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

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 projective curve over the complex numbers 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 the Jacobian of a smooth projective curve over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture 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 ${\widehat {A}}$ (see [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 and 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 $\iota _\ast \colon {{\rm H}}_{2}(C, \mathbb {Z}) \to {\rm H}_{2}(A,\mathbb {Z}) \cong {\rm H}^{2g-2}(A, \mathbb {Z})$ defines a cohomology class $[C] \in {\rm H}^{2g-2}(A, \mathbb {Z})$. This construction extends to one-cycles and factors modulo rational equivalence. As such, it induces a canonical homomorphism, called the cycle class map,

\[ cl\colon \text{CH}_1(A) \to \text{Hdg}^{2g-2}(A, \mathbb{Z}), \]

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

The liftability of the Fourier transform turns out to have important consequences for the image of the cycle class map. Recall that an element $\alpha \in {\rm H}^{\bullet }(A, \mathbb {Z})$ 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 ${\rm H}^{2g-2k}(A,\mathbb {Z})$ are algebraic. Although the integral Hodge conjecture fails in general [AH62, Tre92, Tot97], it is an open question for abelian varieties. Our main result is as follows.

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. (i) The cohomology class $c_1(\mathcal P_A)^{2g-1}/(2g-1)! \in {\rm H}^{4g-2}(A \times {\widehat {A}}, \mathbb {Z})$ is algebraic.

2. (ii) The Chern character $\text {ch}(\mathcal P_A) = \exp (c_1(\mathcal P_A)) \in {\rm H}^\bullet (A \times {\widehat {A}}, \mathbb {Z})$ is algebraic.

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

Any of these statements implies the following.

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

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

1. (v) The minimal cohomology class $\gamma _\theta := \theta ^{g-1}/(g-1)! \in {\rm H}^{2g-2}(A, \mathbb {Z})$ is algebraic.

2. (vi) The cohomology class $c_1(\mathcal P_A)^{2g-2}/(2g-2)! \in {\rm H}^{4g-4}(A \times {\widehat {A}}, \mathbb {Z})$ is algebraic.

3. (vii) For every algebraic cohomology class $\alpha \in {\rm H}^{> 0}(A, \mathbb {Z})$ and every $i \in \mathbb {Z}_{\geqslant 1}$, the cohomology class $\alpha ^i/i! \in {\rm H}^{\bullet }(A, \mathbb {Z})$ is algebraic.

Then statements (i)–(vii) are equivalent.

We remark that condition (v) 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(i) 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 § 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 $g\geqslant 2$, 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, cf. Remark 4.6.

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 $i \geqslant 0$ be an integer. Poincaré duality induces a canonical isomorphism

\[ \varphi\colon {\rm H}^{2i}(A, \mathbb{Z}) \cong {\rm H}^{2g-2i}(A, \mathbb{Z})^\vee \cong {\rm H}^{2g-2i}({\widehat{A}}, \mathbb{Z}). \]

The map $\varphi$ respects the Hodge structures and, thus, induces an isomorphism $\text {Hdg}^{2i}(A, \mathbb {Z}) \cong \text {Hdg}^{2g-2i}({\widehat {A}}, \mathbb {Z})$. 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 $\Gamma \in \text {CH}(A \times {\widehat {A}})$. In particular, one then has

\[ {\rm Z}^{2i}(A) := \text{Hdg}^{2i}(A, \mathbb{Z}) / {\rm H}^{2i}(A, \mathbb{Z})_{\text{alg}} \cong {\rm Z}^{2g-2i}({\widehat{A}}). \]

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 and Polishchuk stating 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 § 5, we consider an abelian variety $A_{/\mathbb {C}}$ of dimension $g$ and ask: if $n \in \mathbb {Z}_{\geqslant 1}$ is such that $n \cdot c_1(\mathcal P_A)^{2g-1}/(2g-1)! \in {\rm H}^{4g-2}(A \times {\widehat {A}}, \mathbb {Z})_{\text {alg}}$, can we bound the order of ${\rm Z}^{2g-2}(A)$ in terms of $g$ and $n$? For a smooth complex projective $d$-dimensional variety $X$, ${\rm Z}^{2d-2}(X)$ is called the degree $2d-2$ Voisin group of $X$ (see [Per22]), is a stably birational invariant [Voi16, Lemma 2.20], and related to the unramified cohomology groups by Colliot-Thélène and Voisin [CTV12] and Schreieder [Sch21]. We prove that if $n \cdot c_1(\mathcal P_A)^{2g-1}/(2g-1)!$ is algebraic, then $\gcd ( n^2, (2g-2)!) \cdot {\rm Z}^{2g-2}(A) = (0)$. In particular, $(2g-2)! \cdot {\rm Z}^{2g-2}(A) = (0)$ for any $g$-dimensional complex abelian variety $A$. Moreover, if $A$ is principally polarized by $\theta \in \text {NS}(A)$ and if $n \cdot \gamma _\theta \in {\rm H}^{2g-2}(A,\mathbb {Z})$ is algebraic, then $n \cdot c_1(\mathcal P_A)^{2g-1}/(2g-1)!$ is algebraic. As 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 5.2).

Theorem 1.5 Let $A$ be a $g$-dimensional Prym variety over $\mathbb {C}$. Then $4 \cdot {\rm Z}^{2g-2}(A) = (0)$.

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 § 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 $\text {char}(k)$ and for some finitely generated field of definition $k_0 \subset k$ of $X$, the cycle class map

(1)\begin{equation} cl\colon \text{CH}_1(X)_{\mathbb{Z}_\ell} = \text{CH}_1(X)\otimes_\mathbb{Z}{\mathbb{Z}_\ell} \to \bigcup_U{\rm H}_{\unicode{x00E9}\text {t}}^{2d-2}(X, \mathbb{Z}_\ell(d-1))^U \end{equation}

is surjective, where $U$ ranges over the open subgroups of $\text {Gal}(k/k_0)$.

Theorem 1.6 Let $A$ be an abelian variety of dimension $g$ over the separable closure $k$ of a finitely generated field. The following assertions are true.

1. (i) The abelian variety $A$ satisfies the integral Tate conjecture for one-cycles over $k$ if the cohomology class

   \[ c_1(\mathcal P_A)^{2g-1}/(2g-1)! \in {\rm H}_{\unicode{x00E9}\text {t}}^{4g-2}(A \times {\widehat{A}}, \mathbb{Z}_\ell(2g-1)) \]
   is the class of a one-cycle with $\mathbb {Z}_\ell$-coefficients for every prime number $\ell < (2g-1)!$ unequal to $\text {char}(k)$.

2. (ii) Suppose that $A$ is principally polarized and let $\theta _\ell \in {\rm H}^2_{\unicode{x00E9}\text {t}}(A, \mathbb {Z}_\ell (1))$ be the class of the polarization. The map (1) is surjective if $\gamma _{\theta _\ell } := \theta _\ell ^{g-1}/(g-1)! \in {\rm H}_{{\unicode{x00E9} \text t}}^{2g-2}(A, \mathbb {Z}_\ell (g-1))$ 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 $\text {char}(k)$.

Theorem 1.6 implies that products of Jacobians of smooth projective curves over $k$ satisfy the integral Tate conjecture for one-cycles over $k$. Moreover, for a principally polarized 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 $A_{/\mathcal {O}_K}$ of $A_K$ at any prime ideal $\mathfrak p \subset \mathcal {O}_K$ 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 ${\mathsf {A}}_{g}$.

2. Notation

We use the following notation.

1. − 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 $\text {CH}(X)$ be the Chow group of $X$ and define $\text {CH}(X)_\mathbb {Q} = \text {CH}(X) \otimes \mathbb {Q}$, $\text {CH}(X)_{\mathbb {Q}_\ell } = \text {CH}(X) \otimes {\mathbb {Q}_\ell }$ and $\text {CH}(X)_{\mathbb {Z}_\ell } = \text {CH}(X) \otimes {\mathbb {Z}_\ell }$. We let ${\rm H}_{\unicode{x00E9}\text {t}}^i(X_{k_s}, \mathbb {Z}_\ell (a))$ be the $i$th-degree étale cohomology group with coefficients in $\mathbb {Z}_\ell (a)$, $a \in \mathbb {Z}$.

2. − Often, $A$ will denote an abelian variety of dimension $g$ over $k$, with dual abelian variety ${\widehat {A}}$ and (normalized) Poincaré bundle $\mathcal P_A$ on $A \times {\widehat {A}}$. The abelian group $\text {CH}(A)$ 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:

   \[ \star \colon \text{CH}(A) \times \text{CH}(A) \to \text{CH}(A), \quad x \star y = m_{\ast} (\pi_1^{\ast}(x) \cdot \pi_2^{\ast}(y)). \]
   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 $A$. There is a similar Pontryagin product $\star$ on étale cohomology, and on Betti cohomology if $k = \mathbb {C}$.

3. − For any abelian group $M$ and any element $x \in M$, we 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 relate such lifts to the integral Hodge conjecture when $k = \mathbb {C}$. In § 4 we 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 $\mathcal F_A \colon \text {CH}(A)_{\mathbb {Q}} \to \text {CH}({\widehat {A}})_{\mathbb {Q}}$ preserves the subgroups of integral cycles modulo torsion. A sufficient condition for this to hold is that $\mathcal F_A$ lifts to a homomorphism $\text {CH}(A) \to \text {CH}({\widehat {A}})$. In this section, we outline a second consequence of such a lift $\text {CH}(A) \to \text {CH}({\widehat {A}})$ 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

\[ \mathcal F_A \colon \text{CH}(A)_\mathbb{Q} \to \text{CH}({\widehat{A}})_\mathbb{Q} \]

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

\[ \mathscr{F}_{A}\colon {\rm H}_{{\unicode{x00E9}\text t}}^\bullet(A_{k_s}, \mathbb{Q}_\ell(\bullet)) \to {\rm H}_{{\unicode{x00E9}\text t}}^\bullet({\widehat{A}}_{k_s}, \mathbb{Q}_\ell(\bullet)). \]

In fact, $\mathscr{F}_A$ preserves the integral cohomology classes and induces, for each integer $j$ with $0 \leqslant j \leqslant 2g$, an isomorphism (see [Bea83, Proposition 1] and [Tot21, p. 18]):

\[ \mathscr{F}_{A}\colon {\rm H}_{{\unicode{x00E9}\text t}}^j(A_{k_s}, \mathbb{Z}_\ell(a)) \to {\rm H}_{{\unicode{x00E9}\text t}}^{2g-j}({\widehat{A}}_{k_s}, \mathbb{Z}_\ell(a+g-j)). \]

Similarly, if $k = \mathbb {C}$, then $\text {ch}(\mathcal P_A)$ induces, for each integer $i$ with $0 \leqslant i \leqslant 2g$, an isomorphism of Hodge structures:

(2)\begin{equation} \mathscr{F}_A\colon {\rm H}^i(A, \mathbb{Z}) \to {\rm H}^{2g-i}({\widehat{A}}, \mathbb{Z})(g-i). \end{equation}

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 consider similar homomorphisms $\text {CH}(A) \to \text {CH}({\widehat {A}})$. However, to make their existence easier to verify (cf. Theorem 3.8) we relax some of the above conditions.

Definition 3.1 Let $A_{/k}$ be an abelian variety and let $\mathcal F\colon \text {CH}(A) \to \text {CH}({\widehat {A}})$ be a group homomorphism. We call $\mathcal F$ a weak integral Fourier transform if the following diagram commutes.

(3)

A group homomorphism $\mathcal F \colon \text {CH}(A) \to \text {CH}({\widehat {A}})$ is an integral Fourier transform up to homology if the following diagram commutes.

(4)

Similarly, a $\mathbb {Z}_\ell$-module homomorphism $\mathcal F_\ell \colon \text {CH}(A)_{\mathbb {Z}_\ell } \to \text {CH}({\widehat {A}})_{\mathbb {Z}_\ell }$ 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.

Remarks 3.1 (i) Let $\Gamma \in \text {CH}(A \times {\widehat {A}})$ (respectively, $\Gamma _\ell \in \text {CH}(A \times {\widehat {A}})_{\mathbb {Z}_\ell })$ such that

\[ cl(\Gamma) = \text{ch}(\mathcal P_A) \quad \big({\text{respectively, }} cl(\Gamma_\ell) = \text{ch}(\mathcal P_A) \big) \quad{\text{in}} \quad \bigoplus_{r \geqslant 0}{\rm H}_{{\unicode{x00E9} \text t}}^{2r}((A \times {\widehat{A}})_{k_s}, \mathbb{Z}_\ell(r)). \]

Then $\mathcal F = \Gamma _\ast \colon \text {CH}(A) \to \text {CH}({\widehat {A}})$ (respectively, $\mathcal F_\ell = (\Gamma _\ell )_\ast \colon \text {CH}(A)_{\mathbb {Z}_\ell } \to \text {CH}({\widehat {A}})_{\mathbb {Z}_\ell })$ is an integral Fourier transform up to homology (respectively, an $\ell$-adic integral Fourier transform up to homology). Similarly, any cycle $\Gamma \in \text {CH}(A \times {\widehat {A}})$ that satisfies $\Gamma _\mathbb {Q} = \text {ch}(\mathcal P_A) \in \text {CH}(A \times {\widehat {A}})_{\mathbb {Q}}$ induces a weak integral Fourier transform $\mathcal F = \Gamma _\ast \colon \text {CH}(A) \to \text {CH}({\widehat {A}})$.

(ii) If $\mathcal F\colon \text {CH}(A) \to \text {CH}({\widehat {A}})$ is a weak integral Fourier transform, then $\mathcal F$ is an integral Fourier transform up to homology, the $\mathbb {Z}_\ell$-module $\bigoplus _{r \geqslant 0}{\rm H}_{{\unicode{x00E9} \text t}}^{2r}({\widehat {A}}_{k_s}, \mathbb {Z}_\ell (r))$ being torsion-free. If $k = \mathbb {C}$, then $\mathcal F\colon \text {CH}(A) \to \text {CH}({\widehat {A}})$ is an integral Fourier transform up to homology if and only if $\mathcal F$ is compatible with the Fourier transform $\mathscr{F}_A\colon {\rm H}^\bullet (A, \mathbb {Z}) \to {\rm H}^\bullet ({\widehat {A}},\mathbb {Z})$ on Betti cohomology.

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

Lemma 3.2 Let $A$ be a complex abelian variety and $\mathcal F \colon \text {CH}(A) \to \text {CH}({\widehat {A}})$ an integral Fourier transform up to homology.

1. (i) For each $i \in \mathbb {Z}_{\geqslant 0}$, the integral Hodge conjecture for degree $2i$ classes on $A$ implies the integral Hodge conjecture for degree $2g-2i$ classes on ${\widehat {A}}$.

2. (ii) If $\mathcal F = \Gamma _\ast$ for some $\Gamma \in \text {CH}(A \times {\widehat {A}})$ with $cl(\Gamma ) = \text {ch}(\mathcal P_A)$, then $\mathscr{F}_A$ induces a group isomorphism ${\rm Z}^{2i}(A) \xrightarrow {\sim } {\rm Z}^{2g-2i}({\widehat {A}})$ 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.

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

(5)

Thus, the surjectivity of $cl^i$ implies the surjectivity of $cl_i$. Moreover, if $\mathcal F$ is induced by some $\Gamma \in \text {CH}(A \times {\widehat {A}})$, then replacing $A$ by ${\widehat {A}}$ and ${\widehat {A}}$ by $\skew {5.5}\widehat {\widehat {A}}$ in the argument above shows that the images of $cl^i$ and $cl_i$ are identified under the isomorphism $\mathscr{F}_A\colon \text {Hdg}^{2i}(A, \mathbb {Z}) \xrightarrow {\sim } \text {Hdg}^{2g-2i}({\widehat {A}}, \mathbb {Z})$ in diagram (

5

).

3.2 Properties of the Fourier transform on rational Chow groups

Let $A$ be a complex abelian variety. Observe that, for any $j \in \mathbb {Z}_{\geqslant 1}$ and $x \in {\rm H}^{2j}(A,\mathbb {Z})$, one has

\[ \frac{x^i}{i!} \in {\rm H}^{2ij}(A,\mathbb{Z}) \subset {\rm H}^{2ij}(A, \mathbb{Q}) \quad {\text{ for all }} \ i \in \mathbb{Z}_{\geqslant 1}. \]

In particular, the ideal $\bigoplus _{j > 0}{\rm H}^{2j}(A, \mathbb {Z}) \subset {\rm H}^{2\bullet }(A, \mathbb {Z})$ admits a PD-structure [Sta18, Tag 07GM]. The analogue of this statement in $\ell$-adic étale cohomology holds when $A$ is an abelian variety over a separably closed field.

Lemma 3.2 suggests that to prove Theorem 1.1, one needs to show that for a complex abelian variety of dimension $g$ whose minimal Poincaré class $c_1(\mathcal P_A)^{2g-1}/(2g-1)! \in {\rm H}^{4g-2}(A \times {\widehat {A}}, \mathbb {Z})$ is algebraic, all classes of the form $c_1(\mathcal P_A)^{i}/i! \in {\rm H}^{2i}(A \times {\widehat {A}}, \mathbb {Z})$ are algebraic. With this goal in mind, we study Fourier transforms on rational Chow groups in § 3.2, and investigate how these relate to $\text {ch}(\mathcal P_A) \in \text {CH}(A \times {\widehat {A}})_{\mathbb {Q}}$. In turns out that the cycles $c_1(\mathcal P_A)^{i}/i! \in \text {CH}(A \times {\widehat {A}})_{\mathbb {Q}}$ satisfy several relations that are very similar to those proved by Beauville for the cycles $\theta ^{i}/i! \in \text {CH}(A)_{\mathbb {Q}}$ in case $A$ is principally polarized, see [Bea83]. As we need these results in any characteristic in order to prove Theorem 1.6, we work over our general field $k$, see § 2.

Let $A$ be an abelian variety over $k$. For $a \in \text {CH}(A)_{\mathbb {Q}}$, define $\mathrm {E}(a) \in \text {CH}(A)_{\mathbb {Q}}$ as the $\star$-exponential of $a$:

\[ \mathrm{E}(a) := \sum_{n \geqslant 0} \frac{a^{\star n}}{n!} \in \text{CH}(A)_{\mathbb{Q}}, \]

where $a^{\star n}$ denotes the $n$-fold Pontryagin product of $a$ (see § 2). Lemma 3.4 is the key to our proof of Theorem 1.1. To prove it, we first need to show the following.

Lemma 3.3 With respect to the Fourier transform $\mathcal F_{A \times {\widehat {A}}}\colon \text {CH}(A \times {\widehat {A}})_{\mathbb {Q}} \to \text {CH}({\widehat {A}} \times A)_{\mathbb {Q}}$, one has

\[ \mathcal F_{A \times {\widehat{A}}}\big(\exp(c_1(\mathcal P_A)) \big) = (-1)^g \cdot \exp( - c_1(\mathcal P_{{\widehat{A}}}) ) \in \text{CH}({\widehat{A}} \times A)_{\mathbb{Q}}. \]

Proof. We lift the desired equality in the rational Chow group of ${\widehat {A}} \times A$ to an isomorphism in the derived category ${\rm D}^b({\widehat {A}} \times A)$ 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 $\pi _{13}^\ast \mathcal {P}_A \otimes \pi _{24}^\ast \mathcal {P}_{{\widehat {A}}}$. Let

\[ \Phi_{\mathcal P_X} \colon {\rm D}^b(A \times {\widehat{A}}) \to {\rm D}^b({\widehat{A}} \times A) \]

be the Fourier–Mukai transform attached to $\mathcal P_X \in {\rm D}^b(X \times {\widehat {X}})$ as in [Huy06, Definition 5.1]. Evaluating it at $\mathcal P_A$ gives the object

\[ \Phi_{{\mathcal P}_{X}}(\mathcal P_A) \cong \pi_{34,\ast} \big(\pi_{13}^\ast\mathcal P_A \otimes \pi_{24}^\ast \mathcal P_{{\widehat{A}}} \otimes \pi_{12}^\ast\mathcal P_A \big) \in {\rm D}^b({\widehat{A}}\times A), \]

whose Chern character is exactly $\mathcal {F}_X(\exp (c_1(\mathcal P_A)))$. Consider the permutation map

\[ (123) \colon A \times {\widehat{A}} \times {\widehat{A}} \times A \cong {\widehat{A}} \times A \times {\widehat{A}} \times A, \]

with inverse $(321)$. We have

\begin{align*} \pi_{34,\ast} \big(\pi_{13}^\ast\mathcal P_A \otimes \pi_{24}^\ast \mathcal P_{{\widehat{A}}} \otimes \pi_{12}^\ast\mathcal P_A \big) &\cong \pi_{34,\ast} \big( \pi_{31}^\ast\mathcal P_{{\widehat{A}}} \otimes \pi_{12}^\ast\mathcal P_A \otimes \pi_{24}^\ast \mathcal P_{{\widehat{A}}} \big) \\ & \cong \pi_{14,\ast} \big( (123)_\ast \big( \pi_{31}^\ast\mathcal P_{{\widehat{A}}} \otimes \pi_{12}^\ast\mathcal P_A \otimes \pi_{24}^\ast \mathcal P_{{\widehat{A}}} \big) \big) \\ & \cong \pi_{14,\ast} \big( (321)^\ast \big( \pi_{31}^\ast\mathcal P_{{\widehat{A}}} \otimes \pi_{12}^\ast\mathcal P_A \otimes \pi_{24}^\ast \mathcal P_{{\widehat{A}}} \big) \big) \\ & \cong \pi_{14,\ast} \big( \pi_{12}^\ast\mathcal P_{{\widehat{A}}} \otimes \pi_{23}^\ast\mathcal P_A \otimes \pi_{34}^\ast \mathcal P_{{\widehat{A}}} \big). \end{align*}

We conclude that $\Phi _{{\mathcal P}_{X}}(\mathcal P_A) \cong \pi _{14,\ast }\big ( \pi _{12}^\ast \mathcal {P}_{{\widehat {A}}} \otimes \pi _{23}^\ast \mathcal {P}_A \otimes \pi _{34}^\ast \mathcal {P}_{{\widehat {A}}} \big )$. The latter is isomorphic to the Fourier–Mukai kernel of the composition

\[ \Phi_{\mathcal{P}_{{\widehat{A}}}} \circ \Phi_{\mathcal{P}_A} \circ \Phi_{\mathcal{P}_{{\widehat{A}}}}. \]

As $\Phi _{\mathcal {P}_{A}}\circ \Phi _{\mathcal {P}_{{\widehat {A}}}}$ 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 $\mathcal {P}_{{\widehat {A}}}^\vee [-g] \in {\rm D}^b({\widehat {A}} \times A)$. By uniqueness of the Fourier–Mukai kernel of an equivalence [Orl97, Theorem 2.2], it follows that $\Phi _{\mathcal P_X}(\mathcal P_A) \cong \mathcal {P}_{{\widehat {A}}}^\vee [-g]$. The Chern character of $\mathcal {P}_{{\widehat {A}}}^\vee [-g]$ equals $(-1)^{g}\cdot \exp (-c_1(\mathcal P_{{\widehat {A}}})) \in \text {CH}({\widehat {A}} \times A)_{\mathbb {Q}}$, and we are done.

Let $A$ be an abelian variety over $k$. Define

\begin{align*} {\mathscr{R}}_A &= c_1(\mathcal P_A)^{2g-1}/(2g-1)! \in \text{CH}_1(A\times {\widehat{A}})_{\mathbb{Q}},\\ {\mathscr{R}}_{{\widehat{A}}} &= c_1(\mathcal P_{{\widehat{A}}})^{2g-1}/(2g-1)! \in \text{CH}_1({\widehat{A}}\times A)_{\mathbb{Q}}. \end{align*}

Lemma 3.4 We have $\text {ch}(\mathcal P_A) = \exp ({c_1(\mathcal P_A)}) = (-1)^g\cdot {\rm E}((-1)^g\cdot {\mathscr {R}}_A) \in \text {CH}( A \times {\widehat {A}})_{\mathbb {Q}}$.

Proof. We claim that $(-1)^g \cdot \mathcal F_{{\widehat {A}} \times A}(-c_1(\mathcal P_{{\widehat {A}}}))= {\mathscr {R}}_A$. To see this, recall that for each integer $i$ with $0 \leqslant i \leqslant g$, there is a canonical Beauville decomposition $\text {CH}^i(A)_\mathbb {Q} = \bigoplus _{j = i-g}^i\text {CH}^{i,j}(A)_{\mathbb {Q}}$ (see [Bea86, DM91]). As the Poincaré bundle $\mathcal P_A$ is symmetric, we have $c_1(\mathcal P_A) \in \text {CH}^{1,0}(A \times {\widehat {A}})_{\mathbb {Q}}$ and, hence, $c_1(\mathcal P_A)^{i} \in \text {CH}^{i,0}(A \times {\widehat {A}})_{\mathbb {Q}}$. In particular, we have ${\mathscr {R}}_A \in \text {CH}^{2g-1,0}(A \times {\widehat {A}})_\mathbb {Q}$. The fact that $\mathcal P_{A}$ is symmetric also implies, via Lemma 3.3, that we have

\[ \mathcal F_{{\widehat{A}} \times A}\big((-1)^g\cdot \exp(- c_1(\mathcal P_{{\widehat{A}}})\big) = \exp(c_1(\mathcal P_A)). \]

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]. As $\mathcal F_{{\widehat {A}} \times A}$ identifies $\text {CH}^{i,0}({\widehat {A}} \times A)_{\mathbb {Q}}$ with $\text {CH}^{g-i,0}(A \times {\widehat {A}})$ (see [DM91, Lemma 2.18]), we must indeed have

(6) \begin{equation} (-1)^g \cdot \mathcal F_{{\widehat{A}} \times A}\big({-}c_1(\mathcal P_{{\widehat{A}}})\big)= \mathcal F_{{\widehat{A}} \times A}\big((-1)^{g+1}\cdot c_1(\mathcal P_{{\widehat{A}}})\big) = \frac{c_1(\mathcal P_A)^{2g-1}}{(2g-1)!} = {\mathscr{R}}_A, \end{equation}

which proves our claim. For a $g$-dimensional abelian variety $X$ and any $x,y \in \text {CH}(X)_{\mathbb {Q}}$, one has

\[ \mathcal F_{X}(x \cdot y) = (-1)^g\cdot \mathcal F_{X}(x) \star \mathcal F_{X}(y)\in \text{CH}({\widehat{X}})_{\mathbb{Q}}. \]

Indeed, in [Mur00, Theorem 4.5] this is proved when $k$ is algebraically closed, but holds over general $k$ (and even for abelian schemes; see, e.g., forthcoming work by Edixhoven, van der Geer and Moonen). This implies (see also [MP10, § 3.7]) that if $a$ is a cycle on $X$ with $\mathcal F_{X}(a) \in \text {CH}_{>0}({\widehat {X}})_{\mathbb {Q}}$, then $\mathcal F_{X}(\exp (a)) =(-1)^g\cdot {\rm E}((-1)^g\cdot \mathcal F_{X}(a))$. This allows us to conclude that

\begin{align*} \exp(c_1(\mathcal P_A)) &=\mathcal F_{{\widehat{A}} \times A} \big( (-1)^g \cdot \exp(-c_1(\mathcal P_{{\widehat{A}}}))\big) \\ &= (-1)^{g} \cdot{\rm E}\big(\mathcal F_{{\widehat{A}} \times A}(-c_1(\mathcal P_{{\widehat{A}}}))\big) \\ &= (-1)^g\cdot {\rm E}((-1)^g\cdot {\mathscr{R}}_A), \end{align*}

which finishes the proof of Lemma 3.4.

Next, assume that $A$ is equipped with a principal polarization $\lambda \colon A \xrightarrow {\sim } {\widehat {A}}$, and define

(7)\begin{equation} \Theta = \tfrac{1}{2}\cdot (\text{id}, \lambda)^\ast c_1(\mathcal P_A) \in \text{CH}^1(A)_\mathbb{Q} \quad {\text{and}}\quad \widehat\Theta = \tfrac{1}{2}\cdot (\lambda^{-1}, \text{id})^\ast c_1(\mathcal P_{A}) \in \text{CH}^1({\widehat{A}})_\mathbb{Q}. \end{equation}

Here $(\text {id},\lambda )$ (respectively, $(\lambda ^{-1}, \text {id})$) is the morphism $(\text {id},\lambda ) \colon A \to A\times {\widehat {A}}$ (respectively, $(\lambda ^{-1}, \text {id}) \colon {\widehat {A}} \to A \times {\widehat {A}}$). One can understand the relation between

\[ \Gamma_\Theta := \Theta^{g-1}/(g-1)! \in \text{CH}_1(A)_{\mathbb{Q}} \]

and ${\mathscr {R}}_A = {c_1(\mathcal P_A)}^{2g-1}/(2g-1)! \in \text {CH}_1(A \times {\widehat {A}})_{\mathbb {Q}}$ 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. Define a one-cycle $\tau$ on $A \times {\widehat {A}}$ as follows:

\[ \tau := j_{1, \ast} (\Gamma_\Theta) + j_{2, \ast}(\Gamma_{{\widehat{\Theta}}}) - (\text{id}, \lambda)_\ast (\Gamma_\Theta) \in \text{CH}_1( A \times {\widehat{A}} )_\mathbb{Q}. \]

Lemma 3.5 One has $\tau = (-1)^{g+1}\cdot {\mathscr {R}}_A \in \text {CH}_1(A \times {\widehat {A}})_{\mathbb {Q}}$.

Proof. Identify $A$ and ${\widehat {A}}$ via $\lambda$. This gives $c_1(\mathcal P_A) = m^\ast (\Theta ) - \pi _1^\ast (\Theta ) - \pi _2^\ast (\Theta )$, and the Fourier transform becomes an endomorphism $\mathcal F_A \colon \text {CH}(A)_\mathbb {Q} \to \text {CH}(A)_\mathbb {Q}$. We claim that

\[ \tau = (-1)^g \cdot \big( \Delta_\ast \mathcal F_A(\Theta) - j_{1, \ast} \mathcal F_A(\Theta) - j_{2, \ast} \mathcal F_A(\Theta) \big). \]

For this, it suffices to show that $\mathcal F_A(\Theta ) = (-1)^{g-1}\cdot \Theta ^{g-1}/(g-1)! \in \text {CH}_1(A)_\mathbb {Q}$. Now $\mathcal F_A(\exp (\Theta )) = \exp ({-\Theta })$ by Lemma 3.6. Moreover, because $\Theta$ is symmetric, we have $\Theta \in \text {CH}^{1,0}(A)_{\mathbb {Q}}$, hence $\Theta ^{i}/i! \in \text {CH}^{i,0}(A)_\mathbb {Q}$ for each $i \geqslant 0$. Therefore, $\mathcal F_A\big (\Theta ^{i}/i!\big ) \in \text {CH}^{g-i,0}(A)_\mathbb {Q}$ by [DM91, Lemma 2.18]. This implies that, in fact, $\mathcal F_A\big (\Theta ^i/i!\big ) =(-1)^{g-i}\cdot \Theta ^{g-i}/(g-i)! \in \text {CH}^{g-i,0}(A)_\mathbb {Q}$ for every $i$. In particular, the claim follows.

Next, recall that $\mathcal F_{A\times A}(c_1(\mathcal P_A)) = (-1)^{g+1}\cdot {\mathscr {R}}_A$, see Lemma 3.3. Thus, at this point, it suffices to prove the identity

\[ \mathcal F_{A\times A}(c_1(\mathcal P_A)) = (-1)^g \cdot \big( \Delta_\ast \mathcal F_A(\Theta) - j_{1, \ast} \mathcal F_A(\Theta) - j_{2, \ast} \mathcal F_A(\Theta) \big). \]

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)]:

(8)\begin{equation} ({\widehat{f}})^\ast \circ \mathcal F_X = \mathcal F_Y \circ f_\ast, \quad \mathcal F_X \circ f^\ast = (-1)^{\dim X - \dim Y}\cdot ({\widehat{f}})_\ast \circ \mathcal F_Y. \end{equation}

As $c_1(\mathcal P_A) = m^\ast \Theta - \pi _1^\ast \Theta - \pi _2^\ast \Theta$, it follows from (8) that

\begin{align*} \mathcal F_{A \times A}(c_1(\mathcal P_A)) &= \mathcal F_{A \times A}\big( m^\ast \Theta\big) - \mathcal F_{A \times A}\big(\pi_1^\ast \Theta\big) - \mathcal F_{A \times A}\big(\pi_2^\ast \Theta\big) \\ &= (-1)^g\cdot \big(\Delta_\ast \mathcal F_A(\Theta) - j_{1, \ast} \mathcal F_A(\Theta) - j_{2, \ast} \mathcal F_A(\Theta) \big). \end{align*}

Lemma 3.6 (Beauville)

Let $A$ be an abelian variety over $k$, principally polarized by $\lambda \colon A \xrightarrow {\sim } \widehat {A}$, and define $\Theta = \frac {1}{2}\cdot ({\rm {id}}, \lambda )^\ast c_1(\mathcal {P}_A) \in {\rm CH}^1(A)_{\mathbb {Q}}$. Identify $A$ and $\widehat {A}$ via $\lambda$. With respect to the Fourier transform $\mathcal {F}_A\colon {\rm CH}(A)_{\mathbb {Q}} \xrightarrow {\sim } {\rm CH}(A)_{\mathbb {Q}}$, one has $\mathcal {F}_A(\exp (\Theta )) = \exp ({-\Theta })$.

Proof. Our proof follows the proof of [Bea83, Lemme 1], but has to be adapted, because $\Theta$ does not necessarily come from a symmetric ample line bundle on $A$. As one still has $c_1(\mathcal P_A) = m^\ast \Theta - \pi _1^\ast \Theta - \pi _2^\ast \Theta$, the argument can be made to work: one has

\begin{align*} \mathcal F_A(\exp(\Theta)) &= \pi_{2,\ast}\big(\!\exp(c_1(\mathcal P_A)) \cdot \pi_1^\ast \exp({\Theta}) \big) \\ &=\pi_{2,\ast}\big(\!\exp({m^\ast \Theta - \pi_2^\ast\Theta}) \big)\\ &= \exp({-\Theta}) \cdot \pi_{2,\ast}(m^\ast \exp({\Theta})) \in \text{CH}(A)_{\mathbb{Q}}. \end{align*}

For codimension reasons, one has $\pi _{2,\ast }(m^\ast \exp (\Theta )) = \pi _{2,\ast }m^\ast (\Theta ^g/g!)= \deg (\Theta ^g/g!) \in \text {CH}^0(A)_{\mathbb {Q}} = \mathbb {Q}$. Pull back $\Theta ^g/g!$ along $A_{k_s} \to A$ to see that $\deg (\Theta ^g/g!) = 1 \in \text {CH}^0(A)_{\mathbb {Q}} \cong \text {CH}^0(A_{k_s})_{\mathbb {Q}}$, because 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 $\text {CH}_0(X)$. However, the case of algebraically closed fields remains open [MP10, § 3.2]. What we do know, is as follows.

Theorem 3.7 (Moonen–Polishchuk)

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

In particular, for each element $x \in \text {CH}_{>0}(A)$ and each $n \in \mathbb {Z}_{\geqslant 1}$, there is a canonical element $x^{[n]} \in \text {CH}_{>0}(A)$ such that $n!x^{[n]} = x^{\star n}$, see [Sta18, Tag 07GM]. For $x \in \text {CH}_{>0}(A)$, we may then define $\mathrm {E}(x) = \sum _{n \geqslant 0} x^{[n]} \in \text {CH}(A)$ as the $\star$-exponential of $x$ in terms of its divided powers.

Together with the results of § 3.2, Theorem 3.7 enables us to provide several criteria for the existence of a 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 $\Theta \in \text {CH}^1(A)_\mathbb {Q}$ to be the symmetric ample class attached to the polarization $\lambda$, see (7).

Theorem 3.8 Let $A_{/k}$ be an abelian variety of dimension $g$. The following are equivalent.

1. (i) The one-cycle ${\mathscr {R}}_A = c_1(\mathcal P_A)^{2g-1}/(2g-1)! \in \text {CH}(A \times {\widehat {A}})_{\mathbb {Q}}$ lifts to a one-cycle in $\text {CH}(A \times {\widehat {A}})$.

2. (ii) The cycle $\text {ch}(\mathcal P_A) \in \text {CH}(A \times {\widehat {A}})_\mathbb {Q}$ lifts to a cycle in $\text {CH}(A \times {\widehat {A}})$.

3. (iii) The cycle $\text {ch}(\mathcal P_{A \times {\widehat {A}}}) \in \text {CH}(A \times {\widehat {A}} \times {\widehat {A}} \times A)_\mathbb {Q}$ lifts to a cycle in $\text {CH}(A \times {\widehat {A}} \times {\widehat {A}} \times A)$.

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. (iv) The two-cycle $c_1(\mathcal P_A)^{2g-2}/(2g-2)! \in \text {CH}(A \times {\widehat {A}})_{\mathbb {Q}}$ lifts to a two-cycle in $\text {CH}(A \times {\widehat {A}})$.

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

3. (vi) The abelian variety $A$ admits a weak integral Fourier transform.

4. (vii) The Fourier transform $\mathcal F_A$ satisfies $\mathcal F_A\big ( \text {CH}(A)/\text {torsion}\big ) \subset \text {CH}({\widehat {A}})/\text {torsion}$.

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

Proof. Suppose that statement (i) holds, and let $\Gamma \in \text {CH}_1(A \times {\widehat {A}})$ be a cycle such that $\Gamma _\mathbb {Q} = {\mathscr {R}}_A$. Then consider the cycle $(-1)^g\cdot {\rm E}((-1)^g\cdot \Gamma ) \in \text {CH}(A \times {\widehat {A}})$. By Lemma 3.4, we have

\[ (-1)^g\cdot {\rm E}((-1)^g\cdot \Gamma)_{\mathbb{Q}} = (-1)^g\cdot {\rm E}((-1)^g\cdot \Gamma_{\mathbb{Q}}) = (-1)^g\cdot {\rm E}((-1)^g\cdot {\mathscr{R}}_A) = \text{ch}(\mathcal P_A) \in \text{CH}(A \times {\widehat{A}})_{\mathbb{Q}}. \]

Thus statement (ii) holds. We claim that statement (iii) 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

(9) \begin{align} {\mathscr{R}}_{A \times {\widehat{A}}} &= \frac{1}{(4g-1)!}\cdot \big( \pi_{13}^\ast c_1(\mathcal P_A) + \pi_{24}^\ast c_1(\mathcal P_{{\widehat{A}}}) \big)^{4g-1} \nonumber\\ &=\frac{1}{(2g)!(2g-1)!} \cdot \big( \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}\big) \nonumber\\ &=\frac{1}{(2g)!(2g-1)!} \cdot \big( \pi_{13}^\ast c_1(\mathcal P_A)^{2g-1} \cdot \pi_{24}^\ast \big((2g)! \cdot [0]_{A \times {\widehat{A}}}\big)\nonumber\\ &\quad + \pi_{13}^\ast \big( (2g)! \cdot [0]_{{\widehat{A}} \times A} \big) \cdot \pi_{24}^\ast c_1(\mathcal P_{{\widehat{A}}})^{2g-1} \big) \nonumber\\ &= \pi_{13}^\ast \bigg( \frac{c_1(\mathcal P_A)^{2g-1}}{(2g-1)!} \bigg)\cdot \pi_{24}^\ast ([0]_{A \times {\widehat{A}}}) + \pi_{13}^\ast ( [0]_{{\widehat{A}} \times A} ) \cdot \pi_{24}^\ast \bigg( \frac{c_1(\mathcal P_{{\widehat{A}}})^{2g-1}}{(2g-1)!} \bigg) \in \text{CH}_1(X)_\mathbb{Q}. \end{align}

We conclude that ${\mathscr {R}}_{A \times {\widehat {A}}}$ lifts to $\text {CH}_1(X)$ which, by the implication $[({\rm i}) \implies ({\rm {ii}})]$ (that has already been proved), implies that statement (iii) holds. The implication $[({\rm iii})\implies ({\rm i})]$ follows from the fact that $(-1)^g \cdot \mathcal F_{{\widehat {A}} \times A}(- c_1(\mathcal P_{{\widehat {A}}})) = {\mathscr {R}}_A$ (see (6)). Therefore, we have $[({\rm i})\iff ({\rm ii})\iff ({\rm iii})]$.

From now on let us 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 identify ${\widehat {A}}$ and $A$ via $\lambda$.

Suppose that statement (iv) holds and let $S_{ A} \in \text {CH}_2(A\times A) = \text {CH}^{2g-2}(A \times A)$ be such that

\[ (S_{A})_\mathbb{Q} = c_1(\mathcal P_A)^{2g-2}/(2g-2)! \in \text{CH}_2(A \times A)_\mathbb{Q}. \]

Define $\text {CH}^{1,0}(A) := \text {Pic}^{\text {sym}}(A)$ to be the group of isomorphism classes of symmetric line bundles on $A$. Then $S_{A}$ induces a homomorphism $\mathcal F \colon \text {CH}^{1,0}(A) \to \text {CH}_1( A)$ defined as the composition

\[ \mathcal F \colon \text{CH}^{1,0}(A) \xrightarrow{\pi_1^\ast} \text{CH}^1(A \times A) \xrightarrow{\cdot S_{A}} \text{CH}^{2g-1}(A \times A) = \text{CH}_1(A \times A) \xrightarrow{\pi_{2,\ast}} \text{CH}_1(A). \]

As $\mathcal F_A \big (\text {CH}^{1,0}(A)_\mathbb {Q}\big ) \subset \text {CH}_1( A)_\mathbb {Q}$ (see [DM91, Lemma 2.18]) we see that the following diagram commutes.

(10)

On the other hand, because the line bundle $\mathcal L$ is symmetric, we have

(11)\begin{equation} \Theta = \tfrac{1}{2}\cdot \Delta^\ast c_1(\mathcal P_A) = \tfrac{1}{2}\cdot c_1\big( \Delta^\ast\mathcal P_A\big) = \tfrac{1}{2} \cdot c_1(\mathcal L \otimes \mathcal L) = c_1(\mathcal L) \in \text{CH}^1(A)_{\mathbb{Q}}. \end{equation}

The class $c_1(\mathcal L) \in \text {CH}^{1,0}(A)$ of the line bundle $\mathcal L$ thus lies above $\Theta \in \text {CH}^1(A)_\mathbb {Q}$. Therefore, $\mathcal F(c_1(\mathcal L)) \in \text {CH}_1(A)$ lies above $\Gamma _\Theta = (-1)^{g-1}\mathcal F_{A}(\Theta )$ by the commutativity of (

10

), and statement (v) holds.

Suppose that statement (v) holds. Then statement (i) follows readily from Lemma 3.5. Moreover, if statement (ii) holds, then $\text {ch}(\mathcal P_A) \in \text {CH}(A \times A)_\mathbb {Q}$ lifts to $\text {CH}(A \times A)$, hence, in particular, statement (iv) holds. As we have already proved that statement (i) implies (ii), we conclude that $[({\rm iv}) \implies ({\rm v}) \implies ({\rm i}) \implies ({\rm ii}) \implies ({\rm iv})]$.

The implications $[({\rm ii})\implies ({\rm vi})\implies ({\rm vii})]$ are trivial. Assume that statement (vii) holds. By (11), $\Theta \in \text {CH}^1(A)_{\mathbb {Q}}$ lifts to $\text {CH}^1(A)$, hence $\mathcal F_{A}(\Theta ) = (-1)^{g-1}\cdot \Gamma _\Theta$ lifts to $\text {CH}_1(A)$, i.e. statement (v) holds.

Assume that statement (vii) holds. The fact that $\mathcal F_{A}\big ( \text {CH}(A)/\text {torsion} \big ) \subset \text {CH}(A)/\text {torsion}$ implies that

\[ \text{CH}(A)/\text{torsion} = \mathcal F_{A}\big( \mathcal F_{ A} \big( \text{CH}( A)/\text{torsion} \big) \big) \subset \mathcal F_{A}\big( \text{CH}(A)/\text{torsion} \big) \subset \text{CH}(A)/\text{torsion}. \]

Thus, the restriction of the Fourier transform $\mathcal F_{A}$ to $\text {CH}(A)/\text {torsion}$ defines an isomorphism

\[ \mathcal F_{A}\colon \text{CH}(A)/\text{torsion} \xrightarrow{\sim} \text{CH}( A)/\text{torsion}. \]

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 $I/\text {torsion} \subset R/\text {torsion}$. Consequently, the ideal $\text {CH}_{>0}(A)/\text {torsion} \subset \text {CH}(A)/\text {torsion}$ admits a PD-structure for the Pontryagin product $\star$ by Theorem 3.7. As $\mathcal F_A$ exchanges the Pontryagin and intersection product (up to a sign, see [Bea83, Proposition 3(ii)]), it follows that statement (viii) holds. As statement (viii) trivially implies statement (v), we are done.

Question 3.9 (Moonen and 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. (i) The class $c_1(\mathcal P_A)^{2g-1}/(2g-1)! \in {\rm H}^{4g-2}_{\unicode{x00E9}\text {t}}((A \times {\widehat {A}})_{k_s}, \mathbb {Z}_\ell (2g-1))$ lifts to $\text {CH}_1(A \times {\widehat {A}})$.

2. (ii) The class $\text {ch}(\mathcal P_A) \in \bigoplus _{r \geqslant 0}{\rm H}_{{\unicode{x00E9} \text t}}^{2r}((A \times {\widehat {A}})_{k_s}, \mathbb {Z}_\ell (r))$ lifts to a cycle in $\text {CH}(A \times {\widehat {A}})$.

3. (iii) The class $\text {ch}(\mathcal P_{A \times {\widehat {A}}}) \in \bigoplus _{r \geqslant 0}{\rm H}_{ {\unicode{x00E9} \text t}}^{2r}((A \times {\widehat {A}} \times {\widehat {A}} \times A)_{k_s}, \mathbb {Z}_\ell (r))$ lifts to a cycle in $\text {CH}(A \times {\widehat {A}} \times {\widehat {A}} \times A)$.

Moreover, if $A$ carries an 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. (iv) The class $c_1(\mathcal P_A)^{2g-2}/(2g-2)! \in {\rm H}_{\unicode{x00E9}\text {t}}^{4g-4}((A \times {\widehat {A}})_{k_s}, \mathbb {Z}_\ell (2g-2))$ lifts to $\text {CH}_2(A \times {\widehat {A}})$.

2. (v) The class $\gamma _\theta = \theta ^{g-1}/(g-1)! \in {\rm H}^{2g-2}_{\unicode{x00E9}\text {t}}(A_{k_s}, \mathbb {Z}_\ell (g-1))$ lifts to a cycle in $\text {CH}_1(A)$.

3. (vi) The abelian variety $A$ admits an integral Fourier transform up to homology.

4. (vii) The Fourier transform $\mathscr{F}_A$ satisfies

   \[ \mathscr{F}_A\big( {\rm H}^{2\bullet}_{\unicode{x00E9}\text {t}}(A_{k_s}, \mathbb{Z}_\ell(\bullet))_{{\text{alg}}}\big) \subset {\rm H}^{2\bullet}_{\unicode{x00E9}\text {t}}({\widehat{A}}_{k_s}, \mathbb{Z}_\ell(\bullet))_{{\text{alg}}}. \]

5. (viii) There exists a PD-structure on the ideal

   \[ \bigoplus_{j > 0} {\rm H}^{2j}_{\unicode{x00E9}\text {t}}(A_{k_s}, \mathbb{Z}_\ell(j))_{{\text{alg}}} \subset {\rm H}^{2\bullet}_{\unicode{x00E9}\text {t}}(A_{k_s}, \mathbb{Z}_\ell(\bullet))_{{\text{alg}}}. \]

Here, ${\rm H}_{ {\unicode{x00E9} \text t}}^{2\bullet }(A_{k_s}, \mathbb {Z}_\ell (\bullet ))_{{\text {alg}}}$ denotes the image of the cycle class map $\text {CH}^\bullet (A) \to {\rm H}_{{\unicode{x00E9} \text t}}^{2\bullet }(A_{k_s}, \mathbb {Z}_\ell (\bullet ))$.

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

Proposition 3.12

1. (i) If $k = \mathbb {C}$, then each of the statements (i)–(viii) in Proposition 3.11 is equivalent to the same statement with étale cohomology replaced by Betti cohomology.

2. (ii) Proposition 3.11 remains valid if one replaces integral Chow groups by their tensor product with $\mathbb {Z}_\ell$, ‘integral Fourier transform up to homology’ by ‘$\ell$-adic integral Fourier transform up to homology’, and ${\rm H}_{{\unicode{x00E9} \text t}}^{2\bullet }(A_{k_s}, \mathbb {Z}_\ell (\bullet ))_{{\text {alg}}}$ by the image of $\text {CH}^\bullet (A) \otimes \mathbb {Z}_\ell \to {\rm H}_{ {\unicode{x00E9} \text t}}^{2\bullet }(A_{k_s}, \mathbb {Z}_\ell (\bullet ))$.

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

\[ {\rm H}^{2i}_{\unicode{x00E9}\text {t}}(A, \mathbb{Z}_\ell) \xrightarrow{\sim} {\rm H}^{2i}(A(\mathbb{C}), \mathbb{Z}_\ell) \]

[AGV71, III, Exposé XI] is compatible with the cycle class map. As the map ${\rm H}^{2i}(A(\mathbb {C}), \mathbb {Z}) \to {\rm H}_{\unicode{x00E9}\text {t}}^{2i}(A, \mathbb {Z}_\ell )$ is injective, a class $\beta \in {\rm H}^{2i}(A(\mathbb {C}), \mathbb {Z})$ is in the image of $cl\colon \text {CH}^i(A) \to {\rm H}^{2i}(A(\mathbb {C}),\mathbb {Z})$ if and only if its image $\beta _\ell \in {\rm H}^{2i}_{\unicode{x00E9}\text {t}}(A, \mathbb {Z}_\ell )$ is in the image of $cl\colon \text {CH}^i(A) \to {\rm H}^{2i}_{\unicode{x00E9}\text {t}}(A, \mathbb {Z}_\ell )$.

(ii) Indeed, for an abelian variety $A$ over $k$, the PD-structure on $\text {CH}_{>0}(A) \subset (\text {CH}(A), \star )$ induces a PD-structure on $\text {CH}_{>0}(A) \otimes \mathbb {Z}_\ell \subset (\text {CH}(A)_{\mathbb {Z}_\ell }, \star )$ by [Sta18, Tag 07H1], because the ring map $(\text {CH}(A),\star ) \to (\text {CH}(A)_{\mathbb {Z}_\ell },\star )$ 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 § 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 statement (i) holds. Then statement (ii) holds by Propositions 3.11 and 3.12(i). Suppose that statement (ii) holds. Then statement (iv) follows from Lemma 3.2. Thus, we have $[({\rm i}) \iff ({\rm ii}) \implies ({\rm iv})]$.

For a complex abelian variety $X$ of dimension $g$, define

\[ \rho_X = c_1(\mathcal P_X)^{2g-1}/(2g-1)! \in {\rm H}^{4g-2}(X \times {\widehat{X}}, \mathbb{Z}). \]

If statement (i) holds, then $\rho _A = c_1(\mathcal P_A)^{2g-1}/(2g-1)! \in {\rm H}^{4g-2}(A \times {\widehat {A}}, \mathbb {Z})$ is algebraic, which implies that $\rho _{{\widehat {A}}} \in {\rm H}^{4g-2}({\widehat {A}} \times A, \mathbb {Z})$ is algebraic. Therefore,

\[ \rho_{A \times {\widehat{A}}} \in {\rm H}^{8g-2}(A \times {\widehat{A}} \times {\widehat{A}} \times A, \mathbb{Z}) \]

is algebraic by (9). We then apply the implication $[({\rm i})\implies ({\rm iv})]$ to the abelian variety $A \times {\widehat {A}}$, which shows that statement (iii) holds. As $[({\rm iii})\implies ({\rm i})]$ is trivial, we have proven $[({\rm i})\iff ({\rm ii})\iff ({\rm iii})\implies ({\rm iv})]$.

Next, assume that $A$ is principally polarized by $\theta \in \text {NS}(A) \subset {\rm H}^2(A, \mathbb {Z})$. The directions $[({\rm iv})\implies ({\rm v})]$ and $[({\rm ii})\implies ({\rm vi})]$ are trivial and $[({\rm v})\implies ({\rm i})]$ follows from Propositions 3.11 and 3.12(i). We claim that statement (vi) implies statement (iv). Define

\[ \sigma_A = c_1(\mathcal P_A)^{2g-2}/(2g-2)! \in {\rm H}^{4g-4}(A \times {\widehat{A}},\mathbb{Z}) \]

and let $S \in \text {CH}_2(A \times {\widehat {A}})$ be such that $cl(S) = \sigma _A$. The squares in the following diagram commute.

(12)

As $\mathscr{F}_A = \pi _{2,\ast }\big ( \text {ch}(\mathcal P_A) \cdot \pi _1^\ast (-)\big )$ restricts to an isomorphism $\mathscr{F}_A\colon {\rm H}^2(A,\mathbb {Z}) \xrightarrow {\sim } {\rm H}^{2g-2}({\widehat {A}},\mathbb {Z})$ by [Bea83, Proposition 1], the composition $\pi _{2,\ast } \circ (- \cdot \sigma _A) \circ \pi _1^\ast$ on the bottom row of (

12

) is an isomorphism. Thus, by Lefschetz $(1,1)$, $cl\colon \text {CH}_1({\widehat {A}}) \to \text {Hdg}^{2g-2}({\widehat {A}},\mathbb {Z})$ is surjective. Finally, the equivalence $[({\rm v})\iff ({\rm vii})]$ follows directly from Propositions 3.11 and 3.12(i).

Corollary 4.1 Let $A$ and $B$ be complex abelian varieties of respective dimensions $g_A$ and $g_B$.

1. (i) The Hodge classes

   \[ c_1(\mathcal P_A)^{2g_A-1}/(2g_A-1)! \in {\rm H}^{4g_A-2}(A \times {\widehat{A}}, \mathbb{Z}) \]
   and
   \[ c_1(\mathcal P_B)^{2g_B-1}/(2g_B-1)! \in {\rm H}^{4g_B-2}(B \times {\widehat{B}}, \mathbb{Z}) \]
   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.

2. (ii) 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 (9). 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 $cl(\iota (C)) = \gamma _\theta \in {\rm H}^{2g-2}(J(C), \mathbb {Z})$ by Poincaré's formula [ACGH85], so $\gamma _\theta$ is algebraic and the result follows from Theorem 1.1.

Remarks 4.2 (i) 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 [dBa02, Corollary 5] which says that for each $n \in \mathbb {Z}_{\geqslant 1}$, there is a natural polarization $\eta$ on the $n$-fold symmetric product $C^{(n)}$ such that for any $i \in \mathbb {Z}_{\geqslant 0}$, the map

\[ \eta^{n-i}\cup (-) \colon {\rm H}^i(C^{(n)}, \mathbb{Z}) \to {\rm H}^{2n-i}(C^{(n)}, \mathbb{Z}) \]

is an isomorphism. In particular, the variety $C^{(n)}$ satisfies the integral Hodge conjecture for one-cycles for any positive integer $n$.

(ii) 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

\[ C_1 \times \cdots \times C_n \]

of smooth projective curves $C_i$ over $\mathbb {C}$. Indeed, this follows readily from the Künneth formula.

(iii) Let $C$ be a smooth projective complex curve of genus $g$. Our proof of Theorem 1.1 provides an explicit description of $\text {Hdg}^{2g-2}(J(C),\mathbb {Z})$ depending on $\text {Hdg}^2(J(C),\mathbb {Z})$. More generally, let $(A,\theta )$ be a principally polarized abelian variety of dimension $g$, and identify $A$ and ${\widehat {A}}$ via the polarization. Then $c_1(\mathcal P_A) = m^\ast (\theta ) - \pi _1^\ast (\theta ) - \pi _2^\ast (\theta )$, which implies that

\[ \frac{1}{(2g-2)!} \cdot c_1(\mathcal P_A)^{2g-2} = \sum_{\substack{i,j,k \geqslant 0\\ i + j + k = 2g-2}}^{2g-2} (-1)^{j+k}\cdot m^\ast\bigg( \frac{\theta^i}{i!}\bigg)\cdot \pi_1^\ast\bigg(\frac{\theta^j}{j!}\bigg)\cdot \pi_2^\ast \bigg(\frac{\theta^{k}}{k!}\bigg). \]

On the other hand, any $\beta \in \text {Hdg}^{2g-2}(A,\mathbb {Z})$ is of the form

\[ \beta = \pi_{2,\ast} \bigg( \frac{c_1(\mathcal P_A)^{2g-2}}{(2g-2)!} \cdot \pi_1^\ast [D] \bigg) \in \text{Hdg}^{2g-2}(A,\mathbb{Z}), \]

where we write $[D] = cl(D)$ for a divisor $D$ on $A$, as follows from (12). Therefore, any element $\beta \in \text {Hdg}^{2g-2}(A,\mathbb {Z})$ may be written as

(13)\begin{equation} \beta = \sum_{\substack{i,j,k \geqslant 0\\ i + j + k = 2g-2}}^{2g-2} (-1)^{j+k}\cdot \pi_{2,\ast}\bigg( m^\ast\bigg( \frac{\theta^i}{i!}\bigg)\cdot \pi_1^\ast\bigg(\frac{\theta^j}{j!}\bigg)\cdot \pi_1^\ast[D]\bigg) \cdot \frac{\theta^{k}}{k!}. \end{equation}

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 (13) are effective algebraic cycle classes. Indeed, for $p \in C$ and $d \in \mathbb {Z}_{\geqslant 1}$, the image of the morphism $C^{d} \to J(C)$, $(x_i) \mapsto \mathcal O(\sum _ix_i-d\cdot p)$ defines a subvariety $W_d(C) \subset J(C)$ and by Poincaré's formula [ACGH85, § I.5] one has

\[ cl(W_d(C)) = \theta^{g-d}/(g-d)! \in {\rm H}^{2g-2d}(J(C), \mathbb{Z}). \]

In addition to 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 $X = \bigcup _iZ_i(\mathbb {C})$ 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 $g \in \mathbb {Z}_{\geqslant 0}$ be the relative dimension of $\pi$ and define, for $t \in S(\mathbb {C})$, $\theta _t \in \text {NS}(\mathcal A_t)\subset {\rm H}^2(\mathcal A_t, \mathbb {Z})$ to be the polarization of $\mathcal A_t$. There is a global section $\gamma _\theta \in {\rm R}^{2g-2}\pi _\ast \mathbb {Z}$ such that for each $t \in B$, $\gamma _{\theta _t} = \theta _t^{g-1}/(g-1)! \in {\rm H}^{2g-2}(\mathcal A_t, \mathbb {Z})$. 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, $\gamma _{\theta _t}$ 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 $\sum _{i = 1}^k n_i\cdot cl(C_{i,y}) = \gamma _{\theta _y} \in {\rm H}^{2g-2}(A_y,\mathbb {Z})$. Then, for every $x \in S(\mathbb {C})$, one has $\sum _{i = 1}^k n_i\cdot cl(C_{i,x}) = \gamma _{\theta _x} \in {\rm H}^{2g-2}(A_x,\mathbb {Z})$.

Proof. As 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 $\theta _{\bar t} \in {\rm H}^{2}_{\unicode{x00E9}\text {t}}(A_{\bar t}, \mathbb {Z}_\ell )$ the class of the polarization and $\gamma _{\theta _{\bar t}} = \theta _{\bar t}^{g-1}/(g-1)!$. Let $\eta = \text {Spec } K$ be the generic point of $S$. The elements $\sum _i n_i \cdot cl(C_{i,\bar \eta })$ and $\gamma _{\theta _{\bar \eta }}$ in ${\rm H}^{2g-2}_{\unicode{x00E9}\text {t}}(A_{\bar \eta }, \mathbb {Z}_\ell )$ both map to $\sum _i n_i \cdot cl(C_{i,y}) = \gamma _{\theta _{y}} \in {\rm H}^{2g-2}_{\unicode{x00E9}\text {t}}(A_{y}, \mathbb {Z}_\ell )$ under the specialization homomorphism $s\colon {\rm H}^{2g-2}_{\unicode{x00E9}\text {t}}(A_{\bar \eta }, \mathbb {Z}_\ell ) \to {\rm H}^{2g-2}_{\unicode{x00E9}\text {t}}(A_{y}, \mathbb {Z}_\ell )$ by [Ful98, Example 20.3.5]. As $s$ is an isomorphism, we have $\sum _i n_i \cdot cl(C_{i,\bar \eta }) = \gamma _{\theta _{\bar \eta }}$, which implies that $\sum _{i}n_i\cdot cl(C_{x,i}) = \gamma _{\theta _x} \in {\rm H}_{\unicode{x00E9}\text {t}}^{2g-2}(A_x,\mathbb {Z}_\ell )$.

4.2 Density of abelian varieties satisfying IHC$_1$

The goal of this section is to prove that conditions (i)–(iii) in Theorem 1.1 are satisfied on a dense subset of the moduli space of complex abelian varieties. To do so, 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 4.1 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 $B$ 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 4.5 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 $\kappa \cdot c_1(\mathcal P_A)/(2g-1)! \in {\rm H}^{4g-2}(A \times {\widehat {A}}, \mathbb {Z})$ is algebraic. Consider the following statements.

1. (i) The abelian variety $A$ satisfies the integral Hodge conjecture for one-cycles.

2. (ii) 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. (iii) 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. (iv) 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. (v) 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.

Then $[({\rm ii})\implies ({\rm iii})\implies ({\rm v}) \implies ({\rm i})]$ and $[({\rm ii})\implies ({\rm iv}) \implies ({\rm v})]$. Moreover, if $A$ is principally polarized by $\theta _A \in \text {NS}(A)$, then statement (i) is implied by the following.

1. (vi) 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 $m \in \mathbb {Z}_{\geqslant 1}$ with $\gcd (m,p) = 1$.

Finally, if $A$ is principally polarized of Picard rank one, then statements (i)–(vi) are equivalent.

Proof. Step one: $[({\rm ii})\implies ({\rm iii}) \implies ({\rm v})]$ and $[({\rm ii}) \implies ({\rm iv}) \implies ({\rm v})]$. These implications are trivial.

Step two: $[({\rm v})\implies ({\rm i})]$. Let $g$ be the dimension of $A$. We want to prove that the class $c_1(\mathcal P_A)^{2g-1}/(2g-1)! \in {\rm H}^{4g-2}(A \times {\widehat {A}}, \mathbb {Z})$ is algebraic. Let $p$ be any prime number that divides $\kappa$. Then by condition (v), there exists an abelian variety $B$ and an isogeny $\alpha \colon A \times B \to Y$ to the product $Y = \prod _iJ(C_i)$ 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

\[ [m_p^{2h}]^{\ast} = (m_p^{N_p} \cdot (-) ) \colon {\rm H}^{4h-2}(X \times {\widehat{X}}, \mathbb{Z}) \to {\rm H}^{4h-2}(X \times {\widehat{X}}, \mathbb{Z}) \]

factors through ${\rm H}^{4h-2}(Y \times {\widehat {Y}}, \mathbb {Z})$. As $Y \times {\widehat {Y}}$ satisfies the integral Hodge conjecture by Theorem 1.2, the Hodge class $m_p^{N_p} \cdot c_1(P_X)^{2h-1}/(2h-1)! \in {\rm H}^{4h-2}(X \times {\widehat {X}} , \mathbb {Z})$ 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

\[ \frac{c_1(\mathcal P_{X})^{2h-1}}{(2h-1)!} =f^{\ast}\bigg(\frac{\mu^{2g-1}}{(2g-1)!}\bigg) \cdot g^\ast \bigg( \frac{\nu^{2g_B}}{(2g_B)!} \bigg) + f^\ast \bigg( \frac{\mu^{2g}}{(2g)!} \bigg) \cdot g^\ast\bigg(\frac{\nu^{2g_B-1}}{(2g_B-1)!} \bigg). \]

This implies that $f_\ast \big (c_1(\mathcal P_X)^{2h-1}/(2h-1)! \big ) = (-1)^{g_b}\mu ^{2g-1}/(2g-1)!$. In particular, the class $m_p^{N_p} \cdot c_1(\mathcal P_A)^{2g-1}/(2g-1)! \in {\rm H}^{4g-2}(A \times {\widehat {A}}, \mathbb {Z})$ 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

\[ \frac{c_1(\mathcal P_A)^{2g-1}}{(2g-1)!} = a \cdot \kappa \cdot \frac{c_1(\mathcal P_A)^{2g-1}}{(2g-1)!} + \sum_{i = 1}^n b_i \cdot m_{p_i}^{N_{p_i}} \cdot \frac{c_1(\mathcal P_A)^{2g-1}}{(2g-1)!} \in {\rm H}^{4g-2}(A \times {\widehat{A}}, \mathbb{Z}). \]

This proves that $c_1(\mathcal P_A)^{2g-1}/(2g-1)!$ is a $\mathbb {Z}$-linear combination of algebraic classes, hence algebraic. Condition (i) then follows from Theorem 1.1.

Step three: $[({\rm vi})\implies ({\rm i})]$ for $A$ principally polarized by $\theta _A \in \text {NS}(A)$. 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 $m_i \in \mathbb {Z}_{\geqslant 1}$ with $p_i \nmid m$. As $\theta _{J(C_i)}^{g-1}/(g-1)! \in {\rm H}^{2g-2}(J(C_i),\mathbb {Z})$ is algebraic for each $i$, the classes $\phi _i^\ast (\theta _{J(C_i)}^{g-1}/(g-1)!) = m_i^{g-1} \cdot \theta _{A}^{g-1}/(g-1)! \in {\rm H}^{2g-2}(A,\mathbb {Z})$ are algebraic. As $\gcd ((g-1)!, m_1,\dotsc, m_k) = 1$, this implies that $\theta _A^{g-1}/(g-1)!$ is algebraic. Condition (i) follows from Theorem 1.1.

Step four: $[({\rm vi})\impliedby ({\rm i}) \implies ({\rm ii})]$ 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 ${\theta ^{g-1}}/{(g - 1)!} = \sum _{i = 1}^n e_i\cdot [Z_i] \in {\rm H}^{2g - 2}(A, \mathbb {Z})$. Since $\rho (A) = 1$, the group $\text {Hdg}^{2g-2}(A, \mathbb {Z})$ is generated by $\theta ^{g-1}/(g-1)!$. Consequently, we have $[Z_i] = f_i\cdot \big (\theta ^{g-1}/(g-1)!\big )$ 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 that $\sum _{i = 1}^n e_i\cdot f_i = 1$. Now let $p$ be any prime number. Then there exists an integer $i$ with $1 \leqslant i \leqslant n$ 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

(14)

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 $\varphi ^\ast \colon {\widehat {A}} = \text {Pic}^0(A) \to \text {Pic}^0(C_i)$ is the pullback of line bundles along $\varphi \colon C_i \to A$. The natural homomorphism $a\colon \text {Pic}^0(C_i) \to J(C_i)$ is an isomorphism by the Abel–Jacobi theorem. As the triangle on the left in diagram (

14

) commutes and $[Z_i] \in {\rm H}^{2g-2}(A, \mathbb {Z})$ 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 $\varphi ^\ast \colon {\widehat {A}} \to \text {Pic}^0(C_i)$ is finite. As the homomorphism $\iota ^\ast \colon {\widehat {J(C_i)}} \to \text {Pic}^0(C_i)$ induced by the embedding $\iota \colon C_i \to J(C_i)$ is an isomorphism, dualizing the triangle on the left in diagram (

14

) proves our claim. By construction, we have $\varphi _{\ast }[C_i] = [Z_i] = f_i \cdot \theta ^{g-1}/(g-1)! \in {\rm H}^{2g-2}(A, \mathbb {Z})$. By a version of Welters’ criterion (see [BL04, Lemma 12.2.3]), this implies that $\phi ^{\ast }\big (\theta _{J(C_i)}\big ) = f_i \cdot \theta \in {\rm H}^2(A, \mathbb {Z})$, where $\theta _{J(C_i)} \in {\rm H}^2(J(C_i), \mathbb {Z})$ is the canonical principal polarization. In particular, statement (vi) holds.

We claim that statement (ii) also 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 the following commutative diagram.

Let $G$ be the kernel of $\pi$. Define $K = \text {Ker}([f_i]_A) = \text {Ker}(f_i\cdot \lambda _A) \cong (\mathbb {Z}/f_i)^{2g} \subset A$, and $U = \text {Ker}({\widehat {\pi }} \circ \lambda _0) \subset A_0$. Also define $H = \text {Ker}(\lambda _0)$, 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

(15)\begin{equation} h | u | k | f_i \quad \text{and} \quad a | k | f_i. \end{equation}

Then define $B = \text {Ker}( {\widehat {j}} \circ \lambda ) \subset J(C_i)$ 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 is an isogeny. As the degree of an isogeny is multiplicative in compositions, we have $\deg (\alpha ) = \deg \big (\psi \circ (\pi \times \text {id}) \big ) = \deg (\psi ) \cdot \deg (\pi \times \text {id}) = h \cdot \deg (\pi ) = h \cdot a$. In particular, $p$ does not divide $\deg (\alpha )$ because $h$ and $a$ divide $f_i$ by (15).

Proof of Theorem 1.3 According to Theorem 1.1, it suffices to show that the cohomology class $c_1(\mathcal P_A)^{2g-1}/(2g-1)! \in {\rm H}^{4g-2}(A \times {\widehat {A}}, \mathbb {Z})$ is algebraic for $[(A,\lambda )]$ in a dense subset $X$ of ${\mathsf {A}}_{g,\delta }(\mathbb {C})$ as in the statement. Define $D = \text {diag}(\delta _1, \dotsc, \delta _g)$ and define, for each subring $R$ of $\mathbb {C}$, a group

\[ \text{Sp}_{2g}^\delta(R) = \left\{M \in \text{GL}_{2g}(R) \biggm\vert M \begin{pmatrix} 0 & D \\ -D & 0 \end{pmatrix} M^t = \begin{pmatrix} 0 & D \\ -D & 0 \end{pmatrix} \right\}. \]

The isomorphism

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

induces an action of $\text {Sp}_{2g}^\delta (\mathbb {Z})$ on the genus $g$ Siegel space ${\mathbb {H}}_g$, and the period map defines an isomorphism of complex analytic spaces ${\mathsf {A}}_{g,\delta }(\mathbb {C}) \cong \text {Sp}_{2g}^\delta (\mathbb {Z}) \setminus {\mathbb {H}}_g$ (see [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 $\text {Sp}_{2g}^\delta (\mathbb {Z}[1/\ell ]) \cdot x \subset {\mathbb {H}}_g$. Let $(A, \lambda )$ be a polarized abelian variety admitting a period matrix equal to $x$. The image of $\text {Sp}_{2g}^\delta (\mathbb {Z}[1/\ell ]) \cdot x$ 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 $n,m\in \mathbb {Z}_{\geqslant 0}$ and an isomorphism of polarized rational Hodge structures $\phi \colon {\rm H}_1(B, \mathbb {Q}) \xrightarrow {\sim } {\rm H}_1(A, \mathbb {Q})$ 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 non-negative 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 4.1.

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

\[ \pi \colon \prod_{i = 1}^g {\mathsf{A}}_{1,1} \to {\mathsf{A}}_{g,\delta}, \quad \big([(E_1, \lambda_1)], \dotsc, [(E_g, \lambda_g)] \big) \mapsto ([E_1 \times \cdots \times E_g, \delta_1\cdot \lambda_1 \times \cdots \times \delta_g\cdot \lambda_g]) \]

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 $\text {Sp}_{2g}^\delta (\mathbb {Z}[1/\ell ])$ is dense in $\text {Sp}_{2g}(\mathbb {R})$. As $\text {Sp}_{2g}^{\delta }(\mathbb {Q})$ arises as the group of rational points of an algebraic subgroup $\text {Sp}_{2g}^\delta$ of $\text {GL}_{2g}$ over $\mathbb {Q}$ (see [PR94, Chapter 2, § 2.3.2]), which is isomorphic to $\text {Sp}_{2g}$ over $\mathbb {Q}$, this claim follows from the well-known fact that for $S = \{\ell \} \subset \text {Spec } \mathbb {Z}$, the algebraic group $\text {Sp}_{2g}$ satisfies the strong approximation property with respect to $S$ (see [PR94, Chapter 7, § 7.1]; indeed, this is classical and follows from the non-compactness of $\text {Sp}_{2g}(\mathbb {Q}_\ell )$, see [PR94, Theorem 7.12]).

Let $V = \pi \big ( \prod _{i = 1}^g {\mathsf {A}}_{1,1}\big ) \subset {\mathsf {A}}_{g,\delta }$. Then $X' := \text {Sp}_{2g}^\delta (\mathbb {Z}[1/\ell ]) \cdot V = \bigcup _iZ_i \subset {\mathsf {A}}_{g, \delta }(\mathbb {C})$ is a countable union of closed analytic subsets $Z_i \subset {\mathsf {A}}_{g, \delta }(\mathbb {C})$ of dimension $\dim Z_i \geqslant g$ such that $X' \subset {\mathsf {A}}_{g, \delta }(\mathbb {C})$ is dense in the analytic topology and $c_1(\mathcal P_A)^{2g-1}/(2g-1)! \in {\rm H}^{4g-2}( A \times {\widehat {A}}, \mathbb {Z})$ is algebraic for every polarized abelian variety $(A, \lambda )$ of polarization type $\delta$ whose isomorphism class lies in $X'$. 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 $c_1(\mathcal P_A)^{2g-1}/(2g-1)! \in {\rm H}^{4g-2}( A \times {\widehat {A}}, \mathbb {Z})_{\text {alg}}$ is a countable union $W = \bigcup _jY_j \subset {\mathsf {A}}_{g, \delta }(\mathbb {C})$ of closed algebraic subsets $Y_j \subset {\mathsf {A}}_{g, \delta }(\mathbb {C})$.

Indeed, assuming the claim, we see that $X' \subset W$ and because 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 $U' \subset U$, the locus in $U'(\mathbb {C})$ where $c_1(\mathcal P_A)^{2g-1}/(2g-1)!$ is algebraic is a countable union of closed algebraic subvarieties of $U'(\mathbb {C})$.

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 4.6 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 § 5 below), there is a countable union $X = \bigcup _iZ_i \subset {\mathsf {A}}_{g, \delta }(\mathbb {C})$ as in Theorem 1.3 such that $\dim Z_i \geqslant 3g-3$. 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 4.7 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 is as follows.

Definition 5.1 Let $d,k,n \in \mathbb {Z}_{\geqslant 1}$ 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 $X$ (see [Voi16, Per22]):

\[ {\rm Z}^{2d-2k}(X) := \text{Hdg}^{2d-2k}(X, \mathbb{Z})/ {\rm H}^{2d-2k}(X,\mathbb{Z})_{\text{alg}} = \text{Coker}\, \big( \text{CH}_k(X) \to \text{Hdg}^{2d-2k}(X, \mathbb{Z}) \big). \]

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

Lemma 5.1 Let $A$ be a complex abelian variety of dimension $g$.

1. (i) Let $n$ be a positive integer and let $\mathcal F_n\colon \text {CH}^1({\widehat {A}}) \to \text {CH}_1(A)$ be a group homomorphism such that the following diagram commutes.

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

2. (ii) Let $n \in \mathbb {Z}_{\geqslant 1}$ be such that $n \cdot c_1(\mathcal P_A)^{2g-2}/(2g-2)!$ is algebraic. Then a homomorphism $\mathcal F_n$ as in statement (i) exists.

Proof. Statement (i) follows immediately from the fact that $\text {CH}^1({\widehat {A}}) \to \text {Hdg}^2({\widehat {A}},\mathbb {Z})$ is surjective by Lefschetz $(1,1)$. To prove statement (ii), define $\sigma _A \in {\rm H}^{4g-4}(A \times {\widehat {A}},\mathbb {Z})$ as the class $c_1(\mathcal P_A)^{2g-2}/(2g-2)!$ and similarly define $\sigma _{{\widehat {A}}} = c_1(\mathcal P_{{\widehat {A}}})^{2g-2}/(2g-2)! \in {\rm H}^{4g-4}({\widehat {A}} \times A,\mathbb {Z})$. Observe that $n \cdot \sigma _{{\widehat {A}}}$ is algebraic because $n \cdot \sigma _A$ is. Let $\Sigma _n \in \text {CH}_2({\widehat {A}} \times A)$ such that $cl(\Sigma _n) = n \cdot \sigma _{{\widehat {A}}}$. This gives a commutative diagram.

As $\pi _{2,\ast } \circ \big ((-)\cdot n \cdot \sigma _{{\widehat {A}}}\big ) \circ \pi _1^\ast = n \cdot \mathscr{F}_{{\widehat {A}}}$, the homomorphism $\mathcal F_n := \pi _{2,\ast } \circ \big ((-)\cdot \Sigma _n\big ) \cdot \pi _1^\ast$ has the required property.

Theorem 5.2 Consider a complex abelian variety $A$ of dimension $g$.

1. (i) Let $n \in \mathbb {Z}_{\geqslant 1}$ be such that $n \cdot c_1(\mathcal P_A)^{2g-1}/(2g-1)!$ is algebraic. Then $n^2 \cdot c_1(\mathcal P_A)^{2g-2}/ (2g-2)!$ is algebraic. In particular, $A$ satisfies the integral Hodge conjecture up to factor $\gcd (n^2, (2g-2)!)$ in this case.

2. (ii) If $A$ is principally polarized, and $n \in \mathbb {Z}_{\geqslant 1}$ is such that $n \cdot \gamma _\theta \in \text {Hdg}^{2g-2}(A, \mathbb {Z})$ is algebraic, then $n \cdot c_1(\mathcal P_A)^{2g-1}/(2g-1)!\in \text {Hdg}^{4g-2}(A \times {\widehat {A}}, \mathbb {Z})$ is algebraic.

3. (iii) 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. (i) By Lemma 3.4, one has

\[ c_1(\mathcal P_A)^{2g-2}/(2g-2)! = (-1)^g\cdot \big( c_1(\mathcal P_A)^{2g-1}/(2g-1)!\big)^{\star 2}/2! \in {\rm H}^{4g-4}(A \times {\widehat{A}},\mathbb{Z}). \]

By Theorem 3.7, this implies that if $n \cdot c_1(\mathcal P_A)^{2g-1}/(2g-1)!$ is algebraic, then also the element $n^2\cdot c_1(\mathcal P_A)^{2g-2}/(2g-2)!$ is algebraic. As $(2g-2)! \cdot c_1(\mathcal P_A)^{2g-2}/(2g-2)!$ is algebraic, it follows that $\gcd (n^2, (2g-2)!) \cdot c_1(\mathcal P_A)^{2g-2}/(2g-2)!$ is algebraic. We are done by Lemma 5.1.

(ii) This follows from Lemma 3.5.

(iii) This follows from Lemma 5.1, parts (i) and (ii) and the fact that if $A$ is a $g$-dimensional Prym variety with principal polarization $\theta \in \text {Hdg}^2(A,\mathbb {Z})$, then $2 \cdot \gamma _\theta \in {\rm H}^{2g-2}(A,\mathbb {Z})$ 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 $u \in {\rm H}_{ {\unicode{x00E9} \text t}}^{2i}(X, \mathbb {Z}_\ell (i))$ is an integral Tate class if it is fixed by some open subgroup of $\text {Gal}(k/k_0)$. Totaro has shown that for codimension-one cycles on $X$, the Tate conjecture over $k$ implies the integral Tate conjecture over $k$ (see [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, FWG⁺86] and Zarhin [Zar74a, Zar74b]. 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

(16)\begin{equation} \mathscr{F}_A \colon {\rm H}_{{\unicode{x00E9} \text t}}^2(A, \mathbb{Z}_\ell(1)) \xrightarrow{\sim} {\rm H}_{{\unicode{x00E9} \text t}}^{2g-2}({\widehat{A}}, \mathbb{Z}_\ell(g-1)), \end{equation}

see [Tot21, § 7]. As (16) 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 (16) to a homomorphism $\text {CH}^1(A)_{\mathbb {Z}_\ell } \to \text {CH}_1({\widehat {A}})_{\mathbb {Z}_\ell }$.

Proof of Theorem 1.6 Combine the above with Propositions 3.11 and 3.12(ii).

For an abelian variety $X$ over the separable closure $k$ of a finitely generated field, call $\alpha \in {\rm H}^{2\bullet }_{\unicode{x00E9}\text {t}}(X, \mathbb {Z}_\ell (\bullet ))$ algebraic if $\alpha$ is in the image of the cycle class map $\text {CH}(X) \otimes \mathbb {Z}_\ell \to {\rm H}^{2\bullet }_{\unicode{x00E9}\text {t}}(X, \mathbb {Z}_\ell (\bullet ))$.

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. (i) The classes $c_1(\mathcal P_A)^{2g_A-1}/(2g_A-1)!$ in ${\rm H}_{\unicode{x00E9}\text {t}}^{4g_A-2}(A \times {\widehat {A}}, \mathbb {Z}_\ell (2g_A-1))$ and $c_1(\mathcal P_B)^{2g_B-1}/ (2g_B-1)!$ in ${\rm H}_{\unicode{x00E9}\text {t}}^{4g_B-2}(B \times {\widehat {B}}, \mathbb {Z}_\ell (2g_B-1))$ 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. (ii) 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. (iii) Let $g = g_A$ and let $\theta \in {\rm H}^2_{\unicode{x00E9}\text {t}}(A, \mathbb {Z}_\ell (1))$ be the first Chern class of an ample line bundle that induces a principal polarization on $A$. Suppose that $\theta ^{g-1}/(g-1)! \in {\rm H}_{ {\unicode{x00E9} \text t}}^{2g-2}(A, \mathbb {Z}_\ell (g-1))$ is algebraic. Then for every algebraic cohomology class $\alpha \in \bigoplus _{j > 0} {\rm H}_{ {\unicode{x00E9} \text t}}^{2j}(A, \mathbb {Z}_\ell (j)) \subset {\rm H}_{ {\unicode{x00E9} \text t}}^{2\bullet }(A, \mathbb {Z}_\ell (\bullet ))$ and every $i \in \mathbb {Z}_{\geqslant 1}$, the cohomology class $\alpha ^i/i! \in {\rm H}_{ {\unicode{x00E9} \text t}}^{2\bullet }(A, \mathbb {Z}_\ell (\bullet ))$ is algebraic.

Proof. (i) See (9).

(ii) 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.

(iii) This follows from Propositions 3.11 and 3.12(ii).

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 $\theta _\ell \in {\rm H}_{\unicode{x00E9}\text {t}}^{2}(A_{\bar{\mathbb{Q}}}, \mathbb {Z}_\ell (1))$ be the $\ell$-adic étale cohomology class of the polarization of $A_{\bar{\mathbb{Q}}}$. Similarly, define $\theta _\mathbb {C} \in \text {NS}(A_\mathbb {C}) \subset {\rm H}^{2}(A_\mathbb {C}, \mathbb {Z})$ as the polarization of the complex abelian variety $A_\mathbb {C}$. By Theorems 1.1 and 1.6, it suffices to show that $\gamma _{\theta _\mathbb {C}} \in {\rm H}^{2g-2}(A_\mathbb {C}, \mathbb {Z})$ is algebraic if and only if $\gamma _{\theta _\ell } \in {\rm H}_{\unicode{x00E9}\text {t}}^{2g-2}(A_{\bar{\mathbb{Q}}}, \mathbb {Z}_\ell (g-1))$ is in the image of (1) for each prime number $\ell$. The Artin comparison theorem gives an isomorphism of $\mathbb {Z}_\ell$-algebras

\[ \phi\colon {\rm H}^{\bullet}_{\unicode{x00E9}\text {t}}(A_{\bar{\mathbb{Q}}}, \mathbb{Z}_\ell) = {\rm H}^{\bullet}_{\unicode{x00E9}\text {t}}(A_{\mathbb{C}}, \mathbb{Z}_\ell) \cong {\rm H}^{\bullet}(A_\mathbb{C}, \mathbb{Z}) \otimes_\mathbb{Z} \mathbb{Z}_\ell. \]

As $\phi$ is compatible with the cycle class maps $cl_{\bar{\mathbb{Q}}} \colon \text {CH}(A_{\bar{\mathbb{Q}}}) \to {\rm H}^{\bullet }_{\unicode{x00E9}\text {t}}(A_{\bar{\mathbb{Q}}}, \mathbb {Z}_\ell )$ and $cl_\mathbb {C} \colon \text {CH}(A_\mathbb {C}) \to {\rm H}^{\bullet }(A_\mathbb {C}, \mathbb {Z})$, we have $\phi (\gamma _{\theta _\ell }) = \gamma _{\theta _\mathbb {C}}$. Define

\[ {\rm R}^{2g-2}(A) = \text{Coker}\,\big(\text{CH}_1(A_\mathbb{C}) \to {\rm H}^{2g-2}(A_\mathbb{C},\mathbb{Z})\big). \]

Then ${\rm R}^{2g-2}(A)\otimes \mathbb {Z}_\ell = \text {Coker}\,\big (\text {CH}_1(A_\mathbb {C})_{\mathbb {Z}_\ell } \to {\rm H}^{2g-2}(A_\mathbb {C},\mathbb {Z}_\ell )\big )$. Suppose that $\gamma _{\theta _\ell }$ is in the image of (1) for every prime number $\ell$. The image of $\gamma _{\theta _\mathbb {C}}$ in ${\rm R}^{2g-2}(A)\otimes \mathbb {Z}_\ell$ is then zero for each prime $\ell$, which implies that the image of $\gamma _{\theta _\mathbb {C}}$ in ${\rm R}^{2g-2}(A)$ is zero, i.e. $\gamma _{\theta _\mathbb {C}}$ is algebraic. Conversely, suppose that $\gamma _{\theta _\mathbb {C}} = \sum _{i= 1}^k n_i \cdot cl(C_i)$ for some smooth projective curves $C_i$ over $\mathbb {C}$. The Hilbert scheme $\mathcal H = \text {Hilb}_{A_K/K}$ 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 $C_i' \subset A_{\bar{\mathbb{Q}}}$ over $\bar{\mathbb{Q}}$. If $\Gamma = \sum _i n_i \cdot [C_i'] \in \text {CH}_1(A_{\bar{\mathbb{Q}}})$, then we have $cl_\mathbb {C}(\Gamma _\mathbb {C}) = \gamma _{\theta _\mathbb {C}}$ by Lemma 4.4, hence $cl_{\bar{\mathbb{Q}}}(\Gamma ) = \gamma _{\theta _\ell }$.

Another corollary of Theorem 1.6 is that the integral Tate conjecture for one-cycles on principally polarized abelian varieties is stable under specialization. Indeed, one has the following (cf. 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 $S = \text {Spec } \mathcal {O}_K$ and let $A \to S$ be the Néron model of $A_K$. Let $R$ (respectively, $K_{\mathfrak p}$) be the completion of $\mathcal {O}_K$ (respectively, $K$) 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 the following commutative diagram, where the square on the right is provided in [Ful98, Example 20.3.5].

(17)

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, § 1.2, Definition 1] and because 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 $\text {CH}^1(A_{\bar K})_{\mathbb {Z}_\ell }$ of a theta divisor on $A_{\bar K}$ is sent to the class in $\text {CH}^1(A_{\bar \kappa })_{\mathbb {Z}_\ell }$ of a theta divisor on $A_{\bar \kappa }$. Thus, the minimal class $\gamma _{\theta _{\bar K}} \in {\rm H}^{2g-2}_{\unicode{x00E9}\text {t}}(A_{\bar K}, \mathbb {Z}_\ell (g-1))$ is sent to the minimal class $\gamma _{\theta _{\bar \kappa }} \in {\rm H}^{2g-2}_{\unicode{x00E9}\text {t}}(A_{\bar \kappa }, \mathbb {Z}_\ell (g-1))$ by the isomorphism on the bottom of diagram (

17

). 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 non-negative 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 non-negative integer $N$ such that the isogeny $[q^{2g\cdot N}]\colon A_y \to A_y$ factors through $A_x$, hence $[q^{2g\cdot N}]^{\ast }(\gamma _\theta ) = q^{(2g-2)\cdot 2g \cdot N} \cdot \gamma _\theta$ is algebraic for the first Chern class $\theta$ of the principal polarization on $A_y$, because $A_x$ satisfies the integral Tate conjecture for one-cycles over $k$ by Theorem 1.6. This implies that $\gamma _\theta$ is algebraic (as $q > (g-1)!$) so that, by applying Theorem 1.6 again, the claim follows. Now ${\mathscr {G}}_q(z)$ is dense in ${\mathsf {A}}_g$ for any ordinary principally polarized abelian variety $(A_z,\lambda _z)$, see [Cha95, Theorem 2]. Therefore, ${\mathscr {G}}_q(x)$ is dense in ${\mathsf {A}}_g$ and the proof is finished.