The Walker Abel–Jacobi map descends

For a complex projective manifold, Walker has defined a regular homomorphism lifting Griffiths’ Abel–Jacobi map on algebraically trivial cycle classes to a complex abelian variety, which admits a finite homomorphism to the Griffiths intermediate Jacobian. Recently Suzuki gave an alternate, Hodge-theoretic, construction of this Walker Abel–Jacobi map. We provide a third construction based on a general lifting property for surjective regular homomorphisms, and prove that the Walker Abel–Jacobi map descends canonically to any field of definition of the complex projective manifold. In addition, we determine the image of the l-adic Bloch map restricted to algebraically trivial cycle classes in terms of the coniveau filtration.

which is in particular functorial with respect to the action of correspondences between complex projective manifolds. Since algebraically trivial cycles in CH p (X ) are parametrized by smooth projective complex curves, and since the Abel map CH 1 (C) 0 → J (C) := J (H 1 (C, Z(1))) on degree-0 zero-cycle classes on a curve C is an isomorphism, the image of the Abel-Jacobi map restricted to the subgroup A p (X ) ⊆ CH p (X ) of algebraically trivial cycle classes has image a subtorus which is algebraic, i.e., an abelian variety, and called the algebraic intermediate Jacobian.
The resulting (surjective) Abel-Jacobi map defines a regular homomorphism, meaning that for all pointed smooth connected complex varieties (T , t 0 ) and all families of codimensionp cycles Z ∈ CH p (T × X ) the map T (C) → J (X ) will be called the Walker Abel-Jacobi map. It was first constructed by Walker [23] using Lawson homology; recently, Suzuki [21] gave a Hodge-theoretic construction relying solely on Bloch-Ogus theory [9]. That ψ p W is regular is [23,Lem. 7.3] or [21,Cor. 2.6]. In addition, it is shown in [21,Lem. 2.4] that ψ p W is compatible with the action of correspondences. In the case where p = 1, 2, dim X , the usual Abel-Jacobi map ψ p is universal among regular homomorphisms (see [15,Thm. C]), and so the Walker Abel-Jacobi map coincides with the usual Abel-Jacobi map (i.e., the isogeny α is an isomorphism), while in general it differs (see Cor. 4.2]) and hence provides a finer invariant for algebraically trivial cycles.
The first aim of this paper is to provide a new proof of Walker's Theorem A; see Sect. 2.3. Our proof is based on the general lifting Theorem 1.5 for regular homomorphisms (see also Proposition 1.3), which we hope could prove useful in other situations, especially in positive characteristic.
As our main new result, we show that if X is defined over a field K ⊆ C, then the Walker intermediate Jacobian descends to K in such a way that the diagram of Theorem A can be made Aut(C/K )-equivariant: Theorem B (Distinguished model) Let X be a smooth projective variety variety over a field K ⊆ C. Then the isogeny α : J  [2,Thm. 9.1]). We provide two proofs of Theorem B. The first one is presented in Sect. 2.5; it is based on [4, Thm. A], on the universality of the Walker Abel-Jacobi map among lifts of the Abel-Jacobi map along isogenies (Theorem 2.3) and on the general descent statement of our lifting Theorem 1.5. The second one is presented in Sect. 3.3 and builds directly upon [4]. We note also here that, as in [4, Thm. A] and [3,Prop. 3.1], which concern the case of the algebraic intermediate Jacobian, the K -structure in Theorem B for the Walker intermediate Jacobian and Walker Abel-Jacobi map is stable under field extensions K ⊆ L ⊆ C (Remark 3.4), and independent of the embedding of K into C (Remark 3.5). As a consequence, the kernel of the Walker Abel-Jacobi map is independent of the choice of embedding of K into C; the analogous statement for the Abel-Jacobi map on algebraically trivial cycle classes is [3,Rem. 3.4].
From our second approach to proving Theorem B we obtain two applications. First, we obtain the following proposition, which provides further arithmetic significance to the Walker Abel-Jacobi map, by showing that the torsion-free quotient of This result is established in Sect. 4.1. It was established with Q -coefficients in [4,Thm. A] with the model of the algebraic intermediate Jacobian over K in place of that of the Walker intermediate Jacobian. We direct the reader to [5] for more details, and in particular, the connection to a question of Mazur [14].
Second, for any smooth projective variety X over an algebraically closed field and for any prime invertible in X , Bloch [8] has defined a map λ p : ) τ is then obtained by taking Tate modules and making the identification [20, (2.6.5)], and to [5, §A.3.3], for more details. Here, the Tate module associated to an -primary torsion abelian group M is the group . Thanks to our approach to lifting regular homomorphisms along isogenies, together with the existence of the Walker Abel-Jacobi map, we determine the image of T λ p restricted to algebraically trivial cycle classes: Corollary D Let X be a smooth projective variety over a field K of characteristic zero. Then This extends [20,Prop. 5.2] (see also [5,Prop. 2.1]), where the images of the usual Bloch map λ p and of T λ p ⊗ Q , both restricted to algebraically trivial cycle classes, were determined.

An elementary fact
We start with the following elementary fact, which will be used recurringly throughout this note. As a first consequence, note that since for a smooth complex projective variety X one has that A p (X ) is a divisible group (e.g., [9, Lem. 7.10]), there is at most one homomorphism ψ , there is at most one lifting of the Abel-Jacobi map to the Walker intermediate Jacobian.

Lifting homomorphisms of abelian varieties along isogenies
We have the following elementary lemma on lifting morphisms of abelian varieties. (Recall that for an abelian variety A over a field K of positive characteristic, the N -torsion group scheme A[N ] may carry strictly more information than the abstract group of points A If α is separable (equivalently, étale), and /K is any field extension with algebraically closed, then (1) and (2) are also equivalent to each of the following conditions:

For all prime numbers l, we have
Finally, if any of the lifts in (1)-(4) exist, they are unique. In particular, Proof The uniqueness of the lift f follows from Fact 1.1; and (1) clearly implies (2). Moreover, (2) implies (3), and (3) implies (2) over an algebraically closed field of characteristic zero. Conditions (3) and (4) are obviously equivalent; (4) and (5) are equivalent because each T l α is an inclusion.
To show (2) implies (1), suppose there exists a suitable lift of f on torsion schemes. By rigidity of homomorphisms of abelian varieties, we may assume that K is perfect. Using the uniqueness of f and Galois descent, we may and do assume K is algebraically closed.
We start by reducing to the case where f is an isogeny. To this end, consider the diagram Fix a prime l and consider l-primary torsion. Using the lift ( f [l ∞ ]) , we have the diagram: The splitting of the map f conn [l ∞ ] is elementary, since whenever one has a short exact sequence of abelian varieties the induced maps on l-primary torsion give a split exact sequence (taking l-primary torsion is exact since the kernel is divisible, and then free modules are projective). (If l = char(K ), an appeal to Dieudonné modules gives the same conclusion.) Thus, we now assume f and α are isogenies. Suppose briefly that char(K ) = 0; then f and α are étale. The cover f factors through α if and only if the induced map on étale fundamental groups f * : By taking the inverse limit of the maps of finite groups ( f [N ]) (K ), we see that the condition on fundamental groups is equivalent to (2). Now suppose instead that K is algebraically closed of positive characteristic. Then f , while possibly not étale, is at least a torsor over X under the finite commutative group scheme ker( f ). Consequently, it is classified by a quotient of Nori's fundamental group scheme π Nori 1 (A) [17]. Moreover, for an abelian variety D/K , we have π Nori [18]. Consequently, condition (2) is again equivalent to the hypothesis that the cover f factors through α.
Finally, suppose α is étale by hypothesis and that (3) holds. As noted above, it suffices to consider the case where K is algebraically closed of positive characteristic and f : B → A is an isogeny. Now, any isogeny g : D → C of abelian varieties over K admits a canon- Because f i is a universal homeomorphism, the map of groups f tors (K ) descends to a map of groups f´e t,tors (K ) : B i,tors (K ) → A tors (K ). Now f´e t and α are étale isogenies and we may argue using fundamental groups as before, while recalling that (in all charac- The same argument, combined with the canonical (4) implies (1), as well.

Lifting regular homomorphisms along isogenies
From Lemma 1.2 we get the following lifting criterion for regular homomorphisms: Proposition 1.3 Let K be a field, and /K an algebraically closed extension. Let X /K be a smooth projective variety, let A/K be an abelian variety over K , let φ : A p (X ) → A( ) be an Aut( /K )-equivariant regular homomorphism, and let α : A → A be an étale isogeny of abelian varieties over K . Then the following are equivalent:

For all prime numbers l there exists a group-theoretic lift
Finally, if any of the lifts in (1)-(3) exist, then they are unique and Aut( /K )-equivariant.
Proof The uniqueness and Aut( /K )-equivariance of the lifts follows from Fact 1.1. Clearly (1) implies (2), and (2) implies (3) by taking Tate modules. The equivalence of (3) and (4) is obvious since T l α is injective. Thus we will show (3) implies (1). Let (T , t 0 ) be a smooth pointed variety over , and let ∈ CH p (T × X ). Then we have a commutative diagram where the top row is the pointed Albanese, and the right vertical arrow f comes from the universal property of algebraic representatives, together with the facts that Albaneses are algebraic representatives, and that φ • * can easily be confirmed to be a regular homomorphism.
On Tate modules we obtain a diagram where the lift (T l φ) is provided by assumption (3). The isomorphism on Tate modules for the Albanese map comes from Roitman's theorem (see, e.g., [5,Prop. A.29]). Since we assume α is étale, by Lemma 1.2 we obtain a lift f : It follows immediately that if φ lifts to an abstract homomorphism φ : A p (X ) → A ( ), then φ is a regular homomorphism. Thus we have reduced the problem to showing that φ lifts as an abstract homomorphism to a homomorphism φ : Over an algebraically closed field, algebraically trivial cycles are parameterized by smooth projective curves [11,Ex. 10.3.2]. In other words, A p (X ) is covered by the images of * : A 0 (T ) → A p (X ), where T runs through pointed smooth projective curves over and over correspondences in CH p (T × X ). Now since A 0 (T ) is divisible, it follows that * (A 0 (T )) is divisible; therefore, by the uniqueness of lifts (Fact 1.1) it is enough to show that f • alb in (1.3) factors through * (A 0 (T )) in the case where T is a smooth projective curve. In other words, taking T to be a smooth projective curve over , and given any γ ∈ A 0 (T ) such that * (γ ) = 0, we must show that ( f • alb)(γ ) = 0.
The first observation is that this is clear if is the algebraic closure of a finite field. Indeed, in that case A 0 (T ) is a torsion group, since the Albanese map A 0 (T ) → Alb T ( ) is an isomorphism and closed points of an abelian variety over a finite field are torsion. Thus γ is torsion. Decomposing torsion in A 0 (T ) into a direct sum of l-power torsion, we can work one prime at a time. Now we make the following elementary observation: given any homomorphism of groups h : D → G where D is divisible, and any x ∈ D[l ∞ ], we have that h(x) = 0 if for some lift x l of x to T l D (which exists since D is divisible), we have that (T l h)(x l ) = 0. Consequently, taking Tate modules in (1.3) and using the lift (T l φ) (1.2), we see that alb(γ ) = 0.
We now deduce the general case from the case of finite fields, via a specialization argument. For this we use the terminology of regular homomorphisms from [2], which is much better suited to the relative setting. Since all objects considered here are of finite type, the data X , T , , A, A , α and γ descend to a field L which is finitely generated over the prime field. A standard spreading argument produces a smooth ring R, finitely generated as a Z-algebra and with fraction field L, and smooth X, T , A, A over S = Spec(R), as well as γ ∈ A 1 T /S (S), whose generic fibers are the corresponding original data. Let |S| cl be the set of points of S with finite residue fields; then |S| cl is topologically dense in S. Now suppose s ∈ |S| cl . Then pullback of (1.4) yields a diagram of objects over s = Spec(κ(s)), where specialization of cycles is provided by [11, 20.3.5]. We have seen that for each such s, a s = 0 ∈ A (s). Using the density of |S| cl , we see that a = 0, and in particular its generic fiber ( f • alb)(γ ) is zero.

Remark 1.4 (Regular homomorphisms and mini-versal cycle classes) Given a surjective Aut( /K )-equivariant regular homomorphism
there is a cycle class ∈ CH p (A × K X ) (which we call a mini-versal cycle class) such that the associated map ψ : A → A, induced on -points by a → a − 0 → φ( a − 0 ), is given by multiplication by some non-zero integer r [2, Lem. 4.7]. One can immediately see from the definition that given any étale isogeny α : A → A through which φ factors, one has (deg α) | (deg r · Id A ) = r 2 dim A . In particular, if there is a universal cycle class (i.e., r = 1), then φ does not factor through any non-trivial isogeny A → A.
We obtain the following consequence of Proposition 1.3, establishing the existence of a universal lifting of a surjective regular homomorphism along isogenies. Together with Corollary 1.6, this extends [6, Thm. 0.1] to the case of arbitrary fields. Note also that the proof of [6, Thm. 0.1] is incorrect. (On the bottom of [6, p. 362], it is assumed that the map u : B(k) → A q (X ) is a homomorphism, so that the image of u is a subgroup of A q (X ). There, X is a smooth projective variety over an algebraically closed field k, B is an abelian variety over k, and u : b → Z * ([u] − [0]) is the map induced by a cycle Z ∈ CH p (B × k X ). However, this is not the case in general. Indeed, consider the special instance where X = B is an abelian variety of dimension > 1 over an uncountable algebraically closed field k and where Z = B is the diagonal cycle class. Then the map u : is not a homomorphism since by [7,Thm. 3 [16, p. 309].) Theorem 1.5 (Universal lift of surjective regular homomorphisms along étale isogenies) Let K be a field, and /K an algebraically closed extension. Let X /K be a smooth projective variety, let A/K be an abelian variety over K , and let φ : A p (X ) → A( ) be a surjective regular homomorphism. Then there exist an étale isogeny α : A → A , characterized by the condition im(T l α) = im(T l φ) for all primes l, and a surjective regular homomorphism φ : A p (X ) → A( ) which is initial among all regular lifts of φ along étale isogenies A → A .
Moreover, if φ is Aut( /K )-equivariant, then A admits a unique model over K such that φ is Aut( /K )-equivariant, and the isogeny α descends to K .
Proof Using a mini-versal cycle class as in Remark 1.4, one sees that ( T l φ)(A p (X )) has finite index in T l A π´e t 1 (A , 0). Consequently, it determines an étale isogeny A → A over ; by Proposition 1.3(4), there is a surjective regular homomorphismφ : A p (X ) → A( ) which lifts φ and which is initial among all regular lifts of φ along étale isogenies A → A over . Suppose now that φ is Aut( /K )-equivariant, and briefly assume K perfect. The unicity of the model over K follows from the elementary Fact 1.1. Its existence follows from the universality ofφ: for all σ ∈ Aut( /K ), one obtains an isomorphism g σ : A → A σ over , where A σ is the pull-back of A along σ : → , making the following diagram commute Hereφ σ and α σ are obtained from the action of σ on A p (X ) and on A , and from the canonical σ -morphism A σ → A. To conclude, one checks as in the proof of [1,Thm. 4.4] that the isomorphisms g −1 σ for σ ∈ Aut( /K ) define a Galois-descent datum on the isogeny α : A → A .
If K is a non-perfect field, let K perf be the perfect closure of K inside . From what we have seen, since Aut( /K perf ) ⊆ Aut( /K ), A descends to K perf . Because in fact Aut( /K perf ) = Aut( /K ), it suffices to show that α : A → A K perf descends to K . Now, by definition, the homomorphism α factors through the K perf /K -image A → im K perf /K ( A) K perf , which exists due to [10,Thm. 4.3]. Since α : A → A K perf is étale and K perf /K is primary, the canonical map A → im K perf /K ( A) K perf , which always has connected kernel [10,Thm. 4.5(3)], is an isomorphism, and A and α descend canonically to K .
We derive the following characterization of surjective regular homomorphisms that do not lift along non-trivial isogenies in terms of their kernels: Corollary 1.6 Let X be a smooth projective variety over an algebraically closed field and let φ : A p (X ) → A( ) be a surjective regular homomorphism. Then the following statements are equivalent:

φ does not factor through any non-trivial étale isogeny α : A → A.
Proof The argument in the proof of Theorem 1.5 says that (3) and (4) are equivalent (recall from Proposition 1.3 that a group-theoretic lift of a regular homomorphism along an isogeny is a regular homomorphism). The elementary commutative algebra Lemma 1.7 below gives the equivalence of (1) and (3). Finally, since surjective regular homomorphisms are surjective on torsion (see [4,Rem. 3.3]), Lemma 1.7 below also gives that T l φ being surjective for all l is equivalent to ker(φ tors ) being l-divisible for all primes l, i.e., that (2) is equivalent to (3).

Lemma 1.7 Suppose that we have a short exact sequence of abelian groups
with D an l-divisible group. Then the left exact sequence

is right exact if and only if H is l-divisible. If in addition D tors → G tors is surjective, then this is also equivalent to H tors being l-divisible.
Proof Since D is l-divisible, we have for all n > 0 exact sequences Using that A/l n A = 0 and lim ← − 1 n A[l n ] = 0 for any l-divisible abelian group A, we obtain by passing to the inverse limit a short exact sequence Conversely, if H is not l-divisible, let us assume that H /l n H = 0 for all n ≥ n 0 . In particular D[l n ] → G[l n ] is not surjective for every n ≥ n 0 . Now let g n 0 ∈ G[l n 0 ] be an element that is not in the image of the map D[l n 0 ] → G[l n 0 ]. Since G is l-divisible (being the image of the l-divisible group D), we can lift g n to an element (g n ) ∈ T l G. Clearly (g n ) is not the image of any element (d n ) ∈ T l D, since then d n 0 → g n 0 . Thus T l D → T l G is not surjective. This completes the proof of the converse.
Finally assume that D tors → G tors is surjective. Then we can simply replace the short and we have reduced to the previous case, since D divisible implies that D tors is divisible, and T l A = T l (A tors ) for any abelian group A. Remark 1.8 Given a regular homomorphism φ, using that surjective regular homomorphisms are surjective on torsion (see [4,Rem. 3.3]), one can in fact show that ker(φ)/N = ker(φ tors )/N for any non-zero integer N .

The Walker Abel-Jacobi map
The aim of this section is to provide a new construction of the Walker Abel-Jacobi map (Theorem A), based on our general lifting Proposition 1.3.

The Bloch map and the coniveau filtration
Recall that, for any smooth projective variety X over an algebraically closed field and for any prime invertible in X , Bloch [8] has defined a map λ p : In case X is a smooth projective complex variety, we obtain by comparison isomorphism a map λ p : CH p (X )[ ∞ ] → H 2 p−1 (X an , Q /Z ( p)). When restricted to homologically trivial cycles, the Bloch map factors as (see, e.g., [5, §A.5]) where the right-hand side arrow is the canonical inclusion coming from the universal coefficient theorem. The following lemma is due to Suzuki [22]:

Lemma 2.1 Let X be a projective complex manifold. Then the restriction of the Bloch map λ p to algebraically trivial cycles factors uniquely as:
Abel-Jacobi map ψ p :

Descending the Walker Abel-Jacobi map
In this section we provide a second proof of Theorem B. It is based on a factorization of the Bloch map restricted to algebraically trivial cycles. This approach will prove crucial for our applications (Corollaries C and D). For that purpose, we start by recasting the results of §2 in the -adic setting.

The Bloch map and the coniveau filtration, -adically
For lack of a suitable reference, we start with a comparison between the analytic and -adic coniveau filtrations: Lemma 3.1 Let X be a smooth projective variety over a field K ⊆ C. We have canonical identifications Here the limits are taken over all closed subschemes Z of X C of codimension ≤ i. The top two rows are exact by definition of the coniveau filtration, while the third is also exact by flatness of the Z-module Z . The bottom vertical arrows are isomorphisms by flatness of Z and the fact that lim − → commutes with ⊗. The top two vertical arrows are the isomorphisms provided by Artin's comparison theorem. Thus we obtain the desired identification.
The action of Aut(C/K ) on N i H j et (X C , Z ) comes from the fact that the coniveau filtration on X C can be obtained using subvarieties defined over K (as can be seen by spreading out and by using smooth base-change, followed by taking Galois-orbits).
As an immediate consequence of Lemma 3.1, we obtain: Lemma 3.2 Let X be a smooth projective variety over a field K ⊆ C. Then the restriction of the Aut(C/K )-equivariant Bloch map λ p : CH p (X C )[ ∞ ] → H to algebraically trivial cycles factors uniquely into the following commutative diagram of

Second proof of Theorem B
Let X be a smooth projective variety over a field K ⊆ C. Recall that we showed in [4,Thm. A] (see also [2,Thm. 9.1]) that J 2 p−1 a (X C ) admits a unique model over K such that the Abel-Jacobi map ψ p : We are going to show that α descends uniquely to K with respect to the above K -structure on J (X ) then follows from the unicity of ψ p W . To that end, let C be a K -pointed, geometrically integral, smooth projective curve over K , together with a correspondence ∈ CH p (C × K X ) such that the induced homomorphism The existence of such a C and is provided by [4,Prop. 1.1]. We thus obtain a commutative diagram where the homomorphism γ , which is defined by the fact that the Jacobian of a curve together with the Abel map is a universal regular homomorphism, is also induced by the correspondence * : ; see e.g., [5, Prop. 1.1]). We then show that, with respect to the K -structure on J (C C ) given by the Jacobian J (C) of C, the surjective homomorphism γ descends to K . (That α • γ descends to K was established in [4, §2].) For that purpose, by the elementary [4,Lem. 2.3], it suffices to show that, for all primes , the -primary torsion in is stable under the action of Aut(C/K ) on J (C)(C).
For this we take -primary torsion in the commutative diagram (3.3), then use the compatibility of the Bloch map with the Walker Abel-Jacobi map (Lemma 3.3) to obtain the commutative diagram 0 P[ ∞ ] The only things that needs explaining is the middle vertical map: here we are using the fact that the Bloch map is compatible with correspondences, and the fact mentioned above that the correspondence * : which, since is defined over K , is stable under the action of Aut(C/K ). We have thus showed that γ descends to K . Combined with the fact [4, §2] that α • γ also descends to K with respect to the K -structure of J (C C ) given by J (C C ) = J (C) C , we readily obtain that α descends to K (e.g., by the elementary [4, Lem. 2.4]).

(X )
A p (X σ ) J As a consequence, for a smooth projective variety X over a field K of characteristic 0, the kernel of the Walker Abel-Jacobi map associated to X and an embedding of K into C is independent of that embedding.

Modeling coniveau: on a question of Mazur
In this paragraph, we show Corollary C stating that the model of the Walker intermediate Jacobian over K from Theorem B models the torsion-free quotient of N p−1 H 2 p−1 et (X K , Z ).