aa r X i v : . [ m a t h . AG ] F e b Chow rings and gonality of general abelian varieties
Claire VoisinColl`ege de France
Abstract
We study the (covering) gonality of abelian varieties and their orbits of zero-cyclesfor rational equivalence. We show that any orbit for rational equivalence of zero-cyclesof degree k has dimension at most k −
1. Building on the work of Pirola, we show thatvery general abelian varieties of dimension g have covering gonality k ≥ f ( g ) where f ( g )grows like log g . This answers a question asked by Bastianelli, De Poi, Ein, Lazarsfeldand B. Ullery. We also obtain results on the Chow ring of very general abelian varieties,eg. if g ≥ k −
1, for any divisor D ∈ Pic ( A ), D k is not a torsion cycle. The gonality of a projective variety X is defined in this paper as the minimal gonality ofthe normalization of an irreducible curve C ⊂ X . In the case of an abelian variety, thegonality is the same as the covering gonality studied in [2]. One of the main results of thispaper answers affirmatively a question asked in [2] concerning the gonality of a very generalabelian variety A , namely, whether it grows to infinity with g = dim A . Theorem 0.1.
Let A be a very general abelian variety of dimension g . Then if g ≥ k − (2 k −
1) + (2 k − − k − , the gonality of A is at least k + 1 . In other words, a very general abelian variety of dimension ≥ k − (2 k −
1) + (2 k − − k −
2) does not contain a curve of gonality ≤ k . This theorem is presumably not optimal.What seems reasonable is the following bound: Conjecture 0.2.
Let A be a very general abelian variety of dimension g . Then if g ≥ k − ,the gonality of A is at least k + 1 . We will discuss in Section 5 a strategy towards proving this statement and some evidencefor it. Theorem 0.1 generalizes the following result by Pirola [13]:
Theorem 0.3. (Pirola) A very general abelian variety of dimension at least does notcontain a hyperelliptic curve. We will in fact use in the proof of Theorem 0.1 (and also 0.4, 0.8 below) some of thearguments in [13] that we generalize in Section 1. Theorem 0.1 will be obtained as aconsequence of the study of 0-cycles modulo rational equivalence on abelien varieties. Thisgeneralized setting already appears in the paper [1] where some improvements of Theorem0.3 (for example on the non-existence of trigonal curves on very general abelian varieties ofdimension ≥
4) were obtained. In this paper, Chow groups with Q -coefficients of a variety X are denoted CH( X ). Rational equivalence of 0-cycles is not very well understood, despiteMumford’s theorem [12]. The most striking phenomenon is the existence of surfaces (eg. ofGodeaux type, see [18]) which are of general type but have trivial CH -group. In the papers[16], [17], we emphasized nevertheless the geometric importance of the study of orbits | Z | = { Z ′ ∈ X ( k ) , Z ′ rationally equivalent to Z in X } of degree k Z of X under rational equivalence, particularly in the case of K { x } the 0-cycle of a point x ∈ A and 0 A will bethe origin of A . The following results concerning orbits | Z | ⊂ A ( k ) for rational equivalence,and in particular the orbit | k { A }| , can be regarded as a Chow-theoretic version of Theorem0.1. Theorem 0.4. (i) For any abelian variety A and integer k ≥ , any orbit | Z | ⊂ A ( k ) hasdimension ≤ k − .(ii) If A is very general of dimension g ≥ k (2 k −
1) + (2 k − k − , A has no positivedimensional orbit | Z | , with deg Z ≤ k .(iii) If k ≥ and A is very general of dimension g ≥ k − (2 k − k − − k − , A hasno positive dimensional orbit of the form | Z ′ + 2 { A }| , with Z ′ effective and deg Z ′ ≤ k − .(iv) If A is a very general abelian variety of dimension g ≥ k − , the orbit | k { A }| iscountable. In fact, Theorem 0.4, (iii) implies Theorem 0.1, because a k -gonal curve C ⊂ A , withnormalization j : e C → A and divisor D ∈ Pic k e C with h ( e C, D ) ≥ { j ∗ D ′ } D ′ ∈| D | in A ( k ) . We can assume one Weierstrass point c ∈ e C of | D | ,that is, a point c such that h ( e C, D ( − c )) = 0, is mapped to 0 A by j , which provides apositive dimensional orbit of the form | Z ′ + 2 { A }| , with Z ′ effective and deg Z ′ ≤ k − g = 1 because for any degree k divisor D on an elliptic curve E we have | D | = P k − ⊂ E ( k ) . This immediately implies that thestatement is optimal for any g because for abelian varieties A = E × B admitting an ellipticfactor, we have E ( k ) ⊂ A ( k ) . In the case where g = 2, we observe that orbits | Z | ⊂ A ( k ) are contained in the generalized Kummer variety K k − ( A ) constructed by Beauville [6].(More precisely, this is true for the open set of | Z | parameterizing cycles where all pointsappear with multiplicity 1 but this is secondary, cf. [16] for a discussion of cycles withmultiplicities.) This variety is of dimension 2 k − | Z | is totally isotropic, which implies the estimate(i) in the case g = 2. Furthermore they are also orbits for rational equivalence in K k − ( A ),as proved in [11], hence they are as well constant cycles subvarieties in K k − ( A ) in the senseof Huybrechts[10]. The question whether Lagrangian (that is maximal dimension) constantcycles subvarieties exist in hyper-K¨ahler manifolds is posed in [17]. For a general abelianvariety A , choosing a smooth curve C ⊂ A of genus g ′ , we have C ( k ) ⊂ A ( k ) for any k and C ( k ) contains linear systems P k − g ′ , for k ≥ g ′ . So when k tends to infinity, the estimate (i)has optimal growth in k .Theorem 0.4, (iv), which will be proved in Section 2, has the following immediate con-sequence (which is a much better estimate than the one given in Theorem 0.1): Corollary 0.5. If A is a very general abelian variety of dimension g ≥ k − , and C ⊂ A is any curve with normalization e C , one has h ( C, O e C ( kc )) = 1 for any point c ∈ e C . This corollary could be regarded as the right generalization of Theorem 0.3.
Remark 0.6.
Pirola proves in [14] that for a very general abelian variety A of dimension g ≥
4, any curve C ⊂ A has genus ≥ g ( g − + 1. This suggests that Theorem 0.4, (iv) isneither optimal, and that an inequality g ≥ O ( √ k ) should already imply the countability of | k { A }| .We will give two proofs of Theorem 0.4, (iv). One of them will use Theorem 0.8 andProposition 0.9, which are statements of independent interest concerning the Chow ring (asopposed to the Chow groups) of an abelian variety A , that we now describe. Here the Chowring is relative to the intersection product but one can also consider the ring structure givenby the Pontryagin product ∗ defined by z ∗ z ′ = µ ∗ ( z × z ′ )2here µ : A × A → A is the sum map and z × z ′ = pr ∗ z · pr ∗ z ′ for z, z ′ ∈ CH( A ). The tworings are related via the Fourier transform, see [4]. Define A k ⊂ A (1)to be the set of points x ∈ A such that ( { x } − { A } ) ∗ k = 0 in CH ( A ). We can also define b A k ⊂ b A to be the set of D ∈ Pic ( A ) =: b A such that D k = 0 in CH k ( A ). These two setsare in fact related as follows: choose a polarization θ on A , that is an ample divisor. Thepolarization gives an isogeny of abelian varieties A → b A, x D x := θ x − θ. Lemma 0.7.
One has D kx = 0 in CH k ( A ) if and only if ( { x } − { A } ) ∗ k = 0 in CH ( A ) .Proof. This follows from Beauville’s formulas in [4, Proposition 6]. We get in particular, thefollowing equality: θ g − k ( g − k )! D kx = θ g g ! ∗ γ ( x ) ∗ k , (2)where γ ( x ) := { A } − { x } + 12 ( { A } − { x } ) ∗ + . . . + 1 g ( { A } − { x } ) ∗ g = − log( { x } ) ∈ CH ( A ) . Here the logarithm is taken with respect to the Pontryagin product ∗ and the developmentis finite because 0-cycles of degree 0 are nilpotent for the Pontryagin product. If ( { A } −{ x } ) ∗ k = 0, then γ ( x ) ∗ k = 0 and thus D kx = 0 by (2). Conversely, if D kx = 0, then γ ( x ) ∗ k = 0by (2). But then also ( { A } − { x } ) ∗ k = 0 because { x } = exp( − γ ( x )). (Again exp( − γ ( x ))is a polynomial in γ ( x ), hence well-defined, since γ ( x ) is nilpotent for the ∗ -product, see[7]). Theorem 0.8.
Let A be an abelian variety of dimension g . Then(i) dim A k ≤ k − , dim b A k ≤ k − .(ii) If A is very general and g ≥ k − , the sets b A k and A k are countable. Note that in both (i) and (ii), the two statements are equivalent by Lemma 0.7, usingthe fact that A b A is an open map between moduli spaces, so that, if A is very general,so is b A .The fact that Theorem 0.8 implies Theorem 0.4, (iv), uses the following intriguing resultthat does not seem to be written anywhere, although some related results are available, inparticular the results of [8], [9], [15]. Proposition 0.9.
Let A be an abelian variety and let x , . . . , x k be k points of A such that P ki =1 { x i } − k { A } = 0 in CH ( A ) . Then for any i = 1 , . . . , k ( { x i } − { } ) ∗ k = 0 in CH ( A ) . (3) In other words, x i ∈ A k . For the proof of Theorem 0.8, we will show how the dimension estimate provided by(i) implies the non-existence theorem stated in (ii). This is obtained by establishing andapplying Theorem 1.3, that we will present in Section 1. This theorem, which is obtained bya direct generalization of Pirola’s arguments in [13], says that “naturally defined subsets” ofabelian varieties (see Definition 1.1), assuming they are proper subsets for abelian varietiesof a given dimension g , are at most countable for very general abelian varieties of dimension ≥ g − Thanks.
This paper is deeply influenced by the reading of the beautiful Pirola paper [13].I thank the organizers of the Barcelona Workshop on Complex Algebraic Geometry dedicatedto Pirola’s 60th birthday for giving me the opportunity to speak about Pirola’s work, whichled me to thinking to related questions. Naturally defined subsets of abelian varieties
The proof of Theorem 0.3 by Pirola has two steps. First of all, Pirola shows that hyperellipticcurves in an abelian variety A , one of whose Weierstrass points coincides with 0 A , are rigid.Secondly he deduces from this rigidity statement the nonexistence of any hyperelliptic curvein a very general abelian variety of dimension ≥ B × E , that we now extend to cover more situations. Definition 1.1.
We will say that a subset Σ A ⊂ A is natural if it satisfies the followingconditions:(0) Σ A ⊂ A is defined for any abelian variety A and is a countable union of closedalgebraic subsets of A .(i) For any morphism f : A → B of abelian varieties, f (Σ A ) ⊂ Σ B .(ii) For any family A → S , there is a countable union of closed algebraic subsets Σ A ⊂ A such that the set-theoretic fibers satisfy Σ A ,b = Σ A b . Recall that the dimension of a countable union of closed algebraic subsets is defined asthe supremum of the dimensions of its components (which are well defined since we are overthe uncountable field C ). Remark 1.2.
By morphism of abelian varieties
A, B , we mean group morphisms, that is,mapping 0 A to 0 B . Theorem 1.3.
Let Σ A ⊂ A be a naturally defined subset.(i) Assume that for dim A = g , one has Σ A = A . Then for very general A of dimension ≥ g − , Σ A is at most countable.(ii) Assume that dim Σ A ≤ k for any A . Then for very general A of dimension ≥ k + 1 , Σ A is at most countable.(iii) Assume that dim Σ A ≤ k − for a very general abelian variety A of dimension g ≥ k . Then for a very general abelian variety A of dimension ≥ g + k − , Σ A is at mostcountable. Statement (ii) is a particular case of (i) where we do g = k + 1. Both (i) and (iii) willfollow from the following result: Proposition 1.4. (a) If for a very general abelian variety B of dimension g > k , one has < dim Σ B ≤ k then, for a very general abelian variety A of dimension g + 1 , one has dim Σ A ≤ k − . (b) If for a very general abelian variety B of dimension g > , Σ B is countable, then for A very general of dimension ≥ g , Σ A is countable. Indeed, applying Proposition 1.4, (a), we conclude in case (i) that the dimension of Σ A is strictly decreasing with g ≥ g as long as it is not equal to 0, and by assumption it is notgreater than g − g = g . Hence the dimension of Σ A must be 0 for some g ≤ g − A is countable for any g ≥ g − g = k + 1and we conclude similarly that the dimension of Σ A is strictly decreasing with g ≥ g aslong as it is not equal to 0. Furthermore, for g = g , this dimension is equal to k −
1. Hencethe dimension of Σ A must be 0 for some g ≤ g + k − A is countable for any g ≥ g + k −
1. This proves Theorem 1.3 assuming Proposition 1.4that we now prove along the same lines as in [13].4 roof of Proposition 1.4.
Assume that dim Σ = k ′ for a very general abelian variety A ofdimension g +1. From the definition of a naturally defined subset, and by standard argumentsinvolving the properness and countability properties of relative Chow varieties, there exists,for each universal family A → S of polarized abelian varieties with given polarization type θ , a family Σ ′A ⊂ Σ A S ′ ⊂ A S ′ , where S ′ → S is a generically finite dominant base-changemorphism, A S ′ → S ′ is the base-changed family, and the morphism Σ ′A → S ′ is flat, withirreducible fibers of relative dimension k ′ . In other words, we choose one k ′ -dimensionalcomponent of Σ A for each A , and we can do this in families, maybe after passing to agenerically finite cover of a Zariski open set of the base.The main observation is the fact that there is a dense contable union of algebraic subsets S ′ λ ⊂ S ′ along which the fiber A b is isogenous to a product B λ × E where B is a genericabelian variety of dimension g with polarization of type determined by λ and E is an ellipticcurve ( λ also encodes the structure of the isogeny). Along each S ′ λ , using axiom (i) ofDefinition 1.1, possibly after passing to a generically finite cover S ′′ λ , we have a morphism p λ : A S ′′ λ → B S ′′ λ and p λ (Σ ′A S ′′ λ ) ⊂ Σ B S ′′ λ by axiom (i) of Definition 1.1. Lemma 1.5. If Σ B ⊂ B is a proper subset for a very general abelian variety B of dimension g , the morphism p λ, Σ := p λ | Σ ′A b : Σ ′A b → B b is generically finite on its image for any point b of S ′′ λ .Proof. As p λ (Σ ′A b ) ⊂ Σ B b for b ∈ S ′′ λ , and we know by assumption that dim Σ B b < g , weconclude that Σ ′A b ⊂ A b is a proper algebraic subset for b ∈ S ′′ λ , hence also for general b ∈ S ′ . For very general b ∈ S ′ , the cycle class [Σ ′A b ] ∈ H l ( A b , Q ), l := codim Σ ′A b , is anonzero multiple of θ l because the latter generates the space of degree 2 l Hodge classes ofa very general abelian variety with polarizing class θ . We thus conclude that p λ ∗ ([Σ ′A b ]) isnonzero in H l − ( B b , Q ), and as Σ ′A b is irreducible by construction, it follows that p λ, Σ isgenerically finite on its image.Lemma 1.5 applied to the case where Σ B b ⊂ B b is countable while dim B b > B > dim Σ B > B of dimension g . We want to show that dim Σ ′A b < dim Σ B b which, using Lemma 1.5 when b ∈ S ′′ λ for some λ , means that p λ (Σ ′A b ) is not a componentof Σ B b . Lemma 1.6.
In the situation above, the set of varieties (of dimension k ′ = dim Σ ′A b ) Σ ′A b ,p λ := p λ (Σ ′A b ) and morphisms p λ, Σ : Σ ′A b → Σ ′A b ,p λ for all λ ’s is bounded up to bira-tional transformations.Proof. Recall that Σ ′A b ,p λ ⊂ B b is a proper subvariety of a very general abelian varietyof dimension g with polarization of certain type, and Σ ′A b ⊂ A b is the specialization of asubvariety (of codimension at least 2 by Lemma 1.5) of a general abelian variety of dimension g + 1 at a point b which is Zariski dense in S . In both cases, it follows that the Gauss maps g A of Σ ′A b ⊂ A b and g B of Σ ′A b ,p λ ⊂ B b , which take respective values in G ( k ′ , g + 1) =Grass( k ′ , T A b , A b ) and G ( k ′ , g ) = Grass( k ′ , T B b , B b ), are generically finite on their images.We have the commutative diagramΣ ′A b g A / / p λ, Σ (cid:15) (cid:15) G ( k ′ , g + 1) π λ (cid:15) (cid:15) Σ ′A b ,p λ g B / / G ( k ′ , g ) , (4)5here all the maps are rational maps and the rational map π λ : G ( k ′ , g + 1) G ( k ′ , g )is induced by the linear map dp λ : T A b , A b → T B b , B b which is also the quotient map T A b , A b → T A b , A b /T E, . We observe here that the density of the countable union of the S ′ λ in S has a stronger version, namely, the corresponding points [ T E, ] ∈ P ( T A b , ) are Zariskidense in the projectivized bundle P ( T A /S ). The projection π λ above is thus generic and thecomposition π λ ◦ g A is generically finite as is g A and up to shrinking S ′ if necessary, its graphdeforms in a flat way over the space of parameters (namely a Zariski open set of P ( T A /S )).This is now finished because we first restrict to the Zariski dense open set U of P ( T A S/B , )where the rational map π λ ◦ g A is generically finite and its graph deforms in a flat way, andthen there are finitely many generically finite covers of U parameterizing a factorization ofthe rational map π λ ◦ g A . As the diagram (4) shows that there is a factorization of π λ ◦ g A as Σ ′A b p λ → Σ ′A b ,p λ g B → G ( k ′ , g ) , we conclude that all the maps Σ ′A b p λ, Σ → Σ ′A b ,p λ are, up to birational equivalence of the target,members of finitely many families of generically finite dominant rational maps ψ : A b Y b . As a corollary, we conclude using the density of the union of the sets S ′ λ that there is,up to replacing S ′ by a a generically finite cover of it, a family of k ′ -dimensional varietiesΣ ′′A S ′ , together with a dominant generically finite rational map p : Σ ′A S ′ Σ ′′A S ′ (5)identifying birationally to p λ, Σ : Σ ′A b Σ ′A b,pλ along each S ′ λ .We now finish the proof by contradiction. Assume that k ′ = k . Then Σ ′A b ,p λ must be acomponent Γ b of Σ B b . In particular it does not depend on the elliptic curve E . Restricting toa dense Zariski open set S ′′ of S ′ is necessary, we can assume that we have desingularizations ] Σ ′A S ′ ˜ p ] Σ ′′A S ′ (6)with smooth fibers over S ′′ . Let ˜ j : Σ ′A b → A b be the natural map, and consider themorphism ˜ p ∗ ◦ ˜ j ∗ : Pic ( A b ) → Pic ( g Σ ′′A b )which is a group morphism defined at the general point of S ′′ . This morphism is nonzerobecause when b ∈ S ′′ λ for some λ , it is injective modulo torsion on Pic ( B b ) (which maps bythe pull-back p ∗ λ to Pic ( A b ) with finite kernel). Indeed, by the projection formula, denotingby ˜ j ′ : g Σ ′′A b → B the natural map, we have the equality of maps from Pic ( B b ) to Pic ( g Σ ′′A b ):(˜ p ∗ ◦ ˜ j ∗ ) | Pic ( B b ) = ((˜ p λ, Σ ) ∗ ◦ ˜ j ∗ ) | Pic ( B b ) = ((˜ p λ, Σ ) ∗ ◦ (˜ p λ, Σ ) ∗ ◦ ˜ j ′∗ = deg p λ, Σ ˜ j ′∗ . We note here that the morphism ˜ j ′∗ : Pic ( B b ) → Pic ( g Σ ′′A b ) has finite kernel becausedim Im ˜ j ′ = k >
0. As the abelian variety Pic ( A b ) is simple at the very general point of S ′′ ,the nonzero morphism (˜ p λ, Σ ) ∗ ◦ ˜ j ∗ must be injective. But then, by specializing at a point b of S ′′ λ , where λ is chosen in such a way that S ′′ λ = S ′′ ∩ S ′ λ is non-empty, we find that thismorphism is injective on the component Pic ( E b ) of Pic ( A b ). We can now fix the abelianvariety B b and deform the elliptic curve E b . We then get a contradiction, because we knowthat the variety g Σ ′′A b depends (at least birationally) only on B b and not on E b , so that itsPicard variety cannot contain a variable elliptic curve E b .6 Proof of Theorems 0.8 and 0.4, (iv)
Recall that for an abelian variety A and a nonnegative integer k , we denote by A k ⊂ A theset of points x ∈ A such that ( { x } − { A } ) ∗ k = 0 in CH ( A ). The following proves item (i)of Theorem 0.8: Proposition 2.1.
For k > , one has dim A k ≤ k − .Proof. Let g := dim A and let Γ P ontk be the codimension g cycle of A × A such that(Γ P ontk ) ∗ ( x ) = ( { x } − { A } ) ∗ k . for any x ∈ A . As ( { x } − { A } ) ∗ k = P ki =0 ( − k − i (cid:0) ki (cid:1) { ix } , we can takeΓ P ontk = k X i =0 ( − k − i (cid:18) ki (cid:19) Γ i , (7)where Γ i ⊂ A × A is the graph of the map m i of multiplication by i . Let us compute(Γ P ontk ) ∗ η for any holomorphic form on A . Lemma 2.2.
One has (Γ P ontk ) ∗ η = 0 for any holomorphic form η of degree < k on A , and (Γ P ontk ) ∗ η = k ! η for a holomorphic form of degree k on A .Proof. Indeed m ∗ i η = i d η , where d = deg η . By (7), the lemma is thus equivalent to(i) P ki =0 ( − k − i (cid:0) ki (cid:1) i d = 0 , d < k ,(ii) P ki =0 ( − k − i (cid:0) ki (cid:1) i k = k !.From the formula P ki =0 ( − k − i (cid:0) ki (cid:1) X i = ( X − k , we get that the d -th derivative ofthe polynomial P ki =0 ( − k − i (cid:0) ki (cid:1) X i at 1 is 0 for d < k , and is equal to k ! for d = k . Thisimmediately implies (i) by induction on d , using the fact that i ( i − . . . ( i − d + 1) − i d isa degree d − i , and then (ii) by the same argument.This lemma directly implies Proposition 2.1. Indeed, by Mumford’s theorem [12], onehas (Γ P ontk ) ∗ η | A k = 0 for any holomorphic form η of positive degree, and in particular for anyholomorphic k -form. By Lemma 2.2, we conclude that, denoting by A k,reg ⊂ A k the regularlocus of A k , η | A k = 0 for any holomorphic form η of degree k on A , that is, dim A k < k . The following result is almost obvious:
Lemma 2.3.
For any integer k ≥ , the set A k ⊂ A defined in (1) is naturally defined inthe sense of Definition 1.1.Proof. It is known that the set A k ⊂ A is a countable union of closed algebraic subsets.Using the fact that for a morphism f : A → B of abelian varieties, f ∗ : CH ( A ) → CH ( B )is compatible with the Pontryagin product, we conclude that f ∗ ( A k ) ⊂ B k . Finally, given afamily π : A → S of abelian varieties, the set of points x ∈ A such that ( { x } − { A b } ) ∗ k = 0in CH ( A b ), b = π ( x ), is a countable union of closed algebraic subsets of A whose fiber over b ∈ S coincides set-theoretically with A b,k . Proof of Theorem 0.8.
The theorem follows from Proposition 2.1, Lemma 2.3, and Theorem1.3. 7 .3 Proof of Theorem 0.4, (iv)
We first prove the following Proposition (cf. Proposition 0.9).
Proposition 2.4.
Let A be an abelian variety and let x , . . . , x k ∈ A such that X i { x i } = k { A } in CH ( A ) . (8) Then ( { x i } − { A } ) ∗ k = 0 in CH ( A ) (9) for i = 1 , . . . , k .Proof. Let γ l := P | I | = l,I ⊂{ ,...,k } { x I } , where x I := P i ∈ I x i . Then by (8), we have γ = k X i =2 { x i } = −{ x } + k { A } . (10)Furthermore, γ l = 0 for l ≥ k and the following inductive relation is obvious:( k X i =2 { x i } ) ∗ γ l = ( l + 1) γ l +1 + (( m ) ∗ γ ) ∗ γ l − − (( m ) ∗ γ ) ∗ γ l − + . . . , (11)that is: ( l + 1) γ l +1 = l X i =0 ( − i (( m i +1 ) ∗ γ ) ∗ γ l − i , (12)where by (10), ( m i +1 ) ∗ γ = −{ ( i + 1) x } + k { A } . Formula (12) implies inductively that γ l = l X i =0 α l,i { ix } (13)for some rational nonzero coefficients α l,i . As( { x } − { A } ) ∗ i = i X j =0 ( − i − j (cid:18) ij (cid:19) { jx } , the 0-cycles { jx } , ≤ j ≤ l and ( { x } − { A } ) ∗ j , ≤ j ≤ l generate the same subgroupof CH ( A ). The relation γ k = 0 thus provides a nontrivial degree k linear relation with Q -coefficients between the 0-cycles { A } , { x } − { A } , ( { x } − { A } ) ∗ k , or equivalently a polynomial relation in the variable { x }−{ A } for the Pontryagin product,where the scalars are mapped to Q { A } . As we know by [7] that ( { x } − { A } ) ∗ g +1 = 0, weconclude that ( { x } − { A } ) ∗ k = 0.Proposition 2.4 says that if { x } + . . . + { x k } = k { A } in CH ( A ), then x i ∈ A k . Thelocus swept-out by the orbit | k { A }| is thus contained in A k . We thus deduce from Theorem0.8 the following corollary: Corollary 2.5. (Cf. Theorem 0.4, (iv)) For any abelian variety A , the locus swept-out bythe orbit | k { A }| has dimension ≤ k − . For a very general abelian variety A of dimension g ≥ k − , the orbit | k { A }| is countable. In this statement, the locus swept-out by the orbit | k { A }| is the set of points x ∈ A such that a cycle x + Z ′ with Z ′ effective of degree k − | k { A }| . The dimensionof this locus can be much smaller than the dimension of the orbit itself, as shown by theexamples of orbits contained in subvarieties C ( k ) ⊂ A ( k ) for some curve C .8 Proof of Theorem 0.4, (i)
We give in this section the proof of item (i) in Theorem 0.4. We first recall the statement:
Theorem 3.1.
Let A be an abelian variety. The dimension of any orbit | Z | ⊂ A ( k ) forrational equivalence is at most k − .Proof. We will rather work with the inverse image f | Z | of the orbit | Z | in A k . By Mumford’stheorem [12], for any holomorphic i -form α on A with i >
0, one has, along the regular locus f | Z | reg of f | Z | : X j =1 k pr ∗ j α | f | Z | reg = 0 , (14)where the pr j : A k → A are the various projections. Let x = ( x , . . . , x k ) ∈ f | Z | reg and let V := T f | Z | reg ,x ⊂ W k , where W = T A,x = T A, A . One has dim V = dim | Z | , and (14) saysthat:(*) for any α ∈ V i W ∗ with i > , one has ( P j pr ∗ j α ) | V = 0 . Theorem 3.1 thus follows from the following proposition 3.2.
Proposition 3.2.
Let W be a vector space, V ⊂ W k be a vector subspace satisfying property(*). Then dim V ≤ k − . Remark 3.3.
If dim W = 1, the result is obvious, as V ⊂ W k ⊂ W , where W k := Ker ( σ : W k → W ), σ being the sum map. If dim W = 2, the result follows from the fact that,choosing a generator η of V W ∗ , the 2-form P j pr ∗ j η is nondegenerate on W k (which hasdimension 2 k − V satisfying (*) is contained in W k and totally isotropic forthis 2-form, hence has dimension r ≤ k − Proof of Proposition 3.2.
Note that the group Aut W acts on W k , with induced action onGrass( r, W k ) preserving the set of r -dimensional vector subspaces V ⊂ W k satisfying condi-tion (*). Choosing a C ∗ -action on W with finitely many fixed points e , . . . , e n , n = dim W ,the fixed points [ V ] ∈ Grass ( r, W k ) under the induced action of C ∗ on the Grassman-nian are of the form V = h A e , . . . A n e n i , where A i ⊂ ( C k ) ∗ are vector subspaces , with r = P i dim A i . It suffices to prove the inequality r ≤ k − A i have to satisfy the following conditions:(**) For any ∅ 6 = I = { i , . . . , i s } ⊂ { , . . . , n } and for any choices of λ l ∈ A i l , l =1 , . . . , s , k X j =1 ( λ . . . λ s )( f j ) = 0 , where f j is the natural basis of C k . A better way to phrase condition (**) is to use the (standard) pairing h , i on ( C k ) ∗ ,given by h α, β i = k X j =1 α ( f j ) β ( f j ) . Condition (**) when there are only two nonzero spaces A i is the following h α, β i = 0 ∀ α ∈ A , β ∈ A (15) h α, e i = 0 = h e, β i = 0 ∀ α ∈ A , β ∈ A , (16)9here e is the vector (1 , . . . , ∈ ( C k ) ∗ . Indeed, the case s = 2 in (**) provides (15) andthe case s = 1 in (**) provides (16). The fact that the pairing h , i is nondegenerate on( C k ) ∗ := e ⊥ immediately implies that P i dim A i ≤ k − A i arenonzero. By the above arguments, the proof of Proposition 3.2 is finished used the followinglemma: Lemma 3.4.
Let A i ⊂ ( C k ) ∗ , i = 1 , . . . , n , be linear subspaces satisfying conditions (**).Then P i dim A i ≤ k − .Proof. We will use the following result:
Lemma 3.5.
Let A ⊂ C k , B ⊂ C k be vector subspaces satisfying the following conditions:(i) For any a = ( a i ) ∈ A, b = ( b i ) ∈ B, P i a i b i = 0 .(ii) For any a = ( a i ) ∈ A, b = ( b i ) ∈ B, P i a i = 0 , P i b i = 0 .Then dim ( A · B + A + B ) ≥ dim A + dim B , where A · B is the vector subspace of C k generated by the elements ( a i b i ) , a = ( a i ) ∈ A, b = ( b i ) ∈ B . Let us first show how Lemma 3.5 implies Lemma 3.4. Indeed, we can argue induc-tively on the number n of spaces A i . As already noticed, Lemma 3.4 is easy when n = 2.Assuming the statement is proved for n −
1, let A , . . . , A n be as in Lemma 3.4 and let A ′ = A , . . . , A ′ n − = A n − and A ′ n − = A n − · A n + A n − + A n . Then the set of spaces A ′ , . . . , A ′ n − satisfies conditions (**), and on the other hand Lemma 3.5 applies to thepair ( A, B ) = ( A n − , A n ) as they satisfy the desired conditions by (**). Hence we havedim A ′ n − ≥ dim A n − + dim A n and by induction on n , P n − i =1 dim A ′ i + dim A ′ n − ≤ k − P ni =1 dim A i ≤ k − Proof of Lemma 3.5.
Let A := e + A, B = e + B ⊂ C k . Under the conditions (i) and (ii),the multiplication map µ : A × B → C k µ (( a i ) , ( b i )) = ( a i b i )has image in the affine space C k := e + C k , where C k = e ⊥ , and more precisely it generatesthe affine space e + A + B + A · B ⊂ e + C k . It thus suffices to show that the dimension ofthe algebraic set Im µ is at least dim A + dim B . Lemma 3.5 is thus implied by the following: Claim 3.6.
The map µ has finite fiber near the point ( e, e ) ∈ A × B . The proof of the claim is as follows: Suppose µ has a positive dimensional fiber passingthrough ( e, e ). We choose an irreducible curve contained in the fiber, passing through ( e, e )and with normalization C . The curve C admits rational functions σ i , i = 1 , . . . , k mappingit to A such that the functions σ i map C to B . The conditions (i) and (ii) say that X i σ ′ i ( s ) 1 σ i ( t ) = 0as a function of ( s, t ) for any choice of points x, y ∈ C and local coordinates s, t near x ,resp. y , on C . We now do x = y and choose for x a pole (or a zero) of one of the σ l ’s.We assume that the local coordinate s is centered at x , and write σ i ( s ) = s d i f i , with f i aholomorphic function of s which is nonzero at 0. We then get σ ′ i ( s ) 1 σ i ( t ) = d i s d i − t d i φ i ( s, t ) + s d i t d i ψ i ( s, t ) , (17)where φ i ( s, t ) is holomorphic in s, t and takes value 1 at ( x, x ) = (0 ,
0) and ψ i ( s, t ) isholomorphic in s, t . Restricting to a curve D ⊂ C × C defined by the equation s = t l forsome chosen l ≥
2, the function ( σ ′ i ( s ) σ i ( t ) ) | D has order l ( d i − − d i = ( l − d i − l andfirst nonzero coefficient in its Laurent development equal to d i . These orders are differentfor distinct d i and the vanishing P i σ ′ i ( s ) σ i ( t ) = 0 is then clearly impossible: indeed, by10ole order considerations, for the minimal negative value d of d i , hence minimal value of thenumbers ( l − d i − l , the first nonzero coefficient in the Laurent development of ( σ ′ i ( s ) σ i ( t ) ) | D should be also 0 and it is the same as for the sum P i, d i = d ( σ ′ i ( s ) σ i ( t )) | D , which is equal to M d d , where M d is the cardinality of the set { i, d i = d } .The claim is proved.The proof of Proposition 3.2 is thus finished. As a first application, let us give a second proof of Theorem 0.4, (iv). The general dimensionestimate of Theorem 0.4, (i) implies that the locus swept-out by the orbit of | k A | is ofdimension ≤ k − A . This locus is clearly naturally defined. Henceby Theorem 1.3, (ii), it is countable for a very general abelian variety of dimension ≥ k − Theorem 0.4, (iv) has been proved in Section 2.3. We will now prove the following result byinduction on l ∈ { , . . . , k } : Proposition 4.1.
For g ≥ l (2 k −
1) + (2 l − k − , and A a very general abelian varietyof dimension g , any -cycle of the form ( k − l ) { A } + Z , with Z ∈ A ( l ) , has countable orbit. The case l = 0 is Theorem 0.4, (iv) and the case l = k is then Theorem 0.4, (ii). Thecase l = k − ( A ) ⊂ A be the set of points x ∈ A such that the orbit | ( k − { A } + { x }| ⊂ A ( k ) is positive dimensional. The set Σ ( A ) is a countable unionof closed algebraic subsets of A . We would like to show that Σ ( A ) is naturally defined inthe sense of Definition 1.1, and there is a small difficulty here: suppose that p : A → B is a morphism of abelian varieties, and let | Z | ⊂ A ( k ) be a positive dimensional orbit forrational equivalence on A . Then p ∗ ( | Z | ) ⊂ B ( k ) could be zero-dimensional. In the casewhere Z = ( k − { A } + { x } , this prevents a priori proving that Σ ( A ) satisfies axiom (ii)of Definition 1.1. This problem can be circumvented using the following lemma which hasbeen in fact already used in the proof of Theorem 1.3. Let A → S be a generically completefamily of abelian varieties of dimension g . This means that we fixed a polarization type λ and the moduli map S → A g,λ is dominant. Lemma 4.2.
Let
W ⊂ A be a closed algebraic subset which is flat over S of relativedimension k ′ . Then:(i) For any b ∈ S , any morphism p : A b → B of abelian varieties with dim B ≥ k ′ , p ( W b ) ⊂ B has dimension k ′ .(ii) Assume k ′ > . For any b ∈ S , any morphism p : A b → B of abelian varieties with dim B > , p ( W b ) ⊂ B has positive dimension.Proof. (i) Indeed, the locally constant class [ W b ] ∈ H g − k ′ ( A b , Q ) must be a nonzeromultiple of θ g − k ′ λ , since for very generic b ∈ S , these are the only nonzero Hodge classes on A b .We thus have, using our assumption that dim B ≥ k ′ , p ∗ ([ W b ]) = 0 in H B − k ′ ( B, Q ),which implies that dim p ( W b ) = k ′ .Statement (ii) is obtained as an application of (i) in the case k ′ = 1. One first reducesto this case by taking complete intersection curves in W b in order to reduce to the case k ′ = 1.In the following corollary, the orbits for rational equivalence of 0-cycles of X are takenin X l rather than X ( l ) . 11 orollary 4.3. The situation being as in Lemma 4.2, let
W ⊂ A l/S be a family of positivedimensional orbits for rational equivalence in the fibers. Then, up to shrinking S if necessary,for any b ∈ S , any morphism p : A b → B of abelian varieties, where B is an abelian varietyof dimension > , and any i = 1 , . . . , l , p l ( W b ) is a positive dimensional orbit of B .Proof. Indeed, by specialization, W b is a positive dimensional orbit for rational equivalencein A lb . Up to shrinking S , we can assume that the restrictions π | pr i ( W ) : pr ( W ) → S areflat for all i . Our assumption is that for one i , pr i ( W ) has positive relative dimension over S . Lemma 4.2, (ii), then implies that pr i ( p l ( W b )) has positive dimension, so that p l ( W b ) isa positive dimensional orbit for rational equivalence of 0-cycles of B . Proof of Proposition 4.1.
Let now A be a very general abelian variety. This means that forsome generically complete family π : A → S of polarized abelian varieties, A is isomorphicto the fiber over a very general point of S . As A is very general, the locus Σ ( A ) is thespecialization of the corresponding locus Σ ( A /S ) of A , and more precisely, of the unionof its components dominating S . For any fiber A b , let us define the deformable locusΣ ( A ) def as the one which is obtained by specializing to A b the union of the dominatingcomponents of the locus of the relative locus Σ ( A /S ). For a very general abelian variety A ,Σ ( A ) = Σ ( A ) def by definition. Corollary 4.3 essentially says that this locus is naturallydefined. This is not quite true because the definition of Σ ( A ) def depends on choosing afamily A of deformations of A (that is, a polarization on A ). In the axioms of Definition 1.1,we thus should work, not with abelian varieties but with polarized abelian varieties. Axiom(i) should be replaced by its family version, where A → S is locally complete, S ′ ⊂ S is asubvariety, f : A S ′ → B is a morphism of abelian varieties over S ′ , and B → S ′ is locallycomplete. We leave to the reader proving that Theorem 1.3 extends to this context.Assume now g ≥ k −
1. Then Σ ( A ), hence a fortiori Σ ( A ) def , is different from A .Indeed, otherwise, for any x ∈ A , ( k − { A } + { x } has positive dimensional orbit, hencetaking x = 0 A , we get that k { A } has positive dimensional orbit, contradicting Theorem 0.4,(iv). Theorem 1.3, (i) then implies that for g ≥ k − −
1, Σ ( A ) def is countable. Hencethere are only countably many positive dimensional orbits of the form | ( k − { A } + { x }| and the locus they sweep-out forms by Corollary 4.3 a naturally defined locus in A , whichis of dimension ≤ k − g ≥ k −
1) + k −
2, this locus itself is countable, that is, all the orbits | ( k − { A } + { x }| are countable for A very general.The general induction step works exactly in the same way, introducing the locus Σ l ( A ) ⊂ A of points x l ∈ A such that ( k − l )0 A + x + . . . + x l has a positive dimensional orbit forrational equivalence in A for some points x , . . . , x l − ∈ A . It would be nice to improve the estimates in our main theorems. As already mentioned inthe introduction, none of them seems to be optimal. Let us introduce a naturally definedlocus (or the deformation variant of that notion used in the last section) whose study shouldlead to a proof of Conjecture 0.2.
Definition 5.1.
The locus Z A ⊂ A of positive dimensional normalized orbits of degree k isthe set of points x ∈ A such that for some degree k zero-cycle Z = x + Z ′ , with Z ′ effective,one has dim | Z | > , σ ( Z ) = 0 . Here σ : A ( k ) → A is the sum map. It is constant along orbits under rational equivalence.This locus, or rather its deformation version, is naturally defined. Note also that by definitionit is either of positive dimension or empty. The main remaining question is to estimate thedimension of this locus, at least for very general abelian varieties. Conjecture 0.2 wouldfollow from: 12 onjecture 5.2. If A is a very general abelian variety, the locus Z A ⊂ A of positivedimensional normalized orbits of degree k has dimension ≤ k − . Conjecture 5.2 is true for k = 2. Indeed, in this case the normalization condition reads Z = { x } + {− x } for some x ∈ A . The positive dimensional normalized orbits of degree 2are thus also positive dimensional orbits of points in the Kummer variety K ( A ) = A/ ± Id of A . These orbits are rigid because on a surface in K ( A ) swept-out by a continuous familyof such orbits, any holomorphic 2-form on K ( A ) should vanish while Ω K ( A ) reg is generatedby its sections.It would be tempting to try to estimate the dimension of the locus of positive dimensionalnormalized orbits of degree k for any abelian variety. Unfortunately, the following exampleshows that this locus can be the whole of A : Example 5.3.
Let A be an abelian variety which has a degree k − Z ⊂ A ( k − ). Then for each x ∈ A , { x + x } + . . . + { x k − + x } , { x } + . . . + { x k − } ∈ Z isa positive dimensional orbit and thus the set {{ x + x } + . . . + { x k − + x } + {− P i x i − ( k − x } is a positive dimensional normalized orbit of degree k . In this case, the locus of positivedimensional normalized orbits of degree k of A is the whole of A .Nevertheless, we can observe the following small evidence for Conjecture 5.2: Lemma 5.4.
Let O ⊂ A k be a closed irreducible algebraic subset which is a union ofpositive dimensional normalized orbits of degree k . Let Z ∈ O reg and assume the positivedimensional orbit O Z passing through Z has a tangent vector ( u , . . . , u k ) such that thevector space h u , . . . , u k i ⊂ T A, A is of dimension k − . Then the locus swept-out by the pr i ( O ) ⊂ A has dimension ≤ k − . Note that k − h u , . . . , u k i because P i u i = 0. The example above is a case where the vector space h u , . . . , u k i has dimension1. Applying Theorem 1.3, (ii), Conjecture 5.2 in fact implies the following Conjecture 5.5. If A is a very general abelian variety of dimension ≥ k − , the locus ofpositive dimensional normalized orbits of degree k of A is empty. This is a generalization of Conjecture 0.2, because a k -gonal curve ˜ j : e C → A, D ∈ W k ( C ) can always be translated in such a way that σ (˜ j ∗ D ) = 0, hence becomes containedin the locus of positive dimensional normalized orbits of degree k of A .We discussed in this paper only the applications to gonality. The case of higher dimen-sional linea systems would be also interesting to investigate. In a similar but different vein,the following problem is intriguing: Question 5.6.
Let A be a very general abelian variety. Is it true that there is no curve C ⊂ A , whose normalization is a smooth plane curve? If the answer to the above question is affirmative, then one could get examples of surfacesof general type which are not birational to a normal surface in P . Indeed, take a surfacewhose Albanese variety is a general abelian variety as above. If S is birational to a normalsurface S ′ in P , there are plenty of smooth plane curves in S ′ , which clearly map nontriviallyto Alb S , which would be a contradiction. References [1] A. Alzati, G. P. Pirola. Rational orbits on three-symmetric products of abelian varieties.Trans. Amer. Math. Soc. 337 (1993), no. 2, 965-980.[2] F. Bastianelli, P. De Poi, L. Ein, R. Lazarsfeld, B. Ullery. Measures of irrationality forhypersurfaces of large degree. Compos. Math. 153 (2017), no. 11, 2368-2393.133] A. Beauville. Sur l’anneau de Chow d’une vari´et´e ab´elienne. Math. Ann. 273 (1986),no. 4, 647-651.[4] A. Beauville. Quelques remarques sur la transformation de Fourier dans l’anneau deChow d’une vari´et´e ab´elienne. in
Algebraic geometry (Tokyo/Kyoto, 1982), 238-260,Lecture Notes in Math., 1016, Springer, Berlin, 1983.[5] A. Beauville. Algebraic cycles on Jacobian varieties. Compos. Math. 140 (2004), no. 3,683-688.[6] A. Beauville. Vari´et´es K¨ahleriennes dont la premi`ere classe de Chern est nulle. J. Dif-ferential Geom. 18 (1983), no. 4, 755-782 (1984).[7] S. Bloch. Some elementary theorems about algebraic cycles on Abelian varieties. Invent.Math. 37 (1976), no. 3, 215-228.[8] E. Colombo, B. van Geemen. Note on curves in a Jacobian. Compositio Math. 88 (1993),no. 3, 333-353.[9] F. Herbaut. Algebraic cycles on the Jacobian of a curve with a linear system of givendimension. Compos. Math. 143 (2007), no. 4, 883-899.[10] D. Huybrechts. Curves and cycles on K3 surfaces. Algebr. Geom. 1 (2014), no. 1, 69-106.[11] A. Marian, Xiaolei Zhao. On the group of zero-cycles of holomorphic symplectic vari-eties, arXiv:1711.10045.[12] D. Mumford. Rational equivalence of 0-cycles on surfaces, J. Math. Kyoto Univ. 9(1969), 195-204.[13] G. P. Pirola. Curves on generic Kummer varieties, Duke Math. J. 59 (1989), 701-708.[14] G. P. Pirola. Abel-Jacobi invariant and curves on generic abelian varieties, in
Abelianvarieties (Egloffstein, 1993) , 237-249, de Gruyter, Berlin, (1995).[15] C. Voisin. Some new results on modified diagonals. Geom. Topol. 19 (2015), no. 6,3307-3343.[16] C. Voisin. Rational equivalence of 0-cycles on K3 surfaces and conjectures of Huybrechtsand O’Grady in
Recent advances in algebraic geometry , 422-436, London Math. Soc.Lecture Note Ser., 417, Cambridge Univ. Press, Cambridge, 2015.[17] C. Voisin. Remarks and questions on coisotropic subvarieties and 0-cycles of hyper-K¨ahler varieties, in