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

We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, any product of Jacobians of smooth projective curves 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 proper curve of compact type over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture are dense in their moduli space.


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 A. The correspondence attached to the Poincaré bundle P A on A × A defines a powerful duality between the derived categories, rational Chow groups and cohomology of A and A [Muk81; Bea83;Huy06].We shall refer to such morphisms as Fourier transforms.
On the level of cohomology, the Fourier transform preserves integral -adic étale cohomology when k = k s and integral Betti cohomology when k = C.It is thus natural to ask whether the Fourier transform on rational Chow groups preserves integral cycles modulo torsion or, more generally, lifts to a homomorphism between integral Chow groups.This question was raised by Moonen-Polishchuk [MP10] and Totaro [Tot21].More precisely, Moonen and Polishchuk gave a counterexample for abelian varieties over non-closed fields and asked about the case of algebraically closed fields.
In this paper we further investigate this question with a view towards applications concerning the integral Hodge conjecture for one-cycles when A is defined over C. To state our main result, we recall that whenever ι : C → A is a smooth curve, the image of the fundamental class under the pushforward map ι * : H 2 (C, Z) → H 2 (A, Z) ∼ = H 2g−2 (A, Z) defines a cohomology class [C] ∈ H 2g−2 (A, 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 : CH 1 (A) → Hdg 2g−2 (A, Z), which is a direct summand of a natural graded ring homomorphism cl : CH(A) → H • (A, 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 α ∈ H • (A, 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 H 2g−2k (A, Z) are algebraic.Although the integral Hodge conjecture fails in general [AH62;Tre;Tot97], it is an open question for abelian varieties.Our main result is as follows.
Remark that Condition 5 is stable under products, so a product of principally polarized abelian varieties satisfies the integral Hodge conjecture for one-cycles if and only if each of the factors does.More importantly, if J(C) is the Jacobian of a smooth projective curve C of genus g, then every integral Hodge class of degree 2g − 2 on J(C) is a Z-linear combination of curves classes: Theorem 1.2.Let C 1 , . . ., C n be smooth projective curves over C. Then the integral Hodge conjecture for one-cycles holds for the product of Jacobians J(C 1 ) × See Remark 4.2.1 for another approach towards Theorem 1.2 in the case n = 1.A second consequence of Theorem 1.1 is that the integral Hodge conjecture for one-cycles on principally polarized complex abelian varieties is stable under specialization, see Corollary 4.3.An application of somewhat different nature is the following density result, proven in Section 4.2: Theorem 1.3.Let δ = (δ 1 , . . ., δ g ) be positive integers such that δ i |δ i+1 and let A g,δ (C) be the coarse moduli space of polarized abelian varieties over C with polarization type δ.There is a countable union X ⊂ A g,δ (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 δ and is often greater than g.For instance, when δ = 1 and g ≥ 2, there is a set X as in the theorem, whose elements are prime-power isogenous to products of Jacobians of curves.Therefore, the components of X have dimension 3g − 3 in this case, c.f. Remark 4.7.
One could compare Theorem 1.1 with the following statement, proven by Grabowski [Gra04]: if g is a positive integer such that the minimal cohomology class γ θ = θ 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 ≥ 0 be an integer.Then Poincaré duality induces a canonical isomorphism ϕ : The map ϕ respects the Hodge structures and thus induces an isomorphism Hdg 2i (A, Z) ∼ = Hdg 2g−2i ( A, Z).However, it is unclear a priori whether ϕ sends algebraic classes to algebraic classes.We prove that the algebraicity of c 1 (P A ) 2g−1 /(2g − 1)! forces ϕ to be algebraic, i.e. to be induced by a correspondence To prove this, we lift the cohomological Fourier transform to a homomorphism between integral Chow groups whenever c 1 (P A ) 2g−1 /(2g − 1)! is algebraic.For this we use a theorem of Moonen-Polishchuk saying that the ideal of positive dimensional cycles in the Chow ring with Pontryagin product of an abelian variety admits a divided power structure [MP10, Theorem 1.6].
In Section 5, we consider an abelian variety A /C of dimension g and ask: if n ∈ Z ≥1 is such that can we bound the order of Z 2g−2 (A) in terms of g and n?For a smooth complex projective d-dimensional variety X, Z 2d−2 (X) is called the degree 2d−2 Voisin group of X [Per20], is a stably birational invariant [Voi16, Lemma 2.20], and related to the unramified cohomology groups by Colliot-Thélène-Voisin and Schreieder [CTV12;Sch21].We prove that if n Since it is well known that for Prym varieties, the Hodge class 2 • γ θ is algebraic, these observations lead to the following result (see also Theorem 5.3).
Theorem 1.5.Let A be a g-dimensional Prym variety over C. Then 4 For the study of the liftability of the Fourier transform, which was initiated by Moonen and Polishchuk in [MP10], it is more natural to consider abelian varieties defined over arbitrary fields.For this reason we define and study integral Fourier transforms in this generality, see Section 3. We provide, for an abelian variety principally polarized by a symmetric ample line bundle, necessary and sufficient conditions for an integral Fourier transform to exist, see Theorem 3.8.This generality also allows to obtain the analogue of Theorem 1.1 over the separable closure k of a finitely generated field.Recall that a smooth projective variety X of dimension d over k satisfies the integral Tate conjecture for one-cycles over k if, for every prime number different from char(k) and for some finitely generated field of definition k 0 ⊂ k of X, the cycle class map is surjective, where U ranges over the open subgroups of 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.The abelian variety A satisfies the integral Tate conjecture for one-cycles over k if the cohomology class is the class of a one-cycle with Z -coefficients for every prime number < (2g − 1)! unequal to char(k).
2. Suppose that A is principally polarized and let θ ∈ H 2 ét (A, Z (1)) be the class of the polarization.The map (1) is surjective if ) is in its image.In particular, if > (g − 1)! then this always holds.Thus A satisfies the integral Tate conjecture for one-cycles if γ θ is in the image of (1) for every prime number < (g − 1)! unequal to 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 ⊂ C, the integral Hodge conjecture for one-cycles on A C is equivalent to the integral Tate conjecture for one-cycles on A K (Corollary 6.2), which in turn implies the integral Tate conjecture for one-cycles on the geometric special fiber A k(p) of its Néron model of A/O K for any prime p ⊂ 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 A g be the coarse moduli space over k of principally polarized abelian varieties of dimension g over k.The subset of 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 A g .
• We let k be a field with separable closure k s and a prime number different from the characteristic of k.For a smooth projective variety X over k, we let CH(X) be the Chow group of X and define CH(X) ) be the i-th degree étale cohomology group with coeffients in Z (a), a ∈ Z. • Often, A will denote an abelian variety of dimension g over k, with dual abelian variety A and (normalized) Poincaré bundle P A on A × A. The abelian group CH(A) will in that case be equipped with two ring structures: the usual intersection product • as well as the Pontryagin product .Recall that the latter is defined as follows: Here, as well as in the rest of the paper, π i denotes the projection onto the i-th factor, ∆ : A → A × A the diagonal morphism, and m : A × A → A the group law morphism of A. There is a similar Pontryagin product on étale cohomology, and on Betti cohomology if k = C.
• For any abelian group M and any element x ∈ M , we will denote by

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

Integral Fourier transforms and integral Hodge classes
For abelian varieties A over k = k s , it is unknown whether the Fourier transform F A : CH(A) Q → CH( A) Q preserves the subgroups of integral cycles modulo torsion.A sufficient condition for this to hold is that F A lifts to a homomorphism CH(A) → CH( A).In this section we outline a second consequence of such a lift CH(A) → CH( A) when A is defined over the complex numbers: the existence of an integral lift of F A implies the integral Hodge conjecture for one-cycles on A.
Let A be an abelian variety over k.The Fourier transform on the level of Chow groups is the group homomorphism induced by the correspondence ch(P A ) ∈ CH(A × A) Q , where ch(P A ) is the Chern character of P A .Similarly, one defines the Fourier transform on the level of étale cohomology: In fact, F A preserves the integral cohomology classes and induces, for each integer j with 0 ≤ j ≤ 2g, an isomorphism [Bea83, Proposition 1], [Tot21, page 18]: Similarly, if k = C, then ch(P A ) induces, for each integer i with 0 ≤ i ≤ 2g, an isomorphism of Hodge structures In [MP10], Moonen and Polishchuk consider an isomorphism φ : A ∼ − → A, a positive integer d, and define the notion of motivic integral Fourier transform of (A, φ) up to factor d. The definition goes as follows.Let 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 F : h(A) → h(A) in M(k) is a motivic integral Fourier transform of (A, φ) up to factor d if the following three conditions are satisfied: (i) the induced morphism h(A) Q → h(A) Q is the composition of the usual Fourier transform with the isomorphism φ * : h For our purposes, we will consider similar homomorphisms CH(A) → CH( A).However, to make their existence easier to verify (c.f.Theorem 3.8) we relax some of the above conditions: Definition 3.1.Let A /k be an abelian variety and let F : CH(A) → CH( A) be a group homomorphism.We call F a weak integral Fourier transform if the following diagram commutes: A group homomorphism F : CH(A) → CH( A) is an integral Fourier transform up to homology if the following diagram commutes: Similarly, a Z -module homomorphism F : CH(A) Z → CH( A) Z is called an -adic integral Fourier transform up to homology if F is compatible with F A and the -adic cycle class maps. .
) is an integral Fourier transform up to homology (resp.an -adic integral Fourier transform up to homology).Similarly, any cycle 2. If F : CH(A) → CH( A) is a weak integral Fourier transform, then F is an integral Fourier transform up to homology, the Z -module ⊕ r≥0 H 2r ét ( A ks , Z (r)) being torsion-free.If k = C, then F : CH(A) → CH( A) is an integral Fourier transform up to homology if and only if F is compatible with the Fourier transform The relation between integral Fourier transforms and integral Hodge classes is as follows: Lemma 3.3.Let A be a complex abelian variety and F : CH(A) → CH( A) an integral Fourier transform up to homology.
1.For each i ∈ Z ≥0 , the integral Hodge conjecture for degree 2i classes on A implies the integral Hodge conjecture for degree 2g − 2i classes on A.
and, therefore, the integral Hodge conjectures for degree 2i classes on A and degree 2g − 2i classes on A are equivalent for all i.
Proof.We can extend Diagram (4) to the following commutative diagram: The composition H 2i (A, Z) → H 2g−2i ( A, Z) appearing on the bottom line agrees up to a suitable Tate twist with the map F A of Equation (2).Therefore, we obtain a commutative diagram: (5) Thus the surjectivity of cl i implies the surjectivity of cl i .Moreover, if F is induced by some Γ ∈ CH(A × A), then replacing A by A and A by A in the argument above shows that the images of cl i and cl i are identified under the isomorphism

Properties of the Fourier transform on rational Chow groups
Let A be a complex abelian variety.Observe that, for any j ∈ Z ≥1 and x ∈ H 2j (A, Z), one has In particular, the ideal The analogue of this statement in -adic étale cohomology holds when A is an abelian variety over a separably closed field.Lemma 3.3 suggests that to prove Theorem 1.1, one would need to show that for a complex abelian variety of dimension g whose minimal Poincaré class c 1 With this goal in mind we shall study Fourier transforms on rational Chow groups in Section 3.2, and investigate how these relate to ch(P A ) ∈ CH(A × A) Q .In turns out that the cycles c 1 (P A ) i /i! ∈ CH(A × A) Q satisfy several relations that are very similar to those proved by Beauville for the cycles θ i /i! ∈ CH(A) Q in case A is principally polarized, see [Bea83].Since we will need these results in any characteristic in order to prove Theorem 1.6, we will work over our general field k, see Section 2.
Let A be an abelian variety over k.Define where a n denotes the n-fold Pontryagin product of a (see Section 2).The key to Theorem 1.1 will be the following: Proof.The most important ingredient in the proof is the following: To prove Claim ( * ), we lift the desired equality in the rational Chow group of A × A to an isomorphism in the derived category be the Fourier-Mukai transform attached to P X ∈ D b (X × X) as in [Huy06, Definition 5.1].Evaluating it at P A gives the object whose Chern character is exactly F X (exp(c 1 (P A ))). Consider the permutation map with inverse (321).We have We conclude that Φ P X (P A ) ∼ = π 14, * π * 12 P A ⊗ π * 23 P A ⊗ π * 34 P A .The latter is isomorphic to the Fourier-Mukai kernel of the composition This is the Fourier-Mukai transform with kernel . By uniqueness of the Fourier-Mukai kernel of an equivalence [Orl97, Theorem 2.2], it follows that Φ P X (P

Since the Chern character of P
Next, we claim that (−1) g • F A×A (−c 1 (P A )) = R A .To see this, recall that for each integer i with 0 ≤ i ≤ g, there is a canonical Beauville decomposition In particular, we have R A ∈ CH 2g−1,0 (A × A) Q .The fact that P A is symmetric also implies -via Claim ( * ) -that we have For a g-dimensional abelian variety X and any x, y ∈ CH(X) Q , one has 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.[EGM21]).This implies (see also ).This allows us to conclude that which finishes the proof.
Next, assume that A is equipped with a principal polarization λ : A ∼ − → A, and define Here (id, λ) (resp.
One can understand the relation between ) and y → (0, y) respectively.Define a one-cycle τ on A × A as follows: Proof.Identify A and A via λ.This gives c 1 , and the Fourier transform becomes an endomorphism In particular, the claim follows.
Next, recall that F A×A (c 1 (P A )) = (−1) g+1 • R A , see Claim ( * ).So at this point, it suffices to prove the identity 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 : X → Y be a homomorphism with dual homomorphism f : Y → X.We then have the following equalities [MP10, (3.7.1)]: Lemma 3.6 (Beauville).Let A be an abelian variety over k, principally polarized by λ : Proof.Our proof follows the proof of [Bea83, Lemme 1], but has to be adapted, since Θ does not necessarily come from a symmetric ample line bundle on A. Since one still has c 1 , the argument can be made to work: one has For codimension reasons, one has π 2, * (m over k s the cycle Θ becomes the cycle class attached to a symmetric ample line bundle.

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 In particular, for each element x ∈ CH >0 (A) and each n ∈ Z ≥1 , there is a canonical element , we may then define E(x) = n≥0 x [n] ∈ CH(A) as the -exponential of x in terms of its divided powers.
Together with the results of Section 3.2, Theorem 3.7 enables us to provide several criteria for the existence of a weak integral Fourier transform.We recall that for an abelian variety A over k, principally polarized by λ : A ∼ − → A, we defined Θ ∈ CH 1 (A) Q to be the symmetric ample class attached to the polarization λ, see Equation (7).
Theorem 3.8.Let A /k be an abelian variety of dimension g.The following are equivalent:

The cycle ch(P
Moreover, if A carries a symmetric ample line bundle that induces a principal polarization λ : A ∼ − → A, then the above statements are equivalent to the following equivalent statements:

The one-cycle
6.The abelian variety A admits a weak integral Fourier transform.

The Fourier transform
8.There exists a PD-structure on the ideal CH >0 (A)/torsion ⊂ CH(A)/torsion.
Proof.Suppose that 1 holds, and let Thus 2 holds.We claim that 3 holds as well.Indeed, consider the line bundle P A× A on the abelian variety X = A × A × A × A; one has that P A× A ∼ = π * 13 P A ⊗ π * 24 P A , which implies that We conclude that R A× A lifts to CH 1 (X) which, by the implication [1 =⇒ 2] (that has already been proved), implies that 3 holds.The implication [3 =⇒ 1] follows from the fact that (−1) g • F A×A (−c 1 (P A )) = R A (see Equation ( 6)).Therefore, we have [1 ⇐⇒ 2 ⇐⇒ 3].
Let us from now on assume that A is principally polarized by λ : A ∼ − → A, where λ is the polarization attached to a symmetric ample line bundle L on A. Moreover, in what follows we shall identify A and A via λ.
Suppose that 4 holds and let Define CH 1,0 (A) := Pic sym (A) to be the group of isomorphism classes of symmetric line bundes on A. Then S A induces a homomorphism F : CH 1,0 (A) → CH 1 (A) defined as the composition commutes.On the other hand, since the line bundle L is symmetric, we have The class c 1 (L) ∈ CH 1,0 (A) of the line bundle L thus lies above Θ ∈ CH 1 (A) Q .Therefore, F(c 1 (L)) ∈ CH 1 (A) lies above Γ Θ = (−1) g−1 F A (Θ) by the commutativity of (10), and 5 holds.Suppose that 5 holds.Then 1 follows readily from Lemma 3.5.Moreover, if 2 holds, then ch(P A ) ∈ CH(A × A) Q lifts to CH(A × A), hence in particular 4 holds.Since we have already proved that 1 implies 2, we conclude that [4 =⇒ 5 =⇒ 1 =⇒ 2 =⇒ 4].
Assume that 7 holds.The fact that F A (CH(A)/torsion) ⊂ CH(A)/torsion implies that Thus, the restriction of the Fourier transform F A to CH(A)/torsion defines an isomorphism Now if R is a ring and γ is a PD-structure on an ideal I ⊂ R, then γ extends to a PD-structure on I/torsion ⊂ R/torsion.Consequently, the ideal CH >0 (A)/torsion ⊂ CH(A)/torsion admits a PD-structure for the Pontryagin product by Theorem 3.7.Since F A exchanges the Pontryagin and intersection product (up to a sign, see [Bea83, Proposition 3(ii)]), it follows that 8 holds.Since 8 trivially implies 5, we are done.
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:

The class ch(P
Moreover, if A carries an ample line bundle that induces a principal polarization λ : A ∼ − → A, then the above statements are equivalent to the following equivalent statements:

The class γ
) lifts to a cycle in CH 1 (A).

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

The Fourier transform F
Proof. 1.In this case Z (i) = Z and the Artin comparison isomorphism

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

Proof of the main theorem
Let us prove Theorem 1.1.
Corollary 4.1.Let A and B be complex abelian varieties of respective dimensions g A and g B .
1.The Hodge classes are algebraic if and only if A × A, B × B, A × B and A × B satisfy the integral Hodge conjecture for one-cycles.
2. If A and B are principally polarized, then the integral Hodge conjecture for one-cycles holds for A × B if and only if it holds for A and B.
Proof.The first statement follows from Theorem 1.1 and Equation ( 9).The second statement follows from the fact that the minimal cohomology class of the product A × 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 ∈ C and consider the morphism ι : C → J(C) defined by sending a point q to the isomorphism class of the degree zero line bundle O(p − q).Then cl(ι(C)) = γ θ ∈ H 2g−2 (J(C), Z) by Poincaré's formula [Arb+85], so γ θ is algebraic and the result follows from Theorem 1.1.
Remarks 4.2. 1.Let us give another proof of Theorem 1.2 in the case n = 1, i.e. let C be a smooth projective curve of genus g and let us prove the integral Hodge conjecture for one-cycles on J(C) in a way that does not use Fourier transforms.It is classical that any Abel-Jacobi map C (g) → J(C) is birational.On the other hand, the integral Hodge conjecture for one-cycles is a birational invariant, see [Voi07,Lemma 15].Therefore, to prove it for J(C) it suffices to prove it for C (g) .One then uses [Bn02, Corollary 5] which says that for each n ∈ Z ≥1 , there is a natural polarization η on the n-fold symmetric product C (n) such that for any i ∈ Z ≥0 , the map ) is an isomorphism.In particular, the variety C (n) satisfies the integral Hodge conjecture for one-cycles for any positive integer n.
2. Along these lines, observe that the integral Hodge conjecture for one-cycles holds not only for symmetric products of smooth projective complex curves but also for any product Indeed, this follows readily from the Künneth formula.
3. Let C be a smooth projective complex curve of genus g.Our proof of Theorem 1.1 provides an explicit description of Hdg 2g−2 (J(C), Z) depending on Hdg 2 (J(C), Z).More generally, let (A, θ) be a principally polarized abelian variety of dimension g, and identify A and A via the polarization.Then c On the other hand, any β ∈ Hdg 2g−2 (A, Z) is of the form where we write [D] = cl(D) for a divisor D on A, as follows from (12).Therefore, any element β ∈ Hdg 2g−2 (A, Z) may be written as 1.The abelian variety A satisfies the integral Hodge conjecture for one-cycles.
2. For every prime number p, there exists an abelian variety B such that the abelian variety A × B is prime-to-p isogenous to the Jacobian of a smooth projective curve.
3. For every prime number p that divides κ, there exists an abelian variety B such that the abelian variety A × B is prime-to-p isogenous to a Jacobian of a smooth projective curve.
4. For every prime number p, there exists an abelian variety B such that the abelian variety A × B is prime-to-p isogenous to a product of Jacobians of smooth projective curves.
5. For every prime number p dividing κ, there exists an abelian variety B such that the abelian variety A × B is prime-to-p isogenous to a product of Jacobians of smooth projective curves.
Step two: [5 =⇒ 1].Let g be the dimension of A. We want to prove that the class Let p be any prime number that divides κ.Then by Condition 5, there exists an abelian variety B and an isogeny α : , and let m p = deg(α).There exists an isogeny β : Since Y × Y satisfies the integral Hodge conjecture by Theorem 1.2, the Hodge class m be the canonical projections.Then P X ∼ = f * P A ⊗ g * P B .Using this and denoting µ = c 1 (P A ) and ν = c 1 (P B ) we have In particular, the class Let p 1 , . . ., p n be all prime divisors of κ and observe that gcd(κ, m This proves that c 1 (P A ) 2g−1 /(2g − 1)! is a Z-linear combination of algebraic classes, hence algebraic.Condition 1 follows then from Theorem 1.1.
Step four: [6 ⇐= 1 =⇒ 2] for (A, θ A ) principally polarized with ρ(A) = 1.Write θ = θ A .Let Z 1 , . . ., Z n be integral curves Z i ⊂ A and let e 1 , . . ., e n ∈ Z with e i = 0 for all i be such that θ g−1 /(g − 1)! = n i=1 e i • [Z i ] ∈ H 2g−2 (A, Z).Since ρ(A) = 1, the group Hdg 2g−2 (A, Z) is generated by θ g−1 /(g − 1)!.Consequently, we have [Z i ] = f i • θ g−1 /(g − 1)! for some non-zero f i ∈ Z. Hence we can write which implies that n i=1 e i • f i = 1.Now let p be any prime number.Then there exists an integer i with 1 ≤ i ≤ n such that p does not divide f i .Let C i → Z i be the normalization of Z i and let λ A = ϕ θ : A → A be the polarization corresponding to θ.This gives a diagram where ι : is the Abel-Jacobi map (for some p ∈ C), and ϕ * : A = Pic 0 (A) → Pic 0 (C i ) is the pullback of line bundles along ϕ : C i → A. The natural homomorphism a : Pic 0 (C i ) → J(C i ) is an isomorphism by the Abel-Jacobi theorem.Since the triangle on the left in Diagram (14) commutes and As ρ(A) = 1, the map ψ : J(C i ) → A must be surjective, the Picard rank of a non-simple abelian variety being greater than one.Dually, ψ gives rise to a non-zero homomorphism ψ : A → J(C i ), and the simpleness of A implies that ψ is finite onto its image.We claim that the same is true for φ.To prove this, it suffices to show that the kernel of ϕ * : A → Pic 0 (C i ) is finite.Since the homomorphism ι * : J(C i ) → Pic 0 (C i ) induced by the embedding ι : C i → J(C i ) is an isomorphism, dualizing the triangle on the left in Diagram (14) proves our claim.By construction, we have ϕ is the canonical principal polarization.In particular, 6 holds.We claim that also 2 holds.Let j : A 0 → J(C i ) be the embedding of A 0 = φ(A) into J(C i ) and let λ 0 : A 0 → A 0 be the polarization on A 0 induced by j.We have φ * Let G be the kernel of π.
Also define H = Ker(λ 0 ), and observe that H ⊂ U .The exact sequence 0 → G → K → U → 0 shows that if a, k, u and h are the respective orders of G, K, U and H, then one has h|u|k|f i and a|k|f i .
Then define B = Ker( j • λ) ⊂ J(C i ) with inclusion i : B → J(C i ).It is easy to see that B is connected.Moreover, we have A 0 ∩ B = H and, therefore, an exact sequence of commutative group schemes The morphism α : A×B → J(C i ), defined as the composition is an isogeny.Since the degree of an isogeny is multiplicative in compositions, we have deg In particular, p does not divide deg(α) because h and a divide f i by Equation (15).
Proof of Theorem 1.3.According to Theorem 1.1, it suffices to show that the cohomology class c 1 (P A ) 2g−1 /(2g − 1)! ∈ H 4g−2 (A × A, Z) is algebraic for [(A, λ)] in a dense subset X of A g,δ (C) as in the statement.Define D = diag(δ 1 , . . ., δ g ) and define, for each subring R of C, a group induces an action of Sp δ 2g (Z) on the genus g Siegel space H g , and the period map defines an isomorphism of complex analytic spaces A g,δ (C) C), i.e. the set of isomorphism classes of polarized abelian varieties [(B, µ)] ∈ A g,δ (C) for which there exists integers n, m ∈ Z ≥0 and an isomorphism of polarized rational Hodge structures φ : H are morphisms of integral Hodge structures (Hecke orbits were studied in positive characteristic in e.g.[Cha95; CO19]).The degree of the isogeny α = n φ must be k for some nonnegative integer k.In particular, if one abelian variety in a Hecke--orbit happens to be isomorphic to a Jacobian, then every abelian variety in that orbit is -power isogenous to a Jacobian, see Definition 4.5.
The decomposition of a polarized abelian variety into non-decomposable polarized abelian subvarieties is unique [Deb96, Corollaire 2], which implies that the following morphism π : is finite onto its image.Thus A g,δ contains a g-dimensional subvariety on which the integral Hodge conjecture for one-cycles holds.We claim that Sp δ 2g (Z[1/ ]) is dense in Sp 2g (R).Since Sp δ 2g (Q) arises as the group of rational points of an algebraic subgroup Sp δ 2g of GL 2g over Q [PR94, Chapter 2, §2.3.2], which is isomorphic to Sp 2g over Q, this claim follows from the well-known fact that for S = { } ⊂ Spec Z, the algebraic group Sp 2g satisfies the strong approximation property with respect to S [PR94, Chapter 7, §7.1] (indeed, this is classical and follows from the non-compactness of Sp 2g (Q ), see [PR94, Theorem 7.12]). Let is dense in the analytic topology and c 1 (P A ) 2g−1 /(2g − 1)! ∈ H 4g−2 (A × A, Z) is algebraic for every polarized abelian variety (A, λ) of polarization type δ whose isomorphism class lies in X .To prove the theorem, we are reduced to proving that there exists a similar countable union X ⊂ A g,δ (C) whose components are algebraic.For this, it suffices to prove the following claim: C). Indeed, if this holds, then X ⊂ W and since each Z i ⊂ X is irreducible, each Z i is contained in an irreducible component Y j ⊂ W .We may then define X as the union of those Y j ⊂ W that contain some Z i .
To prove the claim, let U → A g,δ be a finite étale cover of the moduli stack A g,δ and let X → U be the pullback of the universal family of abelian varieties along U → A g,δ .This gives an abelian scheme X × X → U carrying a relative Poincaré line bundle P X /U and arguments similar to those used to prove Lemma 4.4 show that indeed, for each irreducible component U ⊂ U , the locus in U (C) where c 1 (P A ) 2g−1 /(2g − 1)! is algebraic is a countable union of closed algebraic subvarieties of U (C).
Finally, Theorem 1.1 implies that for each [(A, λ)] ∈ X, the integral Hodge conjecture for one-cycles holds for the abelian variety A, so we are done.
Remark 4.8.In the principally polarized case, the density in the moduli space of those abelian varieties that satisfy the integral Hodge conjecture for one-cycles admits another proof which might be interesting for comparison.Let A g be the coarse moduli space of principally polarized complex abelian varieties of dimension g and let [(A, θ)] be a closed point of 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 ρ(A) = g 2 , (iii) A is isomorphic to the product E 1 × • • • × 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, θ) for which this holds is dense in A g by [Lan75].For an explicit example in dimension g = 4 of a principally polarized abelian variety (A, θ) that satisfies one of the equivalent conditions above, but which is not isomorphic to a Jacobian, see [Deb87,§5].
5 The integral Hodge conjecture for one-cycles up to factor n In this section, we study a property of a smooth projective complex variety that lies somewhere in between the integral Hodge conjecture and the usual (i.e.rational) Hodge conjecture.The key will be the following: We say that X satisfies the integral Hodge conjecture for k-cycles up to factor n if Z 2d−2k (X) is annihilated by n (in other words, if n • x ∈ H 2d−2k (X, Z) alg for every x ∈ Hdg 2d−2k (X, Z)).
Lemma 5.2.Let A be a complex abelian variety of dimension g.

1.
Let n be a positive integer and let F n : CH 1 ( A) → 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.
Corollary 6.3.Let A K be a principally polarized abelian variety over a number field K and suppose that A K satisfies the integral Tate conjecture for one-cycles over K. Let p be a prime ideal of the ring of integers O K of K at which A K has good reduction and write κ = O K /p.Then the abelian variety A κ over κ satisfies the integral Tate conjecture for one-cycles over κ. ( Now the principal polarization λ K : A K ∼ − → A K extends uniquely to a homomorphism λ : A → A by the Néron mapping property [BLR90, Section 1.2, Definition 1] and since the same is true for the inverse λ −1 K : A K ∼ − → A K we find that λ is an isomorphism.In particular, we see that A κ is principally polarized and that the class in CH 1 (A K ) Z of a theta divisor on A K is sent to the class in CH 1 (A κ) Z of a theta divisor on A κ. Thus, the minimal class γ θ K ∈ H 2g−2 ét (A K , Z (g−1)) is sent to the minimal class γ θκ ∈ H 2g−2 ét (A κ, Z (g − 1)) by the isomorphism on the bottom of Diagram (17).It follows that γ θκ 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 ∈ A g (k), let (A t , λ t ) be a principally polarized abelian variety such that [(A t , λ t )] = t.Let A = E 1 × • • • × E g be the product of g ordinary elliptic curves E i over k and provide A with its natural principal polarization.Let x ∈ 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 G q (x) ⊂ A g (k) be the set of isomorphism classes y = [(A y , λ y )] that admit an isogeny φ : A y → A x with φ * λ x = q N • λ y for some nonnegative integer N .We claim that A y satisfies the integral Tate conjecture for one-cycles over k for any y ∈ G q (x).Indeed, for such y there exists a nonnegative integer N such that the isogeny [q N ] : A y → A y factors through A x .Consequently, q (2g−2)•N • γ θ is algebraic for the first Chern class θ of the principal polarization on A y , which implies that γ θ is algebraic (as q > (g − 1)!).Thus, the claim follows from Theorem 1.6.Now G q (z) is dense in A g for any ordinary principally polarized abelian variety (A z , λ z ) by a result of Chai [Cha95, Theorem 2].Therefore, G q (x) is dense in A g and the proof is finished.
However, the case of algebraically closed fields remains open [MP10, Section 3.2].What we do know, is the following: Theorem 3.7 (Moonen-Polishchuk).Let A be an abelian variety over k.The ring (CH(A), ) admits a canonical PD-structure γ on the ideal CH >0 (A) ⊂ CH(A).If k = k, then γ extends to a PD-structure on the ideal generated by CH >0 (A) and the zero cycles of degree zero.

Proof.
Write S = Spec O K and let A → S be the Néron model of A K .Let R (resp.K p ) be the completion of O K (resp.K) at the prime p.The natural composition K → K p → Kp induces an embedding K → Kp , where Kp is an algebraic closure of K p .This gives a commutative diagram, where the square on the right is provided in[Ful98, Example 20.3.5]:CH(AK ) Z / / CH(A Kp ) Z / / CH(A κ) Z ⊕ r≥0 H 2r ét (A K , Z (r)) ∼ / / ⊕ r≥0 H 2r ét (A Kp , Z (r))∼ / / ⊕ r≥0 H 2r ét (A κ, Z (r)).
•)) alg .8.There exists a PD-structure on the ideal ⊕ j>0 H 2j Proposition 3.12.1.If k = C, then each of the statements 1 − 8 in Proposition 3.11 is equivalent to the same statement with étale cohomology replaced by Betti cohomology.2. Proposition 3.11 remains valid if one replaces integral Chow groups by their tensor product with Z , 'integral Fourier transform up to homology' by ' -adic integral Fourier transform up to homology', and H 2• ét (A ks , Z (•)) alg by the image of the map CH • ét (A ks , Z (j)) alg ⊂ H 2• ét (A ks , Z (•)) alg .Here, H 2• ét (A ks , Z (•)) alg denotes the image of the cycle class map CH • (A) → H 2• ét (A ks , Z (•)).Proof of Proposition 3.11.The proof of Theorem 3.8 can easily be adapted to this situation.