{A}},\mathbb{Z})$
on Betti cohomology.

The relation between integral Fourier transforms and integral Hodge classes is
as follows:

######  Lemma 3.3.

Let $A$ be a complex abelian variety and
${\mathcal{F}}\colon\textnormal{CH}(A)\to\textnormal{CH}({\widehat{A}})$ an
integral Fourier transform up to homology. For each $i\in\mathbb{Z}_{\geq 0}$,
the integral Hodge conjecture for degree $2i$ classes on $A$ implies the
integral Hodge conjecture for degree $2g-2i$ classes on ${\widehat{A}}$. If
${\mathcal{F}}$ is motivic, then ${\mathscr{F}}_{A}$ induces a group
isomorphism
${\mathrm{Z}}^{2i}(A)\xrightarrow{\sim}{\mathrm{Z}}^{2g-2i}({\widehat{A}})$
and, therefore, the integral Hodge conjectures for degree $2i$ classes on $A$
and degree $2g-2i$ classes on ${\widehat{A}}$ are equivalent for all $i$.

###### Proof.

We can extend Diagram (4) to the following commutative diagram:

$\textstyle{\textnormal{CH}^{i}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl^{i}}$$\textstyle{\textnormal{CH}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{F}}}$$\textstyle{\textnormal{CH}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textnormal{CH}_{i}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl_{i}}$$\textstyle{{\mathrm{H}}^{2i}(A,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{\bullet}(A,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathscr{F}}_{A}}$$\textstyle{{\mathrm{H}}^{\bullet}({\widehat{A}},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{2g-2i}({\widehat{A}},\mathbb{Z}).}$

The composition
${\mathrm{H}}^{2i}(A,\mathbb{Z})\to{\mathrm{H}}^{2g-2i}({\widehat{A}},\mathbb{Z})$
appearing on the bottom line agrees up to a suitable Tate twist with the map
${\mathscr{F}}_{A}$ of Equation (2). Therefore, we obtain a commutative
diagram:

$\textstyle{\textnormal{CH}^{i}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl^{i}}$$\textstyle{\textnormal{CH}_{i}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl_{i}}$$\textstyle{\textnormal{Hdg}^{2i}(A,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\textnormal{Hdg}^{2g-2i}({\widehat{A}},\mathbb{Z}).}$
(5)

Thus the surjectivity of $cl^{i}$ implies the surjectivity of $cl_{i}$.
Moreover, if ${\mathcal{F}}$ is motivic, then replacing $A$ by ${\widehat{A}}$
and ${\widehat{A}}$ by ${\widehat{\widehat{A}\mkern 5.5mu}\mkern-5.5mu}{}$ in
the argument above shows that the images of $cl^{i}$ and $cl_{i}$ are
identified under the isomorphism
${\mathscr{F}}_{A}\colon\textnormal{Hdg}^{2i}(A,\mathbb{Z})\xrightarrow{\sim}\textnormal{Hdg}^{2g-2i}({\widehat{A}},\mathbb{Z})$
in Diagram (5). ∎

###  3.2 Properties of the Fourier transform on rational Chow groups

The above suggests that to prove Theorem 1.1, one would need to show that for
a complex abelian variety of dimension $g$ whose minimal Poincaré class
$c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic, all classes of the form
$c_{1}({\mathcal{P}}_{A})^{i}/i!\in{\mathrm{H}}^{2i}(A\times{\widehat{A}},\mathbb{Z})$
are algebraic. With this goal in mind we shall study Fourier transforms on
rational Chow groups in Section 3.2, and investigate how these relate to
$\textnormal{ch}({\mathcal{P}}_{A})\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$.
In turns out that the cycles
$c_{1}({\mathcal{P}}_{A})^{i}/i!\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$
satisfy several relations that are very similar to those proved by Beauville
for the cycles $\theta^{i}/i!\in\textnormal{CH}(A)_{\mathbb{Q}}$ in case $A$
is principally polarized, see [Bea83]. Since we will need these results in any
characteristic in order to prove Theorem 1.6, we will work over our general
field $k$, see Section 2.

Let $A$ be an abelian variety over $k$. Define cycles
$\ell=c_{1}({\mathcal{P}}_{A})\in\textnormal{CH}^{1}(A\times{\widehat{A}})_{\mathbb{Q}}$
and
${\mathscr{R}}_{A}=c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in\textnormal{CH}_{1}(A\times{\widehat{A}})_{\mathbb{Q}}$.
Dually, define
${\widehat{\ell}}=c_{1}({\mathcal{P}}_{{\widehat{A}}})\in\textnormal{CH}^{1}({\widehat{A}}\times
A)_{\mathbb{Q}}$ and
${\mathscr{R}}_{{\widehat{A}}}=c_{1}({\mathcal{P}}_{{\widehat{A}}})^{2g-1}/(2g-1)!\in\textnormal{CH}_{1}({\widehat{A}}\times
A)_{\mathbb{Q}}$. For $a\in\textnormal{CH}(A)_{\mathbb{Q}}$, define
$\mathrm{E}(a)\in\textnormal{CH}(A)_{\mathbb{Q}}$ as the $\star$-exponential
of $a$:

$\mathrm{E}(a)\coloneqq\sum_{n\geq 0}\frac{a^{\star
n}}{n!}\in\textnormal{CH}(A)_{\mathbb{Q}}.$

The key to Theorem 1.1 will be the following:

######  Lemma 3.4.

We have
$\textnormal{ch}({\mathcal{P}}_{A})=e^{\ell}=(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot{\mathscr{R}}_{A})\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$.

###### Proof.

The most important ingredient in the proof is the following:

Claim $(\ast)$: With respect to the Fourier transform
${\mathcal{F}}_{A\times{\widehat{A}}}\colon\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}\to\textnormal{CH}({\widehat{A}}\times
A)_{\mathbb{Q}}$, one has
${\mathcal{F}}_{A\times{\widehat{A}}}(e^{\ell})=(-1)^{g}\cdot
e^{-{\widehat{\ell}}}\in\textnormal{CH}({\widehat{A}}\times A)_{\mathbb{Q}}$.

To prove Claim $(\ast)$, we lift the desired equality in the rational Chow
group of ${\widehat{A}}\times A$ to an isomorphism in the derived category
${\mathrm{D}}^{b}({\widehat{A}}\times A)$ of ${\widehat{A}}\times A$. For
$X=A\times{\widehat{A}}$ the Poincaré line bundle $\mathcal{P}_{X}$ on
$X\times{\widehat{X}}\cong A\times{\widehat{A}}\times{\widehat{A}}\times A$ is
isomorphic to
$\pi_{13}^{\ast}\mathcal{P}_{A}\otimes\pi_{24}^{\ast}\mathcal{P}_{{\widehat{A}}}$.
Consider

$\Phi_{{{\mathcal{P}}}_{X}}({\mathcal{P}}_{A})\cong\pi_{34,\ast}\left(\pi_{13}^{\ast}{\mathcal{P}}_{A}\otimes\pi_{24}^{\ast}{\mathcal{P}}_{{\widehat{A}}}\otimes\pi_{12}^{\ast}{\mathcal{P}}_{A}\right)\in{\mathrm{D}}^{b}({\widehat{A}}\times
A)$ (6)

whose Chern character is exactly $\mathcal{F}_{X}(e^{\ell})$. Applying the
pushforward along the permutation map

$(123)\colon A\times{\widehat{A}}\times{\widehat{A}}\times
A\cong{\widehat{A}}\times A\times{\widehat{A}}\times A$

the object (6) becomes
$\pi_{14,\ast}\left(\pi_{12}^{\ast}\mathcal{P}_{{\widehat{A}}}\otimes\pi_{23}^{\ast}\mathcal{P}_{A}\otimes\pi_{34}^{\ast}\mathcal{P}_{{\widehat{A}}}\right)$
which is isomorphic to the Fourier–Mukai kernel of the composition

$\Phi_{\mathcal{P}_{{\widehat{A}}}}\circ\Phi_{\mathcal{P}_{A}}\circ\Phi_{\mathcal{P}_{{\widehat{A}}}}.$

Since $\Phi_{\mathcal{P}_{A}}\circ\Phi_{\mathcal{P}_{{\widehat{A}}}}$ is
isomorphic to $[-1_{{\widehat{A}}}]^{\ast}\circ[-g]$ by [Muk81, Theorem 2.2],
we have

$\Phi_{\mathcal{P}_{{\widehat{A}}}}\circ\Phi_{\mathcal{P}_{A}}\circ\Phi_{\mathcal{P}_{{\widehat{A}}}}\cong\Phi_{\mathcal{P}_{{\widehat{A}}}}\circ[-1_{{\widehat{A}}}]^{\ast}\circ[-g].$

This is the Fourier–Mukai transform with kernel the object
${\mathcal{E}}=\mathcal{P}_{{\widehat{A}}}^{\vee}[-g]\in{\mathrm{D}}^{b}({\widehat{A}}\times
A)$. By uniqueness of the Fourier–Mukai kernel of an equivalence [Orl97,
Theorem 2.2] and the fact that the Chern character of ${\mathcal{E}}$ equals
$(-1)^{g}\cdot e^{-{\widehat{\ell}}}\in\textnormal{CH}({\widehat{A}}\times
A)_{\mathbb{Q}}$, this finishes the proof of Claim $(\ast)$.

Next, we claim that $(-1)^{g}\cdot{\mathcal{F}}_{{\widehat{A}}\times
A}(-{\widehat{\ell}})={\mathscr{R}}_{A}$. To see this, recall that for each
integer $i$ with $0\leq i\leq g$, there is a canonical Beauville decomposition
$\textnormal{CH}^{i}(A)_{\mathbb{Q}}=\oplus_{j=i-g}^{i}\textnormal{CH}^{i,j}(A)_{\mathbb{Q}}$
[Bea86]. Since the Poincaré bundle ${\mathcal{P}}_{A}$ is symmetric, we have
$\ell\in\textnormal{CH}^{1,0}(A\times{\widehat{A}})_{\mathbb{Q}}$ and hence
$\ell^{i}\in\textnormal{CH}^{i,0}(A\times{\widehat{A}})_{\mathbb{Q}}$. In
particular, we have
${\mathscr{R}}_{A}\in\textnormal{CH}^{2g-1,0}(A\times{\widehat{A}})_{\mathbb{Q}}$.
The fact that ${\mathcal{P}}_{A}$ is symmetric also implies - via Claim
$(\ast)$ \- that we have ${\mathcal{F}}_{{\widehat{A}}\times A}((-1)^{g}\cdot
e^{-{\widehat{\ell}}})=e^{\ell}$. Indeed, ${\mathcal{F}}_{{\widehat{A}}\times
A}\circ{\mathcal{F}}_{A\times{\widehat{A}}}=[-1]^{\ast}\cdot(-1)^{2g}=[-1]^{\ast}$,
see [DM91, Corollary 2.22]. Since ${\mathcal{F}}_{{\widehat{A}}\times A}$
identifies $\textnormal{CH}^{i,0}({\widehat{A}}\times A)_{\mathbb{Q}}$ with
$\textnormal{CH}^{g-i,0}(A\times{\widehat{A}})$ (see [DM91, Lemma 2.18]), we
must indeed have

$(-1)^{g}\cdot{\mathcal{F}}_{{\widehat{A}}\times
A}(-{\widehat{\ell}})={\mathcal{F}}_{{\widehat{A}}\times
A}((-1)^{g+1}\cdot{\widehat{\ell}})=\frac{{\ell}^{2g-1}}{(2g-1)!}={\mathscr{R}}_{A}.$
(7)

For a $g$-dimensional abelian variety $X$ and any
$x,y\in\textnormal{CH}(X)_{\mathbb{Q}}$, one has ${\mathcal{F}}_{X}(x\cdot
y)=(-1)^{g}\cdot{\mathcal{F}}_{X}(x)\star{\mathcal{F}}_{X}(y)\in\textnormal{CH}({\widehat{X}})_{\mathbb{Q}}$,
see [Bea83, Proposition 3]. This implies (see also [MP10, §3.7]) that if $a$
is a cycle on $X$ such that
${\mathcal{F}}_{X}(a)\in\textnormal{CH}_{>0}({\widehat{X}})_{\mathbb{Q}}$,
then
${\mathcal{F}}_{X}(e^{a})=(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot{\mathcal{F}}_{X}(a))$.
This allows us to conclude that

$e^{\ell}={\mathcal{F}}_{{\widehat{A}}\times A}((-1)^{g}\cdot
e^{-{\widehat{\ell}}})=(-1)^{g}\cdot{\mathcal{F}}_{{\widehat{A}}\times
A}(e^{-{\widehat{\ell}}})=(-1)^{g}\cdot{\mathrm{E}}({\mathcal{F}}_{{\widehat{A}}\times
A}(-{\widehat{\ell}}))=(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot{\mathscr{R}}_{A}),$

which finishes the proof. ∎

Next, assume that $A$ is equipped with a principal polarization $\lambda\colon
A\xrightarrow{\sim}{\widehat{A}}$, define $\ell=c_{1}({\mathcal{P}}_{A})$, and
let

$\Theta=\frac{1}{2}\cdot(\textnormal{id},\lambda)^{\ast}\ell\in\textnormal{CH}^{1}(A)_{\mathbb{Q}}$
(8)

be the symmetric ample class corresponding to the polarization. Here
$(\textnormal{id},\lambda)$ is the morphism $(\textnormal{id},\lambda)\colon
A\to A\times{\widehat{A}}$. One can understand the relation between

$\Gamma_{\Theta}\coloneqq\Theta^{g-1}/(g-1)!\in\textnormal{CH}_{1}(A)_{\mathbb{Q}}$

and
${\mathscr{R}}_{A}={\ell}^{2g-1}/(2g-1)!\in\textnormal{CH}_{1}(A\times{\widehat{A}})_{\mathbb{Q}}$
in the following way. Define $j_{1}\colon A\to A\times{\widehat{A}}$ and
$j_{2}\colon{\widehat{A}}\to A\times{\widehat{A}}$ by $x\mapsto(x,0)$ and
$y\mapsto(0,y)$ respectively. Let
${\widehat{\Theta}}\in\textnormal{CH}^{1}({\widehat{A}})_{\mathbb{Q}}$ be the
dual of $\Theta$, and define a one-cycle $\tau$ on $A\times{\widehat{A}}$ as
follows:

$\displaystyle\tau\coloneqq
j_{1,\ast}(\Gamma_{\Theta})+j_{2,\ast}(\Gamma_{{\widehat{\Theta}}})-(\textnormal{id},\lambda)_{\ast}(\Gamma_{\Theta})\in\textnormal{CH}_{1}(A\times{\widehat{A}})_{\mathbb{Q}}.$

######  Lemma 3.5.

One has
$\tau=(-1)^{g+1}\cdot{\mathscr{R}}_{A}\in\textnormal{CH}_{1}(A\times{\widehat{A}})_{\mathbb{Q}}$.

###### Proof.

Identify $A$ and ${\widehat{A}}$ via $\lambda$. This gives
$\ell=m^{\ast}(\Theta)-\pi_{1}^{\ast}(\Theta)-\pi_{2}^{\ast}(\Theta)$, and the
Fourier transform becomes an endomorphism
${\mathcal{F}}_{A}\colon\textnormal{CH}(A)_{\mathbb{Q}}\to\textnormal{CH}(A)_{\mathbb{Q}}$.
We claim that
$\tau=(-1)^{g}\cdot\left(\Delta_{\ast}{\mathcal{F}}_{A}(\Theta)-j_{1,\ast}{\mathcal{F}}_{A}(\Theta)-j_{2,\ast}{\mathcal{F}}_{A}(\Theta)\right)$.
For this, it suffices to show that
${\mathcal{F}}_{A}(\Theta)=(-1)^{g-1}\cdot\Theta^{g-1}/(g-1)!\in\textnormal{CH}_{1}(A)_{\mathbb{Q}}$.
Now ${\mathcal{F}}_{A}(e^{\Theta})=e^{-\Theta}$ by Lemma 3.6 below. Moreover,
since $\Theta$ is symmetric, we have
$\Theta\in\textnormal{CH}^{1,0}(A)_{\mathbb{Q}}$, hence
$\Theta^{i}/i!\in\textnormal{CH}^{i,0}(A)_{\mathbb{Q}}$ for each $i\geq 0$.
Therefore,
${\mathcal{F}}_{A}\left(\Theta^{i}/i!\right)\in\textnormal{CH}^{g-i,0}(A)_{\mathbb{Q}}$
by [DM91, Lemma 2.18]. This implies that in fact,
${\mathcal{F}}_{A}\left(\Theta^{i}/i!\right)=(-1)^{g-i}\cdot\Theta^{g-i}/(g-i)!\in\textnormal{CH}^{g-i,0}(A)_{\mathbb{Q}}$
for every $i$. In particular, the claim follows.

Next, recall that ${\mathcal{F}}_{A\times
A}(\ell)=(-1)^{g+1}\cdot{\mathscr{R}}_{A}$, see Claim $(\ast)$. So at this
point, it suffices to prove the identity ${\mathcal{F}}_{A\times
A}(\ell)=(-1)^{g}\cdot\left(\Delta_{\ast}{\mathcal{F}}_{A}(\Theta)-j_{1,\ast}{\mathcal{F}}_{A}(\Theta)-j_{2,\ast}{\mathcal{F}}_{A}(\Theta)\right)$.
To prove this, we use the following functoriality properties of the Fourier
transform on the level of rational Chow groups. Let $X$ and $Y$ be abelian
varieties and let $f\colon X\to Y$ be a homomorphism with dual homomorphism
${\widehat{f}}\colon{\widehat{Y}}\to{\widehat{X}}$. We then have the following
equalities [MP10, (3.7.1)]:

$({\widehat{f}})^{\ast}\circ{\mathcal{F}}_{X}={\mathcal{F}}_{Y}\circ
f_{\ast},\hskip 8.53581pt{\mathcal{F}}_{X}\circ f^{\ast}=(-1)^{\dim X-\dim
Y}\cdot({\widehat{f}})_{\ast}\circ{\mathcal{F}}_{Y}.$ (9)

Since $\ell=m^{\ast}\Theta-\pi_{1}^{\ast}\Theta-\pi_{2}^{\ast}\Theta$, it
follows from Equation (9) that

$\displaystyle{\mathcal{F}}_{A\times A}(\ell)$
$\displaystyle={\mathcal{F}}_{A\times
A}\left(m^{\ast}\Theta\right)-{\mathcal{F}}_{A\times
A}\left(\pi_{1}^{\ast}\Theta\right)-{\mathcal{F}}_{A\times
A}\left(\pi_{2}^{\ast}\Theta\right)$
$\displaystyle=(-1)^{g}\cdot\left(\Delta_{\ast}{\mathcal{F}}_{A}(\Theta)-j_{1,\ast}{\mathcal{F}}_{A}(\Theta)-j_{2,\ast}{\mathcal{F}}_{A}(\Theta)\right).$

∎

######  Lemma 3.6 (Beauville).

Let $A$ be an abelian variety over $k$, principally polarized by
$\lambda\colon A\xrightarrow{\sim}{\widehat{A}}$, and define
$\Theta=\frac{1}{2}\cdot(\textnormal{id},\lambda)^{\ast}c_{1}({\mathcal{P}}_{A})\in\textnormal{CH}^{1}(A)_{\mathbb{Q}}$.
Identify $A$ and ${\widehat{A}}$ via $\lambda$. With respect to the Fourier
transform
${\mathcal{F}}_{A}\colon\textnormal{CH}(A)_{\mathbb{Q}}\xrightarrow{\sim}\textnormal{CH}(A)_{\mathbb{Q}}$,
one has ${\mathcal{F}}_{A}(e^{\Theta})=e^{-\Theta}$.

###### Proof.

Our proof follows the proof of [Bea83, Lemme 1], but has to be adapted, since
$\Theta$ does not necessarily come from a symmetric ample line bundle on $A$.
Since one still has
$\ell=m^{\ast}\Theta-\pi_{1}^{\ast}\Theta-\pi_{2}^{\ast}\Theta$, the argument
can be made to work: one has

${\mathcal{F}}_{A}(e^{\Theta})=\pi_{2,\ast}\left(e^{\ell}\cdot\pi_{1}^{\ast}e^{\Theta}\right)=\pi_{2,\ast}\left(e^{m^{\ast}\Theta-\pi_{2}^{\ast}\Theta}\right)=e^{-\Theta}\pi_{2,\ast}(m^{\ast}e^{\Theta})\in\textnormal{CH}(A)_{\mathbb{Q}}.$

For codimension reasons, one has
$\pi_{2,\ast}(m^{\ast}e^{\Theta})=\pi_{2,\ast}m^{\ast}(\Theta^{g}/g!)=\deg(\Theta^{g}/g!)\in\textnormal{CH}^{0}(A)_{\mathbb{Q}}\cong\mathbb{Q}$.
Pull back $\Theta^{g}/g!$ along $A_{k_{s}}\to A$ to see that
$\deg(\Theta^{g}/g!)=1\in\textnormal{CH}^{0}(A)_{\mathbb{Q}}\cong\textnormal{CH}^{0}(A_{k_{s}})_{\mathbb{Q}}$,
since over $k_{s}$ the cycle $\Theta$ becomes the cycle class attached to a
symmetric ample line bundle. ∎

###  3.3 Divided powers and integral Fourier transforms

It was asked by Bruno Kahn whether there exists a PD-structure on the Chow
ring of an abelian variety over any field with respect to its usual
(intersection) product. There are counterexamples over non-closed fields: see
[Esn04], where Esnault constructs an abelian surface $X$ and a line bundle
${\mathcal{L}}$ on $X$ such that $c_{1}({\mathcal{L}})\cdot
c_{1}({\mathcal{L}})$ is not divisible by $2$ in $\textnormal{CH}_{0}(X)$.
However, the case of algebraically closed fields remains open [MP10, Section
3.2]. What we do know, is the following:

######  Theorem 3.7 (Moonen–Polishchuk).

Let $A$ be an abelian variety over $k$. The ring
$\left(\textnormal{CH}(A),\star\right)$ admits a canonical PD-structure
$\gamma$ on the ideal $\textnormal{CH}_{>0}(A)\subset\textnormal{CH}(A)$. If
$k=\bar{k}$, then $\gamma$ extends to a PD-structure on the ideal generated by
$\textnormal{CH}_{>0}(A)$ and the zero cycles of degree zero.

In particular, for each element $x\in\textnormal{CH}_{>0}(A)$ and each
$n\in\mathbb{Z}_{\geq 1}$, there is a canonical element
$x^{[n]}\in\textnormal{CH}_{>0}(A)$ such that $n!x^{[n]}=x^{\star n}$, see
[Stacks, Tag 07GM]. For $x\in\textnormal{CH}_{>0}(A)$, we may then define
$\mathrm{E}(x)=\sum_{n\geq 0}x^{[n]}\in\textnormal{CH}(A)$ as the
$\star$-exponential of $x$ in terms of its divided powers.

Together with the results of Section 3.2, Theorem 3.7 enables us to provide
several criteria for the existence of a motivic weak integral Fourier
transform. We recall that for an abelian variety $A$ over $k$, principally
polarized by $\lambda\colon A\xrightarrow{\sim}{\widehat{A}}$, we defined
$\Theta\in\textnormal{CH}^{1}(A)_{\mathbb{Q}}$ to be the symmetric ample class
attached to the polarization $\lambda$, see Equation (8).

######  Theorem 3.8.

Let $A_{/k}$ be an abelian variety of dimension $g$. The following are
equivalent:

  1. 1.

The one-cycle
${\mathscr{R}}_{A}=c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$
lifts to a one-cycle in $\textnormal{CH}(A\times{\widehat{A}})$.

  2. 2.

The abelian variety $A$ admits a motivic weak integral Fourier transform.

  3. 3.

The abelian variety $A\times{\widehat{A}}$ admits a motivic weak integral
Fourier transform.

Moreover, if $A$ carries a symmetric ample line bundle that induces a
principal polarization $\lambda\colon A\xrightarrow{\sim}{\widehat{A}}$, then
the above statements are equivalent to the following equivalent statements:

  1. 4.

The two-cycle
${\mathscr{S}}_{A}=c_{1}({\mathcal{P}}_{A})^{2g-2}/(2g-2)!\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}$
lifts to a two-cycle in $\textnormal{CH}(A\times{\widehat{A}})$.

  2. 5.

The one-cycle
$\Gamma_{\Theta}=\Theta^{g-1}/(g-1)!\in\textnormal{CH}(A)_{\mathbb{Q}}$ lifts
to a one-cycle in $\textnormal{CH}(A)$.

  3. 6.

The abelian variety $A$ admits a weak integral Fourier transform.

  4. 7.

The Fourier transform ${\mathcal{F}}_{A}$ satisfies
${\mathcal{F}}_{A}\left(\textnormal{CH}(A)/\textnormal{torsion}\right)\subset\textnormal{CH}({\widehat{A}})/\textnormal{torsion}$.

  5. 8.

There exists a PD-structure on the ideal
$\textnormal{CH}^{>0}(A)/\textnormal{torsion}\subset\textnormal{CH}(A)/\textnormal{torsion}$.

###### Proof.

Suppose that 1 holds, and let
$\Gamma\in\textnormal{CH}_{1}(A\times{\widehat{A}})$ be a cycle such that
$\Gamma_{\mathbb{Q}}={\mathscr{R}}_{A}$. Then consider the cycle
$(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot\Gamma)\in\textnormal{CH}(A\times{\widehat{A}})$.
By Lemma 3.4, we have

$(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot\Gamma)_{\mathbb{Q}}=(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot\Gamma_{\mathbb{Q}})=(-1)^{g}\cdot{\mathrm{E}}((-1)^{g}\cdot{\mathscr{R}}_{A})=\textnormal{ch}({\mathcal{P}}_{A})\in\textnormal{CH}(A\times{\widehat{A}})_{\mathbb{Q}}.$

Thus 2 holds. We claim that 3 holds as well. Indeed, consider the line bundle
${\mathcal{P}}_{A\times{\widehat{A}}}$ on the abelian variety
$X=A\times{\widehat{A}}\times{\widehat{A}}\times A$; one has that
${\mathcal{P}}_{A\times{\widehat{A}}}\cong\pi_{13}^{\ast}{\mathcal{P}}_{A}\otimes\pi_{24}^{\ast}{\mathcal{P}}_{{\widehat{A}}}$,
which implies that

$\begin{split}{\mathscr{R}}_{A\times{\widehat{A}}}&=\frac{1}{(4g-1)!}\cdot\left(\pi_{13}^{\ast}c_{1}({\mathcal{P}}_{A})+\pi_{24}^{\ast}c_{1}({\mathcal{P}}_{{\widehat{A}}})\right)^{4g-1}\\\
&=\frac{1}{(2g)!(2g-1)!}\cdot\left(\pi_{13}^{\ast}c_{1}({\mathcal{P}}_{A})^{2g-1}\cdot\pi_{24}^{\ast}c_{1}({\mathcal{P}}_{{\widehat{A}}})^{2g}+\pi_{13}^{\ast}c_{1}({\mathcal{P}}_{A})^{2g}\cdot\pi_{24}^{\ast}c_{1}({\mathcal{P}}_{{\widehat{A}}})^{2g-1}\right)\\\
&=\frac{1}{(2g)!(2g-1)!}\cdot\left(\pi_{13}^{\ast}c_{1}({\mathcal{P}}_{A})^{2g-1}\cdot\pi_{24}^{\ast}\left((2g)!\cdot[0]_{A\times{\widehat{A}}}\right)+\pi_{13}^{\ast}\left((2g)!\cdot[0]_{{\widehat{A}}\times
A}\right)\cdot\pi_{24}^{\ast}c_{1}({\mathcal{P}}_{{\widehat{A}}})^{2g-1}\right)\\\
&=\pi_{13}^{\ast}\left(\frac{c_{1}({\mathcal{P}}_{A})^{2g-1}}{(2g-1)!}\right)\cdot\pi_{24}^{\ast}([0]_{A\times{\widehat{A}}})+\pi_{13}^{\ast}([0]_{{\widehat{A}}\times
A})\cdot\pi_{24}^{\ast}\left(\frac{c_{1}({\mathcal{P}}_{{\widehat{A}}})^{2g-1}}{(2g-1)!}\right)\in\textnormal{CH}_{1}(X)_{\mathbb{Q}}.\end{split}$
(10)

We conclude that ${\mathscr{R}}_{A\times{\widehat{A}}}$ lifts to
$\textnormal{CH}_{1}(X)$ which, by the implication
$[\ref{motivicone}\implies\ref{motivictwo}]$ (that has already been proved),
implies that $A\times{\widehat{A}}$ admits a motivic weak integral Fourier
transform. On the other hand, the implication
$[\ref{motivicthree}\implies\ref{motivicone}]$ follows from the fact that
$(-1)^{g}\cdot{\mathcal{F}}_{{\widehat{A}}\times
A}(-{\widehat{\ell}})={\mathscr{R}}_{A}$ (see Equation (7)) and the fact that
an abelian variety admits a motivic weak integral Fourier transform if and
only if its dual abelian variety does. Therefore, we have
$[\ref{motivicone}\iff\ref{motivictwo}\iff\ref{motivicthree}]$.

Let us from now on assume that $A$ is principally polarized by $\lambda\colon
A\xrightarrow{\sim}A$, where $\lambda$ is the polarization attached to a
symmetric ample line bundle ${\mathcal{L}}$ on $A$. Moreover, in what follows
we shall identify ${\widehat{A}}$ and $A$ via $\lambda$.

Suppose that 4 holds and let $S_{A}\in\textnormal{CH}_{2}(A\times
A)=\textnormal{CH}^{2g-2}(A\times A)$ be such that
$(S_{A})_{\mathbb{Q}}={\mathscr{S}}_{A}\in\textnormal{CH}_{2}(A\times
A)_{\mathbb{Q}}$. Define
$\textnormal{CH}^{1,0}(A)\coloneqq\textnormal{Pic}^{\textnormal{sym}}(A)$ to
be the group of isomorphism classes of symmetric line bundes on $A$. Then
$S_{A}$ induces a homomorphism
${\mathcal{F}}\colon\textnormal{CH}^{1,0}(A)\to\textnormal{CH}_{1}(A)$ defined
as the composition

${\mathcal{F}}\colon\textnormal{CH}^{1,0}(A)\xrightarrow{\pi_{1}^{\ast}}\textnormal{CH}^{1}(A\times
A)\xrightarrow{\cdot S_{A}}\textnormal{CH}^{2g-1}(A\times
A)=\textnormal{CH}_{1}(A\times
A)\xrightarrow{\pi_{2,\ast}}\textnormal{CH}_{1}(A).$

Since
${\mathcal{F}}_{A}\left(\textnormal{CH}^{1,0}(A)_{\mathbb{Q}}\right)\subset\textnormal{CH}_{1}(A)_{\mathbb{Q}}$
(see [DM91, Lemma 2.18]) we see that the diagram

$\textstyle{\textnormal{CH}^{1,0}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{F}}}$$\textstyle{\textnormal{CH}_{1}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textnormal{CH}^{1,0}(A)_{\mathbb{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{F}}_{A}}$$\textstyle{\textnormal{CH}_{1}(A)_{\mathbb{Q}}}$
(11)

commutes. On the other hand, since the line bundle ${\mathcal{L}}$ is
symmetric, we have

$\Theta=\frac{1}{2}\cdot(\textnormal{id},\lambda)^{\ast}c_{1}({\mathcal{P}}_{A})=\frac{1}{2}\cdot
c_{1}\left((\textnormal{id},\lambda)^{\ast}{\mathcal{P}}_{A}\right)=\frac{1}{2}\cdot
c_{1}({\mathcal{L}}\otimes{\mathcal{L}})=c_{1}({\mathcal{L}})\in\textnormal{CH}^{1}(A)_{\mathbb{Q}}.$
(12)

The class $c_{1}({\mathcal{L}})\in\textnormal{CH}^{1,0}(A)$ of the line bundle
${\mathcal{L}}$ thus lies above
$\Theta\in\textnormal{CH}^{1}(A)_{\mathbb{Q}}$. Therefore,
${\mathcal{F}}(c_{1}({\mathcal{L}}))\in\textnormal{CH}_{1}(A)$ lies above
$\Gamma_{\Theta}=(-1)^{g-1}{\mathcal{F}}_{A}(\Theta)$ by the commutativity of
(11), and 5 holds.

Suppose that 5 holds. Then 1 follows readily from Lemma 3.5. Moreover, if 2
holds, then $\textnormal{ch}({\mathcal{P}}_{A})\in\textnormal{CH}(A\times
A)_{\mathbb{Q}}$ lifts to $\textnormal{CH}(A\times A)$, hence in particular 4
holds. Since we have already proved that 1 implies 2, we conclude that
$[\ref{motiviczero}\implies\ref{motivicfour}\implies\ref{motivicone}\implies\ref{motivictwo}\implies\ref{motiviczero}]$.

The implications
$[\ref{motivictwo}\implies\ref{motivicfive}\implies\ref{motivicsix}]$ are
trivial. Assume that 7 holds. By Equation (12),
$\Theta\in\textnormal{CH}^{1}(A)_{\mathbb{Q}}$ lifts to
$\textnormal{CH}^{1}(A)$, hence
${\mathcal{F}}_{A}(\Theta)=(-1)^{g-1}\cdot\Gamma_{\Theta}$ lifts to
$\textnormal{CH}_{1}(A)$, i.e. 5 holds.

Assume that 7 holds. The fact that
${\mathcal{F}}_{A}\left(\textnormal{CH}(A)/\textnormal{torsion}\right)\subset\textnormal{CH}(A)/\textnormal{torsion}$
implies that

$\textnormal{CH}(A)/\textnormal{torsion}={\mathcal{F}}_{A}\left({\mathcal{F}}_{A}\left(\textnormal{CH}(A)/\textnormal{torsion}\right)\right)\subset{\mathcal{F}}_{A}\left(\textnormal{CH}(A)/\textnormal{torsion}\right)\subset\textnormal{CH}(A)/\textnormal{torsion}.$

Thus the restriction of the Fourier transform ${\mathcal{F}}_{A}$ to
$\textnormal{CH}(A)/\textnormal{torsion}$ defines an isomorphism
${\mathcal{F}}_{A}\colon\textnormal{CH}(A)/\textnormal{torsion}\xrightarrow{\sim}\textnormal{CH}(A)/\textnormal{torsion}$.
Now if $R$ is a ring and $\gamma$ is a PD-structure on an ideal $I\subset R$,
then $\gamma$ extends to a PD-structure on $I/\textnormal{torsion}\subset
R/\textnormal{torsion}$. Consequently, the ideal
$\textnormal{CH}_{>0}(A)/\textnormal{torsion}\subset\textnormal{CH}(A)/\textnormal{torsion}$
admits a PD-structure for the Pontryagin product $\star$ by Theorem 3.7. Since
${\mathcal{F}}_{A}$ exchanges the Pontryagin and intersection product (up to a
sign, see [Bea83, Proposition 3(ii)]), it follows that 8 holds. Since 8
trivially implies 5, we are done. ∎

######  Question 3.9 (Moonen–Polishchuk [MP10], Totaro [Tot21]).

Let $A$ be any principally polarized abelian variety over $k=\bar{k}$. Are the
equivalent conditions in Theorem 3.8 satisfied for $A$?

######  Remark 3.10.

For Jacobians of hyperelliptic curves the answer to Question 3.9 is "yes"
[MP10].

Similarly, there is a relation between integral Fourier transforms up to
homology and the algebraicity of minimal cohomology classes induced by
Poincaré line bundles and theta divisors.

######  Proposition 3.11.

Let $A_{/k}$ be an abelian variety of dimension $g$. The following are
equivalent:

  1. 1.

The class $\rho_{A}\coloneqq
c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}_{\textnormal{\'{e}t}}((A\times{\widehat{A}})_{k_{s}},\mathbb{Z}_{\ell}(2g-1))$
lifts to $\textnormal{CH}_{1}(A\times{\widehat{A}})$.

  2. 2.

The abelian variety $A$ admits a motivic integral Fourier transform up to
homology.

  3. 3.

The abelian variety $A\times{\widehat{A}}$ admits a motivic integral Fourier
transform up to homology.

Moreover, if $A$ carries a symmetric ample line bundle that induces a
principal polarization $\lambda\colon A\xrightarrow{\sim}{\widehat{A}}$, then
the above statements are equivalent to the following equivalent statements:

  1. 4.

The class $\sigma_{A}\coloneqq
c_{1}({\mathcal{P}}_{A})^{2g-2}/(2g-2)!\in{\mathrm{H}}_{\textnormal{\'{e}t}}^{4g-4}((A\times{\widehat{A}})_{k_{s}},\mathbb{Z}_{\ell}(2g-2))$
lifts to $\textnormal{CH}_{2}(A\times{\widehat{A}})$.

  2. 5.

The class
$\gamma_{\theta}=\theta^{g-1}/(g-1)!\in{\mathrm{H}}^{2g-2}_{\textnormal{\'{e}t}}(A_{k_{s}},\mathbb{Z}_{\ell}(g-1))$
lifts to a cycle in $\textnormal{CH}_{1}(A)$.

  3. 6.

The abelian variety $A$ admits an integral Fourier transform up to homology.

###### Proof.

The proof of Theorem 3.8 can easily be adapted to this situation. ∎

######  Proposition 3.12.

  1. 1.

If $k=\mathbb{C}$, then each of the statements $\ref{motivicone-
new}-\ref{motivicfive-new}$ in Proposition 3.11 is equivalent to the same
statement with étale cohomology replaced by Betti cohomology.

  2. 2.

Proposition 3.11 remains valid if one replaces integral Chow groups in
statements 1, 4 and 5 by their tensor product with $\mathbb{Z}_{\ell}$ as well
as ‘integral Fourier transform up to homology’ by ‘$\ell$-adic integral
Fourier transform up to homology’ in statements 2, 3 and 6.

###### Proof.

  1. 1.

In this case $\mathbb{Z}_{\ell}(i)=\mathbb{Z}_{\ell}$ and the Artin comparison
isomorphism

${\mathrm{H}}^{2i}_{\textnormal{\'{e}t}}(A,\mathbb{Z}_{\ell})\xrightarrow{\sim}{\mathrm{H}}^{2i}(A(\mathbb{C}),\mathbb{Z}_{\ell})$

[AGV71, III, Exposé XI] is compatible with the cycle class map. Since the map
${\mathrm{H}}^{2i}(A(\mathbb{C}),\mathbb{Z})\to{\mathrm{H}}_{\textnormal{\'{e}t}}^{2i}(A,\mathbb{Z}_{\ell})$
is injective, a class $\beta\in{\mathrm{H}}^{2i}(A(\mathbb{C}),\mathbb{Z})$ is
in the image of
$cl\colon\textnormal{CH}^{i}(A)\to{\mathrm{H}}^{2i}(A(\mathbb{C}),\mathbb{Z})$
if and only if its image
$\beta_{\ell}\in{\mathrm{H}}^{2i}_{\textnormal{\'{e}t}}(A,\mathbb{Z}_{\ell})$
is in the image of
$cl\colon\textnormal{CH}^{i}(A)\to{\mathrm{H}}^{2i}_{\textnormal{\'{e}t}}(A,\mathbb{Z}_{\ell})$.

  2. 2.

Indeed, for an abelian variety $A$ over $k$, the PD-structure on
$\textnormal{CH}_{>0}(A)\subset(\textnormal{CH}(A),\star)$ induces a PD-
structure on
$\textnormal{CH}_{>0}(A)\otimes\mathbb{Z}_{\ell}\subset(\textnormal{CH}(A)_{\mathbb{Z}_{\ell}},\star)$
by [Stacks, Tag 07H1], because the ring map
$(\textnormal{CH}(A),\star)\to(\textnormal{CH}(A)_{\mathbb{Z}_{\ell}},\star)$
is flat. The latter follows from the flatness of
$\mathbb{Z}\to\mathbb{Z}_{\ell}$.

∎

##  4 The integral Hodge conjecture for one-cycles on complex abelian
varieties

In this section we use the theory developed in Section 3 to prove Theorem 1.1.
We also prove some applications of Theorem 1.1: the integral Hodge conjecture
for one-cycles on products of Jacobians (Theorem 1.2), the fact that the
integral Hodge conjecture for one-cycles on principally polarized complex
abelian varieties is stable under specialization (Corollary 4.3) and density
of polarized abelian varieties satisfying the integral Hodge conjecture for
one-cycles (Theorem 1.3).

###  4.1 Proof of the main theorem

Let us prove Theorem 1.1.

###### Proof of Theorem 1.1.

Suppose that 1 holds. Then 2 holds by Propositions 3.11 and 3.12.1. Suppose
that 2 holds. Then 4 follows from Lemma 3.3. So we have
$[\ref{introitem:minimalpoincare}\iff\ref{introitem:integralpoincare}\implies\ref{introitem:IHC}]$.
If 1 holds, then
$\rho_{A}=c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic, which implies that
$\rho_{{\widehat{A}}}\in{\mathrm{H}}^{4g-2}({\widehat{A}}\times A,\mathbb{Z})$
is algebraic. Therefore,
$\rho_{A\times{\widehat{A}}}\in{\mathrm{H}}^{8g-2}(A\times{\widehat{A}}\times{\widehat{A}}\times
A,\mathbb{Z})$ is algebraic by Equation (10). We then apply the implication
$[\ref{introitem:minimalpoincare}\implies\ref{introitem:IHC}]$ to the abelian
variety $A\times{\widehat{A}}$, which shows that 3 holds. Since
$[\ref{introitem:integralhodgeforproduct}\implies\ref{introitem:minimalpoincare}]$
is trivial, we have proven
$[\ref{introitem:minimalpoincare}\iff\ref{introitem:integralpoincare}\iff\ref{introitem:integralhodgeforproduct}\implies\ref{introitem:IHC}]$.

Next, assume that $A$ is principally polarized by
$\theta\in\textnormal{NS}(A)\subset{\mathrm{H}}^{2}(A,\mathbb{Z})$. The
directions $[\ref{introitem:IHC}\implies\ref{introitem:minimalclass}]$ and
$[\ref{introitem:integralpoincare}\implies\ref{introitem:minimalpoincare2}]$
are trivial and
$[\ref{introitem:minimalclass}\implies\ref{introitem:minimalpoincare}]$
follows from Propositions 3.11 and 3.12.1. We claim that 6 implies 4. Define
$\sigma_{A}=c_{1}({\mathcal{P}}_{A})^{2g-2}/(2g-2)!\in{\mathrm{H}}^{4g-4}(A\times{\widehat{A}},\mathbb{Z})$
and let $S\in\textnormal{CH}_{2}(A\times{\widehat{A}})$ be such that
$cl(S)=\sigma_{A}$. The squares in the following diagram commute:

$\textstyle{\textnormal{CH}^{1}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl}$$\scriptstyle{\pi_{1}^{\ast}}$$\textstyle{\textnormal{CH}^{1}(A\times{\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl}$$\scriptstyle{\cdot
S}$$\textstyle{\textnormal{CH}^{2g-1}(A\times{\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl}$$\scriptstyle{\pi_{2,\ast}}$$\textstyle{\textnormal{CH}_{1}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl}$$\textstyle{{\mathrm{H}}^{2}(A,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}^{\ast}}$$\textstyle{{\mathrm{H}}^{2}(A\times{\widehat{A}},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot\sigma_{A}}$$\textstyle{{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2,\ast}}$$\textstyle{{\mathrm{H}}^{2g-2}({\widehat{A}},\mathbb{Z}).}$
(13)

Since
${\mathscr{F}}_{A}=\pi_{2,\ast}\left(\textnormal{ch}({\mathcal{P}}_{A})\cdot\pi_{1}^{\ast}(-)\right)$
restricts to an isomorphism
${\mathscr{F}}_{A}\colon{\mathrm{H}}^{2}(A,\mathbb{Z})\xrightarrow{\sim}{\mathrm{H}}^{2g-2}({\widehat{A}},\mathbb{Z})$
by [Bea83, Proposition 1], the composition
$\pi_{2,\ast}\circ(-\cdot\sigma_{A})\circ\pi_{1}^{\ast}$ on the bottom row of
(13) is an isomorphism. Thus, by Lefschetz $(1,1)$,
$cl\colon\textnormal{CH}_{1}({\widehat{A}})\to\textnormal{Hdg}^{2g-2}({\widehat{A}},\mathbb{Z})$
is surjective.

It remains to prove the algebraicity of the classes
$\theta^{i}/i!\in{\mathrm{H}}^{2i}(A,\mathbb{Z})$. This follows from Theorem
3.7 together with the following equality, proven by Beauville [Bea83,
Corollaire 2]):

$\frac{\theta^{i}}{i!}=\frac{\gamma_{\theta}^{\star
j}}{j!},\quad\gamma_{\theta}=\frac{\theta^{g-1}}{(g-1)!}\in{\mathrm{H}}^{2g-2}(A,\mathbb{Z}),\quad
i+j=g.$

Therefore, the proof is finished. ∎

######  Corollary 4.1.

Let $A$ and $B$ be complex abelian varieties of respective dimensions $g_{A}$
and $g_{B}$.

  * •

The Hodge classes
$\rho_{A}\in{\mathrm{H}}^{4g_{A}-2}(A\times{\widehat{A}},\mathbb{Z})$ and
$\rho_{B}\in{\mathrm{H}}^{4g_{B}-2}(B\times{\widehat{B}},\mathbb{Z})$ are
algebraic if and only if $A\times{\widehat{A}}$, $B\times{\widehat{B}}$,
$A\times B$ and ${\widehat{A}}\times{\widehat{B}}$ satisfy the integral Hodge
conjecture for one-cycles.

  * •

If $A$ and $B$ are principally polarized, then the integral Hodge conjecture
for one-cycles holds for $A\times B$ if and only if it holds for $A$ and $B$.

###### Proof.

The first statement follows from Theorem 1.1 and Equation (10). The second
statement follows from the fact that the minimal cohomology class of the
product $A\times B$ is algebraic if and only if the minimal cohomology classes
of the factors $A$ and $B$ are both algebraic. ∎

###### Proof of Theorem 1.2.

By Corollary 4.1 we may assume $n=1$, so let $C$ be a smooth projective curve.
Let $p\in C$ and consider the morphism $\iota\colon C\to J(C)$ defined by
sending a point $q$ to the isomorphism class of the degree zero line bundle
$\mathcal{O}(p-q)$. Then
$cl(\iota(C))=\gamma_{\theta}\in{\mathrm{H}}^{2g-2}(J(C),\mathbb{Z})$ by
Poincaré’s formula [Arb+85], so $\gamma_{\theta}$ is algebraic and the result
follows from Theorem 1.1. ∎

######  Remarks 4.2.

  1. 1.

Let us give another proof of Theorem 1.2 in the case $n=1$, i.e. let $C$ be a
smooth projective curve of genus $g$ and let us prove the integral Hodge
conjecture for one-cycles on $J(C)$ in a way that does not use Fourier
transforms. It is classical that any Abel-Jacobi map $C^{(g)}\to J(C)$ is
birational. On the other hand, the integral Hodge conjecture for one-cycles is
a birational invariant, see [Voi07, Lemma 15]. Therefore, to prove it for
$J(C)$ it suffices to prove it for $C^{(g)}$. One then uses [Bn02, Corollary
5] which says that for each $n\in\mathbb{Z}_{\geq 1}$, there is a natural
polarization $\eta$ on the $n$-fold symmetric product $C^{(n)}$ such that for
any $i\in\mathbb{Z}_{\geq 0}$, the map
$\eta^{n-i}\cup(-)\colon{\mathrm{H}}^{i}(C^{(n)},\mathbb{Z})\to{\mathrm{H}}^{2n-i}(C^{(n)},\mathbb{Z})$
is an isomorphism. In particular, the variety $C^{(n)}$ satisfies the integral
Hodge conjecture for one-cycles for any positive integer $n$.

  2. 2.

Along these lines, observe that the integral Hodge conjecture for one-cycles
holds not only for symmetric products of smooth projective complex curves but
also for any product $C_{1}\times\dotsc\times C_{n}$ of smooth projective
curves $C_{i}$ over $\mathbb{C}$. Indeed, this follows readily from the
Künneth formula.

  3. 3.

Let $C$ be a smooth projective complex curve of genus $g$. Our proof of
Theorem 1.1 provides an explicit description of
$\textnormal{Hdg}^{2g-2}(J(C),\mathbb{Z})$ depending on
$\textnormal{Hdg}^{2}(J(C),\mathbb{Z})$. More generally, let $(A,\theta)$ be a
principally polarized abelian variety of dimension $g$, identify $A$ and
${\widehat{A}}$ via the polarization, and let
$\ell=c_{1}({\mathcal{P}}_{A})\in{\mathrm{H}}^{2}(A\times{\widehat{A}},\mathbb{Z})$.
Then $\ell=m^{\ast}(\theta)-\pi_{1}^{\ast}(\theta)-\pi_{2}^{\ast}(\theta)$,
which implies that

$\displaystyle\sigma_{A}=\frac{\ell^{2g-2}}{(2g-2)!}$
$\displaystyle=\sum_{\begin{subarray}{c}i,j,k\geq 0\\\
i+j+k=2g-2\end{subarray}}^{2g-2}(-1)^{j+k}\cdot
m^{\ast}\left(\frac{\theta^{i}}{i!}\right)\cdot\pi_{1}^{\ast}\left(\frac{\theta^{j}}{j!}\right)\cdot\pi_{2}^{\ast}\left(\frac{\theta^{k}}{k!}\right).$

On the other hand, any $\beta\in\textnormal{Hdg}^{2g-2}(A,\mathbb{Z})$ is of
the form $\pi_{2,\ast}\left(\sigma_{A}\cdot\pi_{1}^{\ast}[D]\right)$, where
$[D]=cl(D)$ for a divisor $D$ on $A$, as follows from (13). Therefore, any
$\beta\in\textnormal{Hdg}^{2g-2}(A,\mathbb{Z})$ may be written as

$\beta=\sum_{\begin{subarray}{c}i,j,k\geq 0\\\
i+j+k=2g-2\end{subarray}}^{2g-2}(-1)^{j+k}\cdot\pi_{2,\ast}\left(m^{\ast}\left(\frac{\theta^{i}}{i!}\right)\cdot\pi_{1}^{\ast}\left(\frac{\theta^{j}}{j!}\right)\cdot\pi_{1}^{\ast}[D]\right)\cdot\frac{\theta^{k}}{k!}.$
(14)

Returning to the case of a Jacobian $J(C)$ of a smooth projective curve $C$ of
genus $g$, the classes $\theta^{i}/i!$ appearing in (14) are effective
algebraic cycle classes. Indeed, for $p\in C$ and $d\in\mathbb{Z}_{\geq 1}$,
the image of the morphism $C^{d}\to J(C)$,
$(x_{i})\mapsto{\mathcal{O}}(\sum_{i}x_{i}-d\cdot p)$ defines a subvariety
$W_{d}(C)\subset J(C)$ and by Poincaré’s formula [Arb+85, §I.5] one has
$cl(W_{d}(C))=\theta^{g-d}/(g-d)!\in{\mathrm{H}}^{2g-2d}(J(C),\mathbb{Z})$.

Besides Theorem 1.2, we obtain the following corollary of Theorem 1.1:

######  Corollary 4.3.

Let $A\to S$ be a principally polarized abelian scheme over a proper, smooth
and connected variety $S$ over $\mathbb{C}$. Let $X\subset S(\mathbb{C})$ be
the set of $x\in S(\mathbb{C})$ such that the abelian variety $A_{x}$
satisfies the integral Hodge conjecture for one-cycles. Then
$X=\cup_{i}Z_{i}(\mathbb{C})$ for some countable union of closed algebraic
subvarieties $Z_{i}\subset S$. In particular, if the integral Hodge conjecture
for one-cycles holds on $U(\mathbb{C})$ for a non-empty open subscheme $U$ of
$S$, then it holds on all of $S(\mathbb{C})$.

###### Proof.

Write ${\mathcal{A}}=A(\mathbb{C})$ and $B=S(\mathbb{C})$ and let
$\pi\colon{\mathcal{A}}\to B$ be the induced family of complex abelian
varieties. Let $g\in\mathbb{Z}_{\geq 0}$ be the relative dimension of $\pi$
and define, for $t\in S(\mathbb{C})$,
$\theta_{t}\in\textnormal{NS}({\mathcal{A}}_{t})\subset{\mathrm{H}}^{2}({\mathcal{A}}_{t},\mathbb{Z})$
to be the polarization of ${\mathcal{A}}_{t}$. There is a global section
$\gamma_{\theta}\in{\mathrm{R}}^{2g-2}\pi_{\ast}\mathbb{Z}$ such that for each
$t\in B$,
$\gamma_{\theta_{t}}=\theta_{t}^{g-1}/(g-1)!\in{\mathrm{H}}^{2g-2}({\mathcal{A}}_{t},\mathbb{Z})$.
Note that $\gamma_{\theta}$ is Hodge everywhere on $B$. For those $t\in B$ for
which $\gamma_{\theta_{t}}$ is algebraic, write $\gamma_{\theta_{t}}$ as the
difference of effective algebraic cycle classes on ${\mathcal{A}}_{t}$. This
gives a countable disjoint union $\phi\colon\sqcup_{ij}H_{i}\times_{S}H_{j}\to
S$ of products of relative Hilbert schemes $H_{i}\to S$. By Lemma 4.4 below,
$\gamma_{\theta_{t}}$ is algebraic precisely for closed points $t$ in the
image $Y\subset S$ of $\phi$. Theorem 1.1 implies that $X=Y$ and the assertion
is proven. ∎

######  Lemma 4.4.

Let $S$ be an integral variety over $\mathbb{C}$, let ${\mathcal{A}}\to S$ be
a principally polarized abelian scheme of relative dimension $g$ over $S$ and
let ${\mathcal{C}}_{i}\subset{\mathcal{A}}$ for $i=1,\dotsc,k$ be relative
curves in ${\mathcal{A}}$ over $S$. Let $n_{1},\dotsc,n_{k}$ be integers and
let $y\in S(\mathbb{C})$ be a point that satisfies $\sum_{i=1}^{k}n_{i}\cdot
cl(C_{i,y})=\gamma_{\theta_{y}}\in{\mathrm{H}}^{2g-2}(A_{y},\mathbb{Z})$. Then
for every $x\in S(\mathbb{C})$, one has $\sum_{i=1}^{k}n_{i}\cdot
cl(C_{i,x})=\gamma_{\theta_{x}}\in{\mathrm{H}}^{2g-2}(A_{x},\mathbb{Z})$.

###### Proof.

Since it suffices to prove the lemma for any open affine $U\subset S$ that
contains $y$, we may assume that $S$ is quasi-projective. Fix $x\in
S(\mathbb{C})$. After replacing $S$ by a suitable base change containing $x$
and $y$, we may assume that $S$ is an open subscheme of a smooth connected
curve. For $t\in S$, denote by
$\theta_{\bar{t}}\in{\mathrm{H}}^{2}_{\textnormal{\'{e}t}}(A_{\bar{t}},\mathbb{Z}_{\ell})$
the class of the polarization and
$\gamma_{\theta_{\bar{t}}}=\theta_{\bar{t}}^{g-1}/(g-1)!$. Let
$\eta=\textnormal{Spec }K$ be the generic point of $S$. The elements
$\sum_{i}n_{i}\cdot cl(C_{i,\bar{\eta}})$ and $\gamma_{\theta_{\bar{\eta}}}$
in
${\mathrm{H}}^{2g-2}_{\textnormal{\'{e}t}}(A_{\bar{\eta}},\mathbb{Z}_{\ell})$
both map to $\sum_{i}n_{i}\cdot
cl(C_{i,y})=\gamma_{\theta_{y}}\in{\mathrm{H}}^{2g-2}_{\textnormal{\'{e}t}}(A_{y},\mathbb{Z}_{\ell})$
under the specialization homomorphism
$s\colon{\mathrm{H}}^{2g-2}_{\textnormal{\'{e}t}}(A_{\bar{\eta}},\mathbb{Z}_{\ell})\to{\mathrm{H}}^{2g-2}_{\textnormal{\'{e}t}}(A_{y},\mathbb{Z}_{\ell})$
by [Ful98, Example 20.3.5]. Since $s$ is an isomorphism, we have
$\sum_{i}n_{i}\cdot cl(C_{i,\bar{\eta}})=\gamma_{\theta_{\bar{\eta}}}$, which
implies that $\sum_{i}n_{i}\cdot
cl(C_{x,i})=\gamma_{\theta_{x}}\in{\mathrm{H}}_{\textnormal{\'{e}t}}^{2g-2}(A_{x},\mathbb{Z}_{\ell})$.
∎

###  4.2 Density of abelian varieties satisfying IHC1

The goal of this section is to prove that Conditions
$\ref{introitem:minimalpoincare}-\ref{introitem:integralhodgeforproduct}$ in
Theorem 1.1 are satisfied on a dense subset of the moduli space of complex
abelian varieties. To do so, will we state yet another criterion that a
complex abelian variety may satisfy. In some sense this criterion provides a
bridge between abelian varieties outside the Torelli locus and those lying
within, thereby implying the integral Hodge conjecture for one-cycles for the
abelian variety under consideration.

######  Definition 4.5.

Let $A$ and $B$ be a complex abelian varieties and let $p$ a prime number. We
say that $A$ is prime-to-$p$ isogenous to a $B$ if there exists an isogeny
$\alpha\colon A\to B$ whose degree $\deg(\alpha)$ is not divisible by $p$. We
say that $A$ is $p$-power isogenous to $B$ if $A$ is isogenous to $B$ for some
isogeny $\alpha$ whose degree is a power of $p$.

The following proposition shows in particular that to prove the density part
of the statement in Theorem 1.3, it suffices to prove that for any prime
number $\ell$, those abelian varieties that are $\ell$-power isogenous to a
product of elliptic curves are dense in their moduli space.

######  Proposition 4.6.

Let $A$ be a complex abelian variety of dimension $g$. Let ${\widehat{A}}$ be
the dual abelian variety and let ${\mathcal{P}}_{A}$ be the Poincaré bundle.
Let $\kappa$ be a non-zero integer such that the cohomology class $\kappa\cdot
c_{1}({\mathcal{P}}_{A})/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic. Consider the following statements:

  1. 1.

The abelian variety $A$ satisfies the integral Hodge conjecture for one-
cycles.

  2. 2.

For every prime number $p$, there exists an abelian variety $B$ such that the
abelian variety $A\times B$ is prime-to-$p$ isogenous to the Jacobian of a
smooth projective curve.

  3. 3.

For every prime number $p$ that divides $\kappa$, there exists an abelian
variety $B$ such that the abelian variety $A\times B$ is prime-to-$p$
isogenous to a Jacobian of a smooth projective curve.

  4. 4.

For every prime number $p$, there exists an abelian variety $B$ such that the
abelian variety $A\times B$ is prime-to-$p$ isogenous to a product of
Jacobians of smooth projective curves.

  5. 5.

For every prime number $p$ dividing $\kappa$, there exists an abelian variety
$B$ such that the abelian variety $A\times B$ is prime-to-$p$ isogenous to a
product of Jacobians of smooth projective curves.

The
$[\ref{itemlemma:two}\implies\ref{itemlemma:twosmall}\implies\ref{itemlemma:threesmall}\implies\ref{itemlemma:one}]$
and
$[\ref{itemlemma:two}\implies\ref{itemlemma:three}\implies\ref{itemlemma:threesmall}]$.
Moreover, if $A$ is principally polarized by
$\theta_{A}\in\textnormal{NS}(A)$, then 1 is implied by

  1. 6.

For any prime number $p|(g-1)!$ there exists a smooth projective curve $C$ and
a morphism of abelian varieties $\phi\colon A\to J(C)$ such that
$\phi^{\ast}\theta_{J(C)}=m\cdot\theta_{A}$ for $m\in\mathbb{Z}_{\geq 1}$ with
$\gcd(m,p)=1$.

Finally, if $A$ is principally polarized of Picard rank one, then the
statements $\ref{itemlemma:one}-\ref{itemlemma:zero}$ are equivalent.

###### Proof.

Step one:
$[\ref{itemlemma:two}\implies\ref{itemlemma:twosmall}\implies\ref{itemlemma:threesmall}]$
and
$[\ref{itemlemma:two}\implies\ref{itemlemma:three}\implies\ref{itemlemma:threesmall}]$.
These implications are trivial.

Step two: $[\ref{itemlemma:threesmall}\implies\ref{itemlemma:one}]$. Let $g$
be the dimension of $A$. We want to prove that the class
$c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic. Let $p$ be any prime number that divides $\kappa$. Then by
Condition 5, there exists an abelian variety $B$ and an isogeny $\alpha\colon
A\times B\to Y$ to the product $Y=\prod_{i}J(C_{i})$ of Jacobians $J(C_{i})$
of smooth projective curves $C_{i}$ such that $\gcd(\deg(\alpha),p)=1$. Define
$X=A\times B$. Let $g_{B}$ be the dimension of $B$, let
$h=g+g_{B}=\dim(X)=\dim(Y)$, and let $m_{p}=\deg(\alpha)$. There exists an
isogeny $\beta\colon Y\to X$ such that $\beta\circ\alpha=[m_{p}]_{X}$. If we
define $n_{p}=\deg(\beta)$ then $m_{p}\cdot
n_{p}=\deg(\alpha)\cdot\deg(\beta)=\deg(\alpha\circ\beta)=m_{p}^{2h}$.
Therefore,
$(\beta\circ\alpha)\times({\widehat{\alpha}}\circ{\widehat{\beta}})=[m_{p}]_{X\times{\widehat{X}}}$.
Consequently, if $N_{p}=2h\cdot(4h-2)$, then the homomorphism
$[m_{p}^{2h}]^{\ast}=(m_{p}^{N_{p}}\cdot(-))\colon{\mathrm{H}}^{4h-2}(X\times{\widehat{X}},\mathbb{Z})\to{\mathrm{H}}^{4h-2}(X\times{\widehat{X}},\mathbb{Z})$
factors through ${\mathrm{H}}^{4h-2}(Y\times{\widehat{Y}},\mathbb{Z})$. Since
$Y\times{\widehat{Y}}$ satisfies the integral Hodge conjecture by Theorem 1.2,
the Hodge class $m_{p}^{N_{p}}\cdot
c_{1}(P_{X})^{2h-1}/(2h-1)!\in{\mathrm{H}}^{4h-2}(X\times{\widehat{X}},\mathbb{Z})$
is algebraic. Let $f\colon A\times B\times{\widehat{A}}\times{\widehat{B}}\to
A\times{\widehat{A}}$ and $g\colon A\times
B\times{\widehat{A}}\times{\widehat{B}}\to B\times{\widehat{B}}$ be the
canonical projections. Then ${\mathcal{P}}_{X}\cong
f^{\ast}{\mathcal{P}}_{A}\otimes g^{\ast}{\mathcal{P}}_{B}$. Using this and
denoting $\mu=c_{1}({\mathcal{P}}_{A})$ and $\nu=c_{1}({\mathcal{P}}_{B})$ we
have

$\frac{c_{1}({\mathcal{P}}_{X})^{2h-1}}{(2h-1)!}=f^{\ast}\left(\frac{\mu^{2g-1}}{(2g-1)!}\right)\cdot
g^{\ast}\left(\frac{\nu^{2g_{B}}}{(2g_{B})!}\right)+f^{\ast}\left(\frac{\mu^{2g}}{(2g)!}\right)\cdot
g^{\ast}\left(\frac{\nu^{2g_{B}-1}}{(2g_{B}-1)!}\right).$

This implies that
$f_{\ast}\left(c_{1}({\mathcal{P}}_{X})^{2h-1}/(2h-1)!\right)=(-1)^{g_{b}}\mu^{2g-1}/(2g-1)!$.
In particular, the class $m_{p}^{N_{p}}\cdot
c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic.

Let $p_{1},\dotsc,p_{n}$ be all prime divisors of $\kappa$ and observe that
$\gcd(\kappa,m_{p_{1}}^{N_{p_{1}}},m_{p_{2}}^{N_{p_{2}}},\dotsc,m_{p_{n}}^{N_{p_{n}}})=1$.
Therefore, there are integers $a,b_{1},\dotsc,b_{n}$ such that
$a\cdot\kappa+\sum_{i=1}^{n}b_{i}\cdot m_{p_{i}}^{N_{p_{i}}}=1$. One obtains

$\frac{c_{1}({\mathcal{P}}_{A})^{2g-1}}{(2g-1)!}=a\cdot\kappa\cdot\frac{c_{1}({\mathcal{P}}_{A})^{2g-1}}{(2g-1)!}+\sum_{i=1}^{n}b_{i}\cdot
m_{p_{i}}^{N_{p_{i}}}\cdot\frac{c_{1}({\mathcal{P}}_{A})^{2g-1}}{(2g-1)!}\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z}).$

This proves that $c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!$ is a
$\mathbb{Z}$-linear combination of algebraic classes, hence algebraic.
Condition 1 follows then from Theorem 1.1.

Step three: [$\ref{itemlemma:zero}\implies\ref{itemlemma:one}]$ for $A$
principally polarized by $\theta_{A}\in\textnormal{NS}(A)$. Let
$p_{1},\dotsc,p_{k}$ be the prime factors of $(g-1)!$ and let
$C_{1},\dotsc,C_{k}$ be smooth proper curves for which there exist
homomorphisms $\phi_{i}\colon A\to J(C_{i})$ such that
$\phi^{\ast}\theta_{J(C_{i})}=m_{i}\cdot\theta_{A}$ for some
$m_{i}\in\mathbb{Z}_{\geq 1}$ with $p_{i}\nmid m$. Since
$\theta_{J(C_{i})}^{g-1}/(g-1)!\in{\mathrm{H}}^{2g-2}(J(C_{i}),\mathbb{Z})$ is
algebraic for each $i$, the classes
$\phi_{i}^{\ast}(\theta_{J(C_{i})}^{g-1}/(g-1)!)=m_{i}^{g-1}\cdot\theta_{A}^{g-1}/(g-1)!\in{\mathrm{H}}^{2g-2}(A,\mathbb{Z})$
are algebraic. Since $\gcd((g-1)!,m_{1},\dotsc,m_{k})=1$, this implies that
$\theta_{A}^{g-1}/(g-1)!$ is algebraic. Condition 1 follows then from Theorem
1.1.

Step four:
[$\ref{itemlemma:zero}\impliedby\ref{itemlemma:one}\implies\ref{itemlemma:two}]$
for $(A,\theta_{A})$ principally polarized with $\rho(A)=1$. Write
$\theta=\theta_{A}$. Let $Z_{1},\dotsc,Z_{n}$ be integral curves $Z_{i}\subset
A$ and let $e_{1},\dotsc,e_{n}\in\mathbb{Z}$ with $e_{i}\neq 0$ for all $i$ be
such that
${\theta^{g-1}}/{(g-1)!}=\sum_{i=1}^{n}e_{i}\cdot[Z_{i}]\in{\mathrm{H}}^{2g-2}(A,\mathbb{Z})$.
Since $\rho(A)=1$, the group $\textnormal{Hdg}^{2g-2}(A,\mathbb{Z})$ is
generated by $\theta^{g-1}/(g-1)!$. Consequently, we have
$[Z_{i}]=f_{i}\cdot\left(\theta^{g-1}/(g-1)!\right)$ for some non-zero
$f_{i}\in\mathbb{Z}$. Hence we can write
${\theta^{g-1}}/{(g-1)!}=\sum_{i=1}^{n}e_{i}\cdot[Z_{i}]=\sum_{i=1}^{n}e_{i}\cdot
f_{i}\cdot\theta^{g-1}/(g-1)!$, which implies $\sum_{i=1}^{n}e_{i}\cdot
f_{i}=1$. Now let $p$ be any prime number. Then there exists an integer $i$
with $1\leq i\leq n$ such that $p$ does not divide $f_{i}$. Let $C_{i}\to
Z_{i}$ be the normalization of $Z_{i}$ and let
$\lambda_{A}=\varphi_{\theta}\colon A\to{\widehat{A}}$ be the polarization
corresponding to $\theta$. This gives a diagram

$\textstyle{C_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{\iota}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{\sim}$$\scriptstyle{\lambda_{A}}$$\textstyle{{\widehat{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{\ast}}$$\textstyle{\textnormal{Pic}^{0}(C_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{\sim}$$\textstyle{J(C_{i}),}$$\textstyle{J(C_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$
(15)

where $\iota\colon C_{i}\to
J(C_{i})=H^{0}(C,\Omega_{C})^{\ast}/H_{1}(C,\mathbb{Z})$ is the Abel–Jacobi
map (for some $p\in C$), and
$\varphi^{\ast}\colon{\widehat{A}}=\textnormal{Pic}^{0}(A)\to\textnormal{Pic}^{0}(C_{i})$
is the pullback of line bundles along $\varphi\colon C_{i}\to A$. The natural
homomorphism $a\colon\textnormal{Pic}^{0}(C_{i})\to J(C_{i})$ is an
isomorphism by the Abel–Jacobi theorem. Since the triangle on the left in
Diagram (15) commutes and $[Z_{i}]\in{\mathrm{H}}^{2g-2}(A,\mathbb{Z})$ is
non-zero, the morphism $\psi\colon J(C_{i})\to A$ is non-zero. As $\rho(A)=1$,
the map $\psi\colon J(C_{i})\to A$ must be surjective, the Picard rank of a
non-simple abelian variety being greater than one. Dually, $\psi$ gives rise
to a non-zero homomorphism
${\widehat{\psi}}\colon{\widehat{A}}\to{\widehat{J(C_{i})}}$, and the
simpleness of ${\widehat{A}}$ implies that ${\widehat{\psi}}$ is finite onto
its image. We claim that the same is true for $\phi$. To prove this, it
suffices to show that the kernel of
$\varphi^{\ast}\colon{\widehat{A}}\to\textnormal{Pic}^{0}(C_{i})$ is finite.
Since the homomorphism
$\iota^{\ast}\colon{\widehat{J(C_{i})}}\to\textnormal{Pic}^{0}(C_{i})$ induced
by the embedding $\iota\colon C_{i}\to J(C_{i})$ is an isomorphism, dualizing
the triangle on the left in Diagram (15) proves our claim. By construction, we
have
$\varphi_{\ast}[C_{i}]=[Z_{i}]=f_{i}\cdot\theta^{g-1}/(g-1)!\in{\mathrm{H}}^{2g-2}(A,\mathbb{Z})$.
By a version of Welters’ Criterion (see [BL04, Lemma 12.2.3]), this implies
that
$\phi^{\ast}\left(\theta_{J(C_{i})}\right)=f_{i}\cdot\theta\in{\mathrm{H}}^{2}(A,\mathbb{Z})$,
where $\theta_{J(C_{i})}\in{\mathrm{H}}^{2}(J(C_{i}),\mathbb{Z})$ is the
canonical principal polarization. In particular, 6 holds.

We claim that also 2 holds. Let $j\colon A_{0}\hookrightarrow J(C_{i})$ be the
embedding of $A_{0}=\phi(A)$ into $J(C_{i})$ and let $\lambda_{0}\colon
A_{0}\to{\widehat{A}}_{0}$ be the polarization on $A_{0}$ induced by $j$. We
have
$\phi^{\ast}(\lambda)=\varphi_{f_{i}\cdot\theta}=f_{i}\cdot\varphi_{\theta}=f_{i}\cdot\lambda_{A}$.
We obtain a commutative diagram

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[f_{i}]_{A}}$$\scriptstyle{\pi}$$\scriptstyle{f_{i}\cdot\lambda_{A}}$$\textstyle{A_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{0}}$$\scriptstyle{j}$$\textstyle{J(C_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\textstyle{A}$$\textstyle{{\widehat{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{{\widehat{A}}}}$$\textstyle{{\widehat{A}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\widehat{\pi}}}$$\textstyle{\widehat{J(C_{i})}.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\widehat{j}}}$

Let $G$ be the kernel of $\pi$. Define
$K=\textnormal{Ker}([f_{i}]_{A})=\textnormal{Ker}(f_{i}\cdot\lambda_{A})\cong(\mathbb{Z}/f_{i})^{2g}\subset
A$, and $U=\textnormal{Ker}({\widehat{\pi}}\circ\lambda_{0})\subset A_{0}$.
Also define $H=\textnormal{Ker}(\lambda_{0})$, and observe that $H\subset U$.
The exact sequence $0\to G\to K\to U\to 0$ shows that if $a,k,u$ and $h$ are
the respective orders of $G$, $K$, $U$ and $H$, then one has

$h|u|k|f_{i}\quad\textnormal{ and }\quad a|k|f_{i}.$ (16)

Then define $B=\textnormal{Ker}({\widehat{j}}\circ\lambda)\subset J(C_{i})$
with inclusion $i\colon B\hookrightarrow J(C_{i})$. It is easy to see that $B$
is connected. Moreover, we have $A_{0}\cap B=H$ and, therefore, an exact
sequence of commutative group schemes

$0\to H\to A_{0}\times B\xrightarrow{\psi}J(C_{i})\to 0.$

The morphism $\alpha\colon A\times B\to J(C_{i})$, defined as the composition
$\textstyle{A\times
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi\times\textnormal{id}}$$\textstyle{A_{0}\times
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{J(C_{i}),}$
is an isogeny. Since the degree of an isogeny is multiplicative in
compositions, we have
$\deg(\alpha)=\deg\left(\psi\circ(\pi\times\textnormal{id})\right)=\deg(\psi)\cdot\deg(\pi\times\textnormal{id})=h\cdot\deg(\pi)=h\cdot
a$. In particular, $p$ does not divide $\deg(\alpha)$ because $h$ and $a$
divide $f_{i}$ by Equation (16). ∎

###### Proof of Theorem 1.3.

According to Theorem 1.1, it suffices to show that the cohomology class
$c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic for $[(A,\lambda)]$ in a dense subset $X$ of
${\mathsf{A}}_{g,\delta}(\mathbb{C})$ as in the statement. Define
$D=\textnormal{diag}(\delta_{1},\dotsc,\delta_{g})$ and define, for each
subring $R$ of $\mathbb{C}$, a group

$\textnormal{Sp}_{2g}^{\delta}(R)=\left\\{M\in\textnormal{GL}_{2g}(R)\mid
M\begin{pmatrix}0&D\\\ -D&0\end{pmatrix}M^{t}=\begin{pmatrix}0&D\\\
-D&0\end{pmatrix}\right\\}.$

The isomorphism

$\textnormal{Sp}_{2g}^{\delta}(\mathbb{R})\to\textnormal{Sp}_{2g}(\mathbb{R}),\quad
M\mapsto\begin{pmatrix}1_{g}&0\\\
0&D\end{pmatrix}^{-1}M\begin{pmatrix}1_{g}&0\\\ 0&D\end{pmatrix}$

induces an action of $\textnormal{Sp}_{2g}^{\delta}(\mathbb{Z})$ on the genus
$g$ Siegel space ${\mathbb{H}}_{g}$, and the period map defines an isomorphism
of complex analytic spaces
${\mathsf{A}}_{g,\delta}(\mathbb{C})\cong\textnormal{Sp}_{2g}^{\delta}(\mathbb{Z})\setminus{\mathbb{H}}_{g}$
[BL04, Theorem 8.2.6]. Pick any prime number $\ell>(2g-1)!$ and consider, for
a period matrix $x\in{\mathbb{H}}_{g}$, the orbit
$\textnormal{Sp}_{2g}^{\delta}(\mathbb{Z}[1/\ell])\cdot
x\subset{\mathbb{H}}_{g}$. Let $(A,\lambda)$ be a polarized abelian variety
admitting a period matrix equal to $x$. The image of
$\textnormal{Sp}_{2g}^{\delta}(\mathbb{Z}[1/\ell])\cdot x$ in
${\mathsf{A}}_{g,\delta}(\mathbb{C})$ is the Hecke-$\ell$-orbit of
$[(A,\lambda)]\in{\mathsf{A}}_{g,\delta}(\mathbb{C})$, i.e. the set of
isomorphism classes of polarized abelian varieties
$[(B,\mu)]\in{\mathsf{A}}_{g,\delta}(\mathbb{C})$ for which there exists
integers $n,m\in\mathbb{Z}_{\geq 0}$ and an isomorphism of polarized rational
Hodge structures
$\phi\colon{\mathrm{H}}_{1}(B,\mathbb{Q})\xrightarrow{\sim}{\mathrm{H}}_{1}(A,\mathbb{Q})$
such that $\ell^{n}\cdot\phi$ and $\ell^{m}\cdot\phi^{-1}$ are morphisms of
integral Hodge structures (Hecke orbits were studied in positive
characteristic in e.g. [Cha95, CO19]). The degree of the isogeny
$\alpha=\ell^{n}\phi$ must be $\ell^{k}$ for some nonnegative integer $k$. In
particular, if one abelian variety in a Hecke-$\ell$-orbit happens to be
isomorphic to a Jacobian, then every abelian variety in that orbit is
$\ell$-power isogenous to a Jacobian, see Definition 4.5.

The decomposition of a polarized abelian variety into non-decomposable
polarized abelian subvarieties is unique [Deb96, Corollaire 2], which implies
that the following morphism

$\pi\colon\prod_{i=1}^{g}{\mathsf{A}}_{1,1}\to{\mathsf{A}}_{g,\delta},\quad\left([(E_{1},\lambda_{1})],\dotsc,[(E_{g},\lambda_{g})]\right)\mapsto([E_{1}\times\cdots\times
E_{g},\delta_{1}\cdot\lambda_{1}\times\cdots\times\delta_{g}\cdot\lambda_{g})]$

is finite onto its image. Thus ${\mathsf{A}}_{g,\delta}$ contains a
$g$-dimensional subvariety on which the integral Hodge conjecture for one-
cycles holds. We claim that
$\textnormal{Sp}_{2g}^{\delta}(\mathbb{Z}[1/\ell])$ is dense in
$\textnormal{Sp}_{2g}(\mathbb{R})$. Since
$\textnormal{Sp}_{2g}^{\delta}(\mathbb{Q})$ arises as the group of rational
points of an algebraic subgroup $\textnormal{Sp}_{2g}^{\delta}$ of
$\textnormal{GL}_{2g}$ over $\mathbb{Q}$ [PR94, Chapter 2, §2.3.2], which is
isomorphic to $\textnormal{Sp}_{2g}$ over $\mathbb{Q}$, this claim follows
from the well-known fact that for $S=\\{\ell\\}\subset\textnormal{Spec
}\mathbb{Z}$, the algebraic group $\textnormal{Sp}_{2g}$ satisfies the strong
approximation property with respect to $S$ [PR94, Chapter 7, §7.1] (indeed,
this is classical and follows from the non-compactness of
$\textnormal{Sp}_{2g}(\mathbb{Q}_{\ell})$, see [PR94, Theorem 7.12]).

Let
$V=\pi\left(\prod_{i=1}^{g}{\mathsf{A}}_{1,1}\right)\subset{\mathsf{A}}_{g,\delta}$.
Then
$X^{\prime}\coloneqq\textnormal{Sp}_{2g}^{\delta}(\mathbb{Z}[1/\ell])\cdot
V=\cup_{i}Z_{i}\subset{\mathsf{A}}_{g,\delta}(\mathbb{C})$ is a countable
union of closed analytic subsets
$Z_{i}\subset{\mathsf{A}}_{g,\delta}(\mathbb{C})$ of dimension $\dim Z_{i}\geq
g$ such that $X^{\prime}\subset{\mathsf{A}}_{g,\delta}(\mathbb{C})$ is dense
in the analytic topology and
$c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})$
is algebraic for every polarized abelian variety $(A,\lambda)$ of polarization
type $\delta$ whose isomorphism class lies in $X^{\prime}$. To prove the
theorem, we are reduced to proving that there exists a similar countable union
$X\subset{\mathsf{A}}_{g,\delta}(\mathbb{C})$ whose components are algebraic.
For this, it suffices to prove the following claim: the locus of
$[(A,\lambda)]\in{\mathsf{A}}_{g,\delta}(\mathbb{C})$ such that
$c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})_{\textnormal{alg}}$
is a countable union
$W=\cup_{j}Y_{j}\subset{\mathsf{A}}_{g,\delta}(\mathbb{C})$ of closed
algebraic subsets $Y_{j}\subset{\mathsf{A}}_{g,\delta}(\mathbb{C})$. Indeed,
if this holds, then $X^{\prime}\subset W$ and since each $Z_{i}\subset X$ is
irreducible, each $Z_{i}$ is contained in an irreducible component
$Y_{j}\subset W$. We may then define $X$ as the union of those $Y_{j}\subset
W$ that contain some $Z_{i}$.

To prove the claim, let $U\to{\mathcal{A}}_{g,\delta}$ be a finite étale cover
of the moduli stack ${\mathcal{A}}_{g,\delta}$ and let ${\mathcal{X}}\to U$ be
the pullback of the universal family of abelian varieties along
$U\to{\mathcal{A}}_{g,\delta}$. This gives an abelian scheme
${\mathcal{X}}\times{\widehat{{\mathcal{X}}}}\to U$ carrying a relative
Poincaré line bundle ${\mathcal{P}}_{{\mathcal{X}}/U}$ and arguments similar
to those used to prove Lemma 4.4 show that indeed, for each irreducible
component $U^{\prime}\subset U$, the locus in $U^{\prime}(\mathbb{C})$ where
$c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!$ is algebraic is a countable union of
closed algebraic subvarieties of $U^{\prime}(\mathbb{C})$.

Finally, Theorem 1.1 implies that for each $[(A,\lambda)]\in X$, the integral
Hodge conjecture for one-cycles holds for the abelian variety $A$, so we are
done. ∎

######  Remark 4.7.

Using level structures one can show that whenever
$\gcd(\prod_{i}\delta_{i},(2g-1)!)=1$ (or, more generally,
$\gcd(\prod_{i}\delta_{i},(2g-2)!)=1$, see Section 5 below), there is a
countable union $X=\cup_{i}Z_{i}\subset{\mathsf{A}}_{g,\delta}(\mathbb{C})$ as
in Theorem 1.3 such that $\dim Z_{i}\geq 3g-3$. Indeed, let
${\mathsf{A}}_{g,\delta_{g}}^{\ast}$ be the moduli space of principally
polarized abelian varieties of dimension $g$ with $\delta_{g}$-level
structure. Then there is a natural morphism
$\phi\colon{\mathsf{A}}_{g,\delta_{g}}^{\ast}\to{\mathsf{A}}_{g,\delta}$ such
that for any
$x=[(A,\lambda)]\in{\mathsf{A}}_{g,\delta_{g}}^{\ast}(\mathbb{C})$ with
$[(B,\mu)]=\phi(x)\in{\mathsf{A}}_{g,\delta}(\mathbb{C})$, there exists an
isogeny $\alpha\colon A\to B$ of degree $\prod_{i=1}^{g}\delta_{i}$, see
[Mum71].

######  Remark 4.8.

In the principally polarized case, the density in the moduli space of those
abelian varieties that satisfy the integral Hodge conjecture for one-cycles
admits another proof which might be interesting for comparison. Let
${\mathsf{A}}_{g}$ be the coarse moduli space of principally polarized complex
abelian varieties of dimension $g$ and let $[(A,\theta)]$ be a closed point of
${\mathsf{A}}_{g}$. Then by [BL04, Exercise 5.6.(10)], the following are
equivalent: (i) $A$ is isogenous to the $g$-fold self-product $E^{g}$ for an
elliptic curve $E$ with complex multiplication, (ii) $A$ has maximal Picard
rank $\rho(A)=g^{2}$, (iii) $A$ is isomorphic to the product
$E_{1}\times\cdots\times E_{g}$ of pairwise isogenous elliptic curves $E_{i}$
with complex multiplication. If any of these conditions is satisfied, then $A$
satisfies the integral Hodge conjecture for one-cycles by Theorem 1.2.
Moreover, the set of isomorphism classes of principally polarized abelian
varieties $(A,\theta)$ for which this holds is dense in ${\mathsf{A}}_{g}$ by
[Lan75]. For an explicit example in dimension $g=4$ of a principally polarized
abelian variety $(A,\theta)$ that satisfies one of the equivalent conditions
above, but which is not isomorphic to a Jacobian, see [Deb87, §5].

##  5 The integral Hodge conjecture for one-cycles up to factor $n$

In this section, we study a property of a smooth projective complex variety
that lies somewhere in between the integral Hodge conjecture and the usual
(i.e. rational) Hodge conjecture. The key will be the following:

######  Definition 5.1.

Let $d,k,n\in\mathbb{Z}_{\geq 1}$ and let $X$ be a smooth projective variety
over $\mathbb{C}$ of dimension $d$. Recall the definition of the degree
$2d-2k$ Voisin group of $X$ [Voi16, Per20]:

${\mathrm{Z}}^{2d-2k}(X)\coloneqq\textnormal{Hdg}^{2d-2k}(X,\mathbb{Z})/{\mathrm{H}}^{2d-2k}(X,\mathbb{Z})_{\textnormal{alg}}=\textnormal{Coker}\left(\textnormal{CH}_{k}(X)\to\textnormal{Hdg}^{2d-2k}(X,\mathbb{Z})\right).$

We say that $X$ satisfies the integral Hodge conjecture for $k$-cycles up to
factor $n$ if ${\mathrm{Z}}^{2d-2k}(X)$ is annihilated by $n$ (in other words,
if $n\cdot x\in{\mathrm{H}}^{2d-2k}(X,\mathbb{Z})_{\textnormal{alg}}$ for
every $x\in\textnormal{Hdg}^{2d-2k}(X,\mathbb{Z})$).

######  Lemma 5.2.

Let $A$ be a complex abelian variety of dimension $g$. Define
$\sigma_{A}\in{\mathrm{H}}^{4g-4}(A\times{\widehat{A}},\mathbb{Z})$ to be the
class $c_{1}({\mathcal{P}}_{A})^{2g-2}/(2g-2)!$.

  1. 1.

Let $n$ be a positive integer and let
${\mathcal{F}}_{n}\colon\textnormal{CH}^{1}({\widehat{A}})\to\textnormal{CH}_{1}(A)$
be a group homomorphism such that the following diagram commutes:

$\textstyle{{\mathrm{\textnormal{CH}}}^{1}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathcal{F}}_{n}}$$\textstyle{\textnormal{CH}_{1}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{H}}^{2}({\widehat{A}},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{n\cdot{\mathscr{F}}_{{\widehat{A}}}}$$\textstyle{{\mathrm{H}}^{2g-2}(A,\mathbb{Z}).}$

Then $A$ satisfies the integral Hodge conjecture for one-cycles up to factor
$n$.

  2. 2.

Let $n\in\mathbb{Z}_{\geq 1}$ be such that $n\cdot\sigma_{A}$ is algebraic.
Then a homomorphism ${\mathcal{F}}_{n}$ as in 1 exists.

###### Proof.

Statement 1 follows immediately from the fact that
$\textnormal{CH}^{1}({\widehat{A}})\to\textnormal{Hdg}^{2}({\widehat{A}},\mathbb{Z})$
is surjective by Lefschetz $(1,1)$. To prove 2, first observe that if
$\sigma_{{\widehat{A}}}\coloneqq
c_{1}({\mathcal{P}}_{{\widehat{A}}})^{2g-2}/(2g-2)!\in{\mathrm{H}}^{4g-4}({\widehat{A}}\times
A,\mathbb{Z})$, then $n\cdot\sigma_{{\widehat{A}}}$ is algebraic since
$n\cdot\sigma_{A}$ is. Let
$\Sigma_{n}\in\textnormal{CH}_{2}({\widehat{A}}\times A)$ be such that
$cl(\Sigma_{n})=n\cdot\sigma_{{\widehat{A}}}$. This gives a commutative
diagram:

$\textstyle{\textnormal{CH}^{1}({\widehat{A}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl}$$\scriptstyle{\pi_{1}^{\ast}}$$\textstyle{\textnormal{CH}^{1}({\widehat{A}}\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl}$$\scriptstyle{\cdot\Sigma_{n}}$$\textstyle{\textnormal{CH}^{2g-1}({\widehat{A}}\times
A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl}$$\scriptstyle{\pi_{2,\ast}}$$\textstyle{\textnormal{CH}_{1}(A)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{cl}$$\textstyle{{\mathrm{H}}^{2}({\widehat{A}},\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}^{\ast}}$$\textstyle{{\mathrm{H}}^{2}({\widehat{A}}\times
A,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cdot
n\cdot\sigma_{{\widehat{A}}}}$$\textstyle{{\mathrm{H}}^{4g-2}({\widehat{A}}\times
A,\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2,\ast}}$$\textstyle{{\mathrm{H}}^{2g-2}(A,\mathbb{Z}).}$

Since $\pi_{2,\ast}\circ\left((-)\cdot
n\cdot\sigma_{{\widehat{A}}}\right)\circ\pi_{1}^{\ast}=n\cdot{\mathscr{F}}_{{\widehat{A}}}$,
the homomorphism
${\mathcal{F}}_{n}\coloneqq\pi_{2,\ast}\circ\left((-)\cdot\Sigma_{n}\right)\cdot\pi_{1}^{\ast}$
has the required property. ∎

######  Theorem 5.3.

Consider a complex abelian variety $A$ of dimension $g$. Define the cycle
$\sigma_{A}\in{\mathrm{H}}^{4g-4}(A\times{\widehat{A}},\mathbb{Z})$ as before
and define
$\rho_{A}=c_{1}({\mathcal{P}}_{A})^{2g-1}/(2g-1)!\in{\mathrm{H}}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})$.

  1. 1.

Let $n\in\mathbb{Z}_{\geq 1}$ be such that $n\cdot\rho_{A}$ is algebraic. Then
$n^{2}\cdot\sigma_{A}$ is algebraic. In particular, $A$ satisfies the integral
Hodge conjecture up to factor $\gcd(n^{2},(2g-2)!)$ in this case.

  2. 2.

If $A$ is principally polarized, and $n\in\mathbb{Z}_{\geq 1}$ is such that
$n\cdot\gamma_{\theta}\in\textnormal{Hdg}^{2g-2}(A,\mathbb{Z})$ is algebraic,
then
$n\cdot\rho_{A}\in\textnormal{Hdg}^{4g-2}(A\times{\widehat{A}},\mathbb{Z})$ is
algebraic.

  3. 3.

We have that $A$ satisfies the integral Hodge conjecture for one-cycles up to
factor $(2g-2)!$, and Prym varieties satisfy the integral Hodge conjecture for
one-cycles up to factor $4$.

###### Proof.

1. By Lemma 3.4, one has

$\sigma_{A}=c_{1}({\mathcal{P}}_{A})^{2g-2}/(2g-2)!=(-1)^{g}\cdot\left(\rho_{A}\right)^{\star
2}/2!\in{\mathrm{H}}^{4g-4}(A\times{\widehat{A}},\mathbb{Z}).$

By Theorem 3.7, this implies that if $n\cdot\rho_{A}$ is algebraic, then
$n^{2}\cdot\sigma_{A}$ is algebraic. Since $(2g-2)!\cdot\sigma_{A}$ is
algebraic, it follows that $\gcd(n^{2},(2g-2)!)\cdot\sigma_{A}$ is algebraic.
Thus we are done by Lemma 5.2.

2. This follows from Lemma 3.5.

3. This follows from Lemma 5.2, parts 1 and 2 and the fact that if $A$ is a $g$-dimensional Prym variety with principal polarization $\theta\in\textnormal{Hdg}^{2}(A,\mathbb{Z})$, then $2\cdot\gamma_{\theta}\in{\mathrm{H}}^{2g-2}(A,\mathbb{Z})$ is algebraic. ∎

##  6 The integral Tate conjecture for one-cycles on abelian varieties over
the separable closure of a finitely generated field

Let $X$ be a smooth projective variety over the separable closure $k$ of a
finitely generated field. Let $k_{0}$ be a finitely generated field of
definition of $X$. A class
$u\in{\mathrm{H}}_{\textnormal{\'{e}t}}^{2i}(X,\mathbb{Z}_{\ell}(i))$ is an
integral Tate class if it is fixed by some open subgroup of
$\textnormal{Gal}(k/k_{0})$. Totaro has shown that for codimension-one cycles
on $X$, the Tate conjecture over $k$ implies the integral Tate conjecture over
$k$ [Tot21, Lemma 6.2]. This means that every integral Tate class is the class
of an algebraic cycle over $k$ with $\mathbb{Z}_{\ell}$-coefficients.

Suppose that $A_{/k}$ is an abelian variety, defined over a finitely generated
field $k_{0}\subset k$ such that $k$ is the separable closure of $k_{0}$. Then
the Tate conjecture for codimension-one cycles holds for $A$ over $k$ by
results of Tate [Tat66], Faltings [Fal83, Fal+86], and Zarhin [Zar74a, Zar74].
By the above, $A$ satisfies the integral Tate conjecture for codimension-one
cycles over $k$. On the other hand, the Fourier transform defines an
isomorphism

${\mathscr{F}}_{A}\colon{\mathrm{H}}_{\textnormal{\'{e}t}}^{2}(A,\mathbb{Z}_{\ell}(1))\xrightarrow{\sim}{\mathrm{H}}_{\textnormal{\'{e}t}}^{2g-2}({\widehat{A}},\mathbb{Z}_{\ell}(g-1)),$
(17)

see [Tot21, Section 7]. Since (17) is Galois-equivariant (the Poincaré bundle
being defined over $k_{0}$) it sends integral Tate classes to integral Tate
classes. Therefore, to prove the integral Tate conjecture for one-cycles on
$A$, it suffices to lift (17) to a homomorphism
$\textnormal{CH}^{1}(A)_{\mathbb{Z}_{\ell}}\to\textnormal{CH}_{1}({\widehat{A}})_{\mathbb{Z}_{\ell}}$.

###### Proof of Theorem 1.6.

This follows from the above together with Proposition 3.12.2. ∎

######  Corollary 6.1.

Let $A$ and $B$ be abelian varieties defined over the separable closure $k$ of
a finitely generated field, of respective dimensions $g_{A}$ and $g_{B}$.

  1. 1.

The classes
$\rho_{A}\in{\mathrm{H}}_{\textnormal{\'{e}t}}^{4g_{A}-2}(A\times{\widehat{A}},\mathbb{Z}_{\ell}(2g_{A}-1))$
and
$\rho_{B}\in{\mathrm{H}}_{\textnormal{\'{e}t}}^{4g_{B}-2}(B\times{\widehat{B}},\mathbb{Z}_{\ell}(2g_{A}-1))$
are algebraic if and only if $A\times{\widehat{A}}$, $B\times{\widehat{B}}$,
$A\times B$ and ${\widehat{A}}\times{\widehat{B}}$ satisfy the integral Tate
conjecture for one-cycles.

  2. 2.

If $A$ and $B$ are principally polarized, then the integral Tate conjecture
for one-cycles holds for $A\times B$ if and only if it holds for both $A$ and
$B$.

  3. 3.

If $\theta\in{\mathrm{H}}^{2}_{\textnormal{\'{e}t}}(A,\mathbb{Z}_{\ell}(1))$
is the first Chern class of an ample line bundle that induces a principal
polarization on $A$, and if the minimal class
$\gamma_{\theta}=\theta^{g-1}/(g-1)!\in{\mathrm{H}}_{\textnormal{\'{e}t}}^{2g-2}(A,\mathbb{Z}_{\ell}(g-1))$
is algebraic, then $\theta^{i}/i!$ in
${\mathrm{H}}_{\textnormal{\'{e}t}}^{2i}(A,\mathbb{Z}_{\ell}(i))$ is algebraic
for each $i\in\mathbb{Z}_{\geq 1}$.

###### Proof.

1. See Equation (10).

2. This is true because the minimal cohomology class of the product is algebraic if and only if the minimal cohomology classes of the factors are algebraic.

3. One has $\theta^{i}/i!=\gamma_{\theta}^{\star(g-i)}/(g-i)!$ by [Bea83, Corollaire 2]. ∎

Combining Theorems 1.1 and 1.6, we obtain:

######  Corollary 6.2.

Let $A_{K}$ be a principally polarized abelian variety over a number field
$K\subset\mathbb{C}$ and let $A_{\mathbb{C}}$ be its base change to
$\mathbb{C}$. Then $A_{\mathbb{C}}$ satisfies the integral Hodge conjecture
for one-cycles if and only if $A_{\bar{K}}$ satisfies the integral Tate
conjecture for one-cycles over $\bar{K}=\bar{\mathbb{Q}}$.

###### Proof.

We view $\bar{\mathbb{Q}}$ as a subfield of $\mathbb{C}$ in a way compatible
with the inclusion $K\hookrightarrow\mathbb{C}$. For a prime number $\ell$,
let
$\theta_{\ell}=c_{1}({\mathcal{L}})\in{\mathrm{H}}_{\textnormal{\'{e}t}}^{2}(A_{\bar{\mathbb{Q}}},\mathbb{Z}_{\ell}(1))$
be the $\ell$-adic étale cohomology class of ${\mathcal{L}}$. On the other
hand, define
$\theta_{\mathbb{C}}\in\textnormal{NS}(A)\subset{\mathrm{H}}^{2}(A_{\mathbb{C}},\mathbb{Z})$
to be the polarization of the complex abelian variety $A_{\mathbb{C}}$. By
Theorems 1.1 and 1.6, it suffices to show that
$\gamma_{\theta_{\mathbb{C}}}\in{\mathrm{H}}^{2g-2}(A_{\mathbb{C}},\mathbb{Z})$
is algebraic if and only if
$\gamma_{\theta_{\ell}}\in{\mathrm{H}}_{\textnormal{\'{e}t}}^{2g-2}(A_{\bar{\mathbb{Q}}},\mathbb{Z}_{\ell}(g-1))$
is in the image of (1) for each prime number $\ell$. We have, by definition,
that
${\mathrm{Z}}^{2g-2}(A)=\textnormal{Coker}\left(\textnormal{CH}_{1}(A_{\mathbb{C}})\to{\mathrm{H}}^{2g-2}(A_{\mathbb{C}},\mathbb{Z})\right)$.
This implies that
${\mathrm{Z}}^{2g-2}(A)\otimes\mathbb{Z}_{\ell}=\textnormal{Coker}\left(\textnormal{CH}_{1}(A_{\mathbb{C}})_{\mathbb{Z}_{\ell}}\to{\mathrm{H}}^{2g-2}(A_{\mathbb{C}},\mathbb{Z}_{\ell})\right)$.

Suppose that $\gamma_{\theta_{\ell}}$ is in the image of (1) for every prime
number $\ell$. Then ${\mathrm{Z}}^{2g-2}(A)\otimes\mathbb{Z}_{\ell}=0$ for
each prime number $\ell$ by Theorem 1.6, which means that
${\mathrm{Z}}^{2g-2}(A)=0$. Conversely, suppose that
$\gamma_{\theta}=\sum_{i=1}^{k}n_{i}\cdot cl(C_{i})$ for some smooth
projective curves $C_{i}$ over $\mathbb{C}$. The Hilbert scheme
${\mathcal{H}}=\textnormal{Hilb}_{A_{K}/K}$ is defined over $K$; for each
$i=1,\dotsc,k$ we pick a $\bar{\mathbb{Q}}$-point in the connected component
of ${\mathcal{H}}$ containing $[C_{i}\subset A]$. This gives smooth projective
curves $C_{i}^{\prime}\subset A_{\bar{\mathbb{Q}}}$ over $\bar{\mathbb{Q}}$
and we define
$\Gamma=\sum_{i}n_{i}\cdot[C_{i}^{\prime}]\in\textnormal{CH}_{1}(A_{\bar{\mathbb{Q}}})$.
On the one hand, we have
$cl(\Gamma_{\mathbb{C}})=\gamma_{\theta_{\mathbb{C}}}$ by Lemma 4.4. On the
other hand, the Artin comparison theorem gives an isomorphism of
$\mathbb{Z}_{\ell}$-algebras

$\phi\colon{\mathrm{H}}^{\bullet}_{\textnormal{\'{e}t}}(A_{\bar{\mathbb{Q}}},\mathbb{Z}_{\ell})={\mathrm{H}}^{\bullet}_{\textnormal{\'{e}t}}(A_{\mathbb{C}},\mathbb{Z}_{\ell})\cong{\mathrm{H}}^{\bullet}(A_{\mathbb{C}},\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{Z}_{\ell}.$

Since $\phi$ is compatible with the cycle class maps
$\textnormal{CH}(A_{\bar{\mathbb{Q}}})\to{\mathrm{H}}^{\bullet}_{\textnormal{\'{e}t}}(A_{\bar{\mathbb{Q}}},\mathbb{Z}_{\ell})$
and
$\textnormal{CH}(A_{\mathbb{C}})\to{\mathrm{H}}^{\bullet}(A_{\mathbb{C}},\mathbb{Z})$,
we have $\phi(\gamma_{\theta_{\ell}})=\gamma_{\theta_{\mathbb{C}}}$ and
$\phi(cl(\Gamma))=cl(\Gamma_{\mathbb{C}})=\gamma_{\theta_{\mathbb{C}}}$.
Therefore, $cl(\Gamma)=\gamma_{\theta_{\ell}}$. ∎

Another corollary of Theorem 1.6 is that the integral Tate conjecture for one-
cycles on abelian varieties is stable under specialization. For example, one
has the following (c.f. Corollary 4.3):

######  Corollary 6.3.

Let $A_{K}$ be a principally polarized abelian variety over a number field $K$
and suppose that $A_{\bar{K}}$ satisfies the integral Tate conjecture for one-
cycles over $\bar{K}$. Let ${\mathfrak{p}}$ be a prime ideal of the ring of
integers $\mathcal{O}_{K}$ of $K$ at which $A_{K}$ has good reduction and
write $\kappa=\mathcal{O}_{K}/{\mathfrak{p}}$. Then the abelian variety
$A_{\bar{\kappa}}$ over $\bar{\kappa}$ satisfies the integral Tate conjecture
for one-cycles over $\bar{\kappa}$.

###### Proof.

Write $S=\textnormal{Spec }\mathcal{O}_{K}$ and let $A\to S$ be the Néron
model of $A_{K}$. Let $R$ (resp. $K_{{\mathfrak{p}}}$) be the completion of
$\mathcal{O}_{K}$ (resp. $K$) at the prime ${\mathfrak{p}}$. The natural
composition $K\to K_{{\mathfrak{p}}}\to\bar{K}_{{\mathfrak{p}}}$ induces an
embedding $\bar{K}\to\bar{K}_{{\mathfrak{p}}}$, where
$\bar{K}_{{\mathfrak{p}}}$ is an algebraic closure of $K_{{\mathfrak{p}}}$.
This gives a commutative diagram, where the square on the right is provided in
[Ful98, Example 20.3.5]:

$\textstyle{\textnormal{CH}(A_{\bar{K}})_{\mathbb{Z}_{\ell}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textnormal{CH}(A_{\bar{K}_{{\mathfrak{p}}}})_{\mathbb{Z}_{\ell}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textnormal{CH}(A_{\bar{\kappa}})_{\mathbb{Z}_{\ell}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\oplus_{r\geq
0}{\mathrm{H}}^{2r}_{\textnormal{\'{e}t}}(A_{\bar{K}},\mathbb{Z}_{\ell}(r))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\oplus_{r\geq
0}{\mathrm{H}}^{2r}_{\textnormal{\'{e}t}}(A_{\bar{K}_{{\mathfrak{p}}}},\mathbb{Z}_{\ell}(r))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\oplus_{r\geq
0}{\mathrm{H}}^{2r}_{\textnormal{\'{e}t}}(A_{\bar{\kappa}},\mathbb{Z}_{\ell}(r)).}$
(18)

Now the principal polarization $\lambda_{K}\colon
A_{K}\xrightarrow{\sim}{\widehat{A}}_{K}$ extends uniquely to a homomorphism
$\lambda\colon A\to{\widehat{A}}$ by the Néron mapping property [BLR90,
Section 1.2, Definition 1] and since the same is true for the inverse
$\lambda_{K}^{-1}\colon{\widehat{A}}_{K}\xrightarrow{\sim}A_{K}$ we find that
$\lambda$ is an isomorphism. In particular, we see that $A_{\bar{\kappa}}$ is
principally polarized and that the class in
$\textnormal{CH}^{1}(A_{\bar{K}})_{\mathbb{Z}_{\ell}}$ of a theta divisor on
$A_{\bar{K}}$ is sent to the class in
$\textnormal{CH}^{1}(A_{\bar{\kappa}})_{\mathbb{Z}_{\ell}}$ of a theta divisor
on $A_{\bar{\kappa}}$. Thus, the minimal class
$\gamma_{\theta_{\bar{K}}}\in{\mathrm{H}}^{2g-2}_{\textnormal{\'{e}t}}(A_{\bar{K}},\mathbb{Z}_{\ell}(g-1))$
is sent to the minimal class
$\gamma_{\theta_{\bar{\kappa}}}\in{\mathrm{H}}^{2g-2}_{\textnormal{\'{e}t}}(A_{\bar{\kappa}},\mathbb{Z}_{\ell}(g-1))$
by the isomorphism on the bottom of Diagram (18). It follows that
$\gamma_{\theta_{\bar{\kappa}}}$ is algebraic which by Theorem 1.6 means that
we are done. ∎

Finally, let us prove Theorem 1.7. The theorem follows from Theorem 1.6
together with a result of Chai on the density of an ordinary isogeny class in
positive characteristic [Cha95].

###### Proof of Theorem 1.7.

For any $t\in A_{g}(k)$, let $(A_{t},\lambda_{t})$ be a principally polarized
abelian variety such that $[(A_{t},\lambda_{t})]=t$. Let
$A=E_{1}\times\cdots\times E_{g}$ be the product of $g$ ordinary elliptic
curves $E_{i}$ over $k$ and provide $A$ with its natural principal
polarization. Let $x\in{\mathsf{A}}_{g}(k)$ be the point corresponding to the
isomorphism class of $A$. Let $q>(g-1)!$ be a prime number different from $p$
and let ${\mathscr{G}}_{q}(x)\subset{\mathsf{A}}_{g}(k)$ be the set of
isomorphism classes $y=[(A_{y},\lambda_{y})]$ that admit an isogeny
$\phi\colon A_{y}\to A_{x}$ with
$\phi^{\ast}\lambda_{x}=q^{N}\cdot\lambda_{y}$ for some nonnegative integer
$N$. We claim that $A_{y}$ satisfies the integral Tate conjecture for one-
cycles over $k$ for any $y\in{\mathscr{G}}_{q}(x)$. Indeed, for such $y$ there
exists a nonnegative integer $N$ such that the isogeny $[q^{N}]\colon A_{y}\to
A_{y}$ factors through $A_{x}$. Consequently, $q^{(2g-2)\cdot
N}\cdot\gamma_{\theta}$ is algebraic for the first Chern class $\theta$ of the
principal polarization on $A_{y}$, which implies that $\gamma_{\theta}$ is
algebraic (as $q>(g-1)!$). Thus, the claim follows from Theorem 1.6. Now
${\mathscr{G}}_{q}(z)$ is dense in ${\mathsf{A}}_{g}$ for any ordinary
principally polarized abelian variety $(A_{z},\lambda_{z})$ by a result of
Chai [Cha95, Theorem 2]. Therefore, ${\mathscr{G}}_{q}(x)$ is dense in
${\mathsf{A}}_{g}$ and the proof is finished. ∎

## References

  * [AGV71] M. Artin, A. Grothendieck and J.-L. Verdier  “Theorie de Topos et Cohomologie Etale des Schemas I, II, III” 269, 270, 305, Lecture Notes in Mathematics  Springer, 1971
  * [AH62] M. Atiyah and F. Hirzebruch  “Analytic cycles on complex manifolds”  In _Topology_ 1, 1962, pp. 25–45
  * [Arb+85] E. Arbarello, M. Cornalba, P. Griffiths and J. Harris  “Geometry of Algebraic Curves: Volume I”, Geometry of Algebraic Curves  Springer, 1985
  * [Bea83] A. Beauville  “Quelques remarques sur la transformation de Fourier dans l’anneau de Chow d’une variété abélienne”  In _Algebraic geometry_ 1016, Lecture Notes in Math.  Springer, Berlin, 1983, pp. 238–260
  * [Bea86] A. Beauville  “Sur l’anneau de Chow d’une variété abélienne”  In _Math. Ann._ 273.4, 1986, pp. 647–651
  * [BL04] C. Birkenhake and H. Lange  “Complex Abelian Varieties” 302, Grundlehren der Mathematischen Wissenschaften  Springer-Verlag, Berlin, 2004
  * [BLR90] S. Bosch, W. Lütkebohmert and M. Raynaud  “Néron Models”, Ergebnisse der Mathematik und ihrer Grenzgebiete 21  Springer-Verlag, 1990
  * [Bn02] S. Baño  “On the Hodge theory of the symmetric powers of a curve”  In _Publ. Mat._ 46.1, 2002, pp. 17–25
  * [Cha95] C.-L. Chai  “Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli”  In _Invent. Math._ 121.3, 1995, pp. 439–479
  * [CO19] C.-L. Chai and F. Oort  “The Hecke orbit conjecture: a survey and outlook”  In _Open problems in arithmetic algebraic geometry_ 46, Adv. Lect. Math. (ALM)  Int. Press, Somerville, MA, 2019, pp. 235–262
  * [CP15] F. Charles and A. Pirutka  “La conjecture de Tate entière pour les cubiques de dimension quatre”  In _Compos. Math._ 151.2, 2015, pp. 253–264
  * [CTV12] J.-L. Colliot-Thélène and C. Voisin  “Cohomologie non ramifiée et conjecture de Hodge entière”  In _Duke Math. J._ 161.5, 2012, pp. 735–801
  * [Deb87] O. Debarre  “Annulation de thêtaconstantes sur les variétés abéliennes de dimension quatre”  In _C. R. Acad. Sci. Paris Sér. I Math._ 305.20, 1987
  * [Deb96] O. Debarre  “Polarisations sur les variétés abéliennes produits”  In _C. R. Acad. Sci. Paris Sér. I Math._ 323.6, 1996, pp. 631–635
  * [DM91] C. Deninger and J. Murre  “Motivic decomposition of abelian schemes and the Fourier transform”  In _J. Reine Angew. Math._ 422, 1991
  * [Esn04] H. Esnault  “Some elementary theorems about divisibility of $0$-cycles on abelian varieties defined over finite fields”  In _Int. Math. Res. Not._ , 2004
  * [Fal83] G. Faltings  “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern.”  In _Invent. Math._ 73, 1983, pp. 349–366
  * [Fal+86] G. Faltings et al.  “Rational points” Papers from the seminar held at the Max-Planck-Institut für Mathematik, 1983/1984, Aspects of Mathematics, E6  Friedr. Vieweg & Sohn, Braunschweig, 1986
  * [Ful98] W. Fulton  “Intersection theory”  Springer-Verlag, Berlin, 1998
  * [Gra04] C. Grabowski  “On the integral Hodge conjecture for 3-folds” Thesis (Ph.D.)–Duke University  ProQuest LLC, Ann Arbor, MI, 2004
  * [Huy06] D. Huybrechts  “Fourier–Mukai transforms in algebraic geometry”, Oxford Mathematical Monographs  Oxford University Press, Oxford, 2006
  * [Lan75] H. Lange  “Modulprobleme exzeptioneller abelscher Varietäten”  In _Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II_ , 1975, pp. 35–42
  * [MP10] B. Moonen and A. Polishchuk  “Divided powers in Chow rings and integral Fourier transforms”  In _Adv. Math._ 224.5, 2010, pp. 2216–2236
  * [Muk81] S. Mukai  “Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves”  In _Nagoya Math. J._ 81, 1981, pp. 153–175
  * [Mum71] D. Mumford  “The structure of the moduli spaces of curves and Abelian varieties”  In _Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1_ , 1971, pp. 457–465
  * [Orl97] D. O. Orlov  “Equivalences of derived categories and $K3$ surfaces” Algebraic geometry, 7  In _J. Math. Sci. (New York)_ 84.5, 1997, pp. 1361–1381
  * [Per20] A. Perry  “The integral Hodge conjecture for two-dimensional Calabi-Yau categories”, 2020  arXiv:2004.03163 [math.AG]
  * [PR94] V. Platonov and A. Rapinchuk  “Algebraic Groups and Number Theory” 139, Pure and applied mathematics  Academic Press Inc., 1994
  * [Sch21] S. Schreieder  “Refined unramified homology of schemes”, 2021  arXiv:2010.05814 [math.AG]
  * [Stacks] The Stacks Project Authors  “Stacks Project”, https://stacks.math.columbia.edu, 2018
  * [Tat66] J. Tate  “Endomorphisms of abelian varieties over finite fields”  In _Invent. Math._ 2, 1966, pp. 134–144
  * [Tot21] B. Totaro  “The integral Hodge conjecture for 3-folds of Kodaira dimension zero”  In _J. Inst. Math. Jussieu_ 20.5, 2021, pp. 1697–1717
  * [Tot97] B. Totaro  “Torsion algebraic cycles and complex cobordism”  In _J. Amer. Math. Soc._ 10.2, 1997, pp. 467–493
  * [Tre] “Trento examples”  In _Classification of irregular varieties (Trento, 1990)_ 1515, Lecture Notes in Math.  Springer, Berlin, 1992, pp. 134–139
  * [Voi06] C. Voisin  “On integral Hodge classes on uniruled or Calabi-Yau threefolds”  In _Moduli spaces and arithmetic geometry_ 45, Adv. Stud. Pure Math.  Math. Soc. Japan, Tokyo, 2006, pp. 43–73
  * [Voi07] C. Voisin  “Some aspects of the Hodge conjecture”  In _Jpn. J. Math._ 2.2, 2007, pp. 261–296
  * [Voi16] C. Voisin  “Stable birational invariants and the Lüroth problem”  In _Surveys in differential geometry 2016. Advances in geometry and mathematical physics_ 21, Surv. Differ. Geom.  Int. Press, Somerville, MA, 2016
  * [Zar74] Y. Zarhin  “A remark on endomorphisms of abelian varieties over function fields of finite characteristic”  In _Izv. Akad. Nauk SSSR Ser. Mat._ 38, 1974
  * [Zar74a] Y. Zarhin  “Isogenies of abelian varieties over fields of finite characteristic”  In _Mat. Sb. (N.S.)_ 95(137), 1974, pp. 461–470, 472