Chow Groups of Abelian Varieties and Beilinson's Conjecture
aa r X i v : . [ m a t h . AG ] D ec Chow Groups of Abelian Varieties and Beilinson’sConjecture
Bogdan Zavyalov
Abstract
In the present paper we introduce the property AA of a subsemigroup of the endomor-phism semigroup of an abelian variety, which holds for semigroup of endomorphisms ofan abelian variety defined over a number field, and show that the orbit of any cycle undera semigroup with property AA in the Chow group ⊗ Q has finite dimensional span. Contents
Introduction
One of the most celebrated conjectures in 20th century algebraic geometry is a conjectureof Beilinson, which in particular predicts that for any variety X defined over a number fieldthe group CH( X ) ⊗ Q has finite dimension. This conjectural picture is completely opposite tothe one in complex setting, e. g., Mumford proved that the Chow group of a smooth complexsurface posessing a nonzero holomorphic 2-form is infinitely generated in any reasonable sense.The most na¨ıve way to produce a counterexample to this conjecture is to build a variety X overa number field together with a cycle on it such that its orbit w. r. t. all the endomorphismsof X in the Chow group ⊗ Q contains infinitely many linearly independent elements.The goal of the present paper is to show that this attempt fails at least for X an abelianvariety. Although this fact is arithmetic in its nature, we show indeed that orbit of any cycleunder any subsemigroup with the property AA (to be defined in the paper) of endomorphismsemigroup (w. r. t. composition) has finite dimensional span in CH ⊗ Q . This has someinteresting applications even in the case of algebraically closed field. The proof of the factthat property AA holds for the entire endomorphism semigroup over number field is due toMordell–Weil theorem.The first part of the paper is a reminder of Fourier–Mukai transform developed by Beauvillein his paper [Bea86]. The prove itself is given in the second part. The Fourier–Mukai trans-form allows us to reduce the case of an arbitrary cycle on A to the case of the Poincar´e linebundle on A × b A , and next we deal with the case of divisors explicitly.1 cknowledgments. We would like to thank Marat Rovinsky for suggesting the problemand numerous helpful discussions. Also, we would like to thank Rodion D´eev for reviewingthe first draft of the paper and Dmitrii Pirozhkov without whose assistance this paper mighthave not been written at all. The research was supported in part by Dobrushin stipend.
Conventions.
1. For any abelian variety A and a k -rational point a ∈ A ( k ) we will denote by t a : A → A the translation of A by a . We will denote the dual of the abelian variety A by b A andthe Poincar´e line bundle by L ∈ Pic( A × b A ) . Moreover, we will denote the first Chernclass of L by l .2. For any smooth projective variety X over a field k we will denote by CH( X ) := L i CH i ( X ) the direct sum of all its Chow groups. The cup product enhances the abeliangroup CH( X ) with a structure of a graded ring. For any proper map f : X → Y we willdenote by f ∗ the pushforward map on Chow rings (resp. groups) f ∗ : CH( X ) → CH( Y )(resp. f ∗ : CH i ( X ) → CH i ( Y )). For any flat morphism g : X → Y we will denoteby g ∗ the pullback map on Chow rings (resp. groups) g ∗ : CH( Y ) → CH( X ) (resp. g ∗ : CH i ( Y ) → CH i ( X )).3. All the products of varieties X × Y will actually mean X × Spec k Y . By p i ,...,i n : Q mj :=1 X j → Q nk :=1 X i k we will always denote the corresponding projection map.4. For any variety A over a field k we will denote by End( A ) the semigroup of its endo-morphisms over Spec( k ) w. r. t. composition. For an abelian variety A we will denoteby End ( A ) its ring of group endomorphisms over a base field. Note that End( A ) hasalso a group structure w. r. t. addition on A , but we will not deal with it (unless theopposite is mentioned explicitly).5. Subscript Q will always mean tensoring by Q . For example, for any abelian group (resp.ring) R , R Q will be a Q -vector space (resp. Q -algebra) R ⊗ Z Q . In this section we will provide the reader with all the well-known facts that will be impor-tant in the paper.
All the results in this section are well-known and could be found in any textbook onabelian varieties, possibly except for Proposition 1 . , which is not always formulated in sucha manner.Let A be an abelian variety of dimension g over a field k . We will need several facts aboutthe structure of its Picard group.Basic fact from algebraic geometry states that there is a short exact sequence0 → Pic ( A ) → Pic( A ) → NS( A ) → , ( A ) is a group of algebraically trivial cycles, and NS( A ) is a group of cycles moduloalgebraic equivalence. Note that NS( A ) is finitely generated Z -module for any smooth projec-tive variety. Also, the semigroup of endomorphisms G := End( A ) acts on each term of thisshort exact sequence making it into a short exact sequence of G -modules. The crucial fact isthat after tensoring by Q it splits as a short exact sequence of G := End ( A )-modules. Definition 1.1.
A divisor D ∈ Pic( A ) is called symmetric , if [ − ∗ D = D. , and antisymmet-ric , if [ − ∗ D = − D. Let us denote the group of symmetric (resp. antisymmetric) divisors byPic + ( A ) (resp. Pic − ( A )).The following lemmata give us a very good characterization of symmetric and antisym-metric divisors. Lemma 1.2.
Let A be an abelian variety over a field k , D ∈ Pic( A ). Then the following areequivalent:1. D is symmetric,2. [ n ] ∗ D = n D for all [ n ] ∈ Z . Proof. [Mil08, Cor. 5.4]
Lemma 1.3.
Let A be an abelian variety over a field k , D ∈ Pic( A ). Then the following areequivalent:1. D is antisymmetric,2. [ n ] ∗ D = nD for all [ n ] ∈ Z ,3. For every f, g ∈ End ( A ) we have ( f + g ) ∗ ( D ) = f ∗ ( D ) + g ∗ ( D ),4. D ∈ Pic ( A ).Additionally, if D is antisymmetric, then for each a ∈ A ( k ) one has t ∗ a ( D ) = D . Proof. [Mil08, Cor. 5.4. + Remark 8.5]These lemmata allows us to construct a splitting of the short exact sequence0 → Pic ( A ) Q → Pic( A ) Q r −→ N S ( A ) Q → Proposition 1.4.
There is a canonical splitting of 1 as G -modules given by a map φ :Pic( A ) Q → Pic ( A ) Q defined as φ ( D ) = ( D − [ − ∗ D ) /
2. Moreover, Pic( A ) Q ≃ Pic + ( A ) Q ⊕ Pic − ( A ) Q as G -modules as well as Pic + ( A ) Q ≃ Pic ( A ) Q , Pic − ( A ) Q ≃ Pic ( A ) Q Proof.
First of all, note that the lemma 1.3 says that a natural inclusion of Pic − ( A ) Q intoPic ( A ) Q gives us an isomorphism of them as G -modules. Secondly, we have two projec-tions p ± : Pic( A ) Q → Pic ± ( A ) Q , namely p ± ( D ) = D ± [ − ∗ D . It is easy to check thatthese morphisms are morphisms of G -modules, so it provides us with a decompositionPic( A ) Q ≃ Pic + ( A ) Q ⊕ Pic − ( A ) Q . This implies that φ is a G -module section of the shortexact sequence. Finally, ker r = Pic ( A ) Q = Pic − ( A ) Q and Pic + ( A ) Q ∩ Pic − ( A ) Q = 0 . Thus,by exactness of 1 and the fact that all morphisms are morphisms of G -modules, we concludethat Pic + ( A ) Q ≃ NS( A ) Q . emark 1.5. In particular this statement claims that [ n ] ∗ ( α ) = n α for every α ∈ NS( A ) Q . Also, the proof actually shows that there is a morphism of G -modules ψ : NS( A ) Q → Pic( A ) Q defined by ψ ( α ) = ( D + [ − ∗ D ) / , where D ∈ Pic( A ) Q is any representative of α ∈ NS( A ) Q . Definition 1.6.
Let A be an abelian variety. The Fourier–Mukai transform is a map F :CH( A ) Q → CH( b A ) Q defined by F ( α ) = p ∗ ( p ∗ ( α ) · exp( l )) for every α ∈ CH( A ) Q . By exp weundestand the map defined by formal power series exp( t ) = 1 + t + t + . . . . Note that exp( l )is well-defined because A is of finite dimension.By b F we will denote the Fourier–Mukai tranform on the dual of abelian variety b F : CH( b A ) Q → CH( A ) Q . The following theorem is essential in the proof of the Main Theorem. Namely, it will allowus to reduce the general case to the case of divisors.
Theorem 1.7.
Let A be an abelian variety of dimension g . Then the following formula holds b F ◦ F = ( − g [ − ∗ Proof. [Bea86, F2 p.647]
Before proving the Main Theorem we have to define some class of subsemigroups of theendomorphism semigroup of abelian variety for which it holds. The definition of such sub-semigroups will be a little bit technical, but we will provide the reader with some interestingexamples of such subsemigroups.Recall that the semigroup of endomorphisms G = End( A ) of any abelian variety is asemidirect product of the semigroup of its group endomorphisms G = End ( A ) and transla-tions. If we denote the latter one by T , then G = T ⋊ G . Definition 2.1.
We say that a subsemigroup H ⊂ G has property AA , if there is a finitenumber of k -rational points a , . . . , a n such that any element of h ∈ H can be written as h = f ◦ t l n a n ◦ · · · ◦ t l a , where f ∈ G and l i ∈ Z . Example 2.2.
1. The subsemigroup G has property AA for trivial reasons.2. Every finitely generated subgroup T ⊂ T has property AA.3. Fix some finitely generated subgroup T ⊂ T . Then a subsemigroup T ⋊ G has propertyAA.4. (the most interesting example) Let A be an abelian variety over a number field k . Thenthe semigroup of all endomorphisms of A satisfies the property AA. The Mordell–Weiltheorem ([Ser89, Chapter 4]) claims that for any abelian variety over number field A ( k )is a finitely generated group. It reduces this example to the third one. Theorem 2.3. [The Main Theorem] Let A be an abelian variety over a field k . Fix somesubsemigroup H ⊂ G that satisfies the property AA. Then for any cycle α ∈ CH p ( A ) Q itsorbit under the action of the semigroup H spans a finite-dimensional Q -vector space.4 roof. We are going to prove the theorem in three steps. Firstly, we will prove it for H = G and p = 1. Secondly, we will reduce the case of any subsemigroup with property AA and p = 1 to the case H = G . Finally, we will reduce the general case to the case p = 1. Step 1.
Assume that p = 1 and H = G . We will prove that for every finite-dimensionalvector space V ⊂ Pic( A ) Q its orbit under the action of G spans finite-dimensional vectorspace.According to Proposition 1.4, we have a split short exact sequence of G -modules0 → Pic ( A ) Q → Pic( A ) Q → NS( A ) Q → . Since this short exact sequence of G -modules splits and NS( A ) Q is of finite dimension, wecan assume that V ⊂ Pic ( A ) Q . From now on we will consider G as a group under additioninstead of considering it as semigroup under composition. Lemma 1.3 tells us that the actionof G is linear (w.r.t. to this group structure) on Pic ( A ) Q . Combined with the fact that G is a finitely generated group ([Mil08, Prop. 10.5 + Lemma 10.6]), we conclude that the orbitof the vector subspace V under the action of G spans a finite-dimensional vector space. Step 2.
Now assume that p = 1, but semigroup H is any subsemigroup of G withthe property AA. According to the definition, we can choose a finite number of k -rationalpoints a , . . . , a n ∈ A ( k ) such that any element of g ∈ H can be written in the following form g = f ◦ t l n a n ◦· · ·◦ t l a with f ∈ G . We know [Mil08, Lemma 8.8] that β i := t ∗ a i ( α ) − α ∈ Pic ( A ) Q for any 0 < i ≤ n. Since all t a i commute and t ∗ a ( β ) − β = 0 for any β ∈ Pic ( A ) Q (Lemma 1.3),we conclude that ( t l n a n ) ∗ ◦ · · · ◦ ( t l a ) ∗ ( α ) = α − l β − · · · − l n β n . In particular, the vector space V generated by the orbit of α under the action of endomorphisms of the form t l n a n ◦ · · · ◦ t l a is of finite dimension. Therefore, since H has property AA, it suffices to show that the orbitof V under the action of G spans finite-dimensional vector space. But it has been alreadydone in the Step 1. Step 3.
Finally, we are going to reduce the general case to the case of a divisor on theabelian variety A × b A. Theorem 1.7 says that α = ( − g [ − ∗ b F ( F ( α )) . Let F ( α ) = P gq =1 η q with η q ∈ CH q ( b A ) Q . Again, choose finite number of k -rational points a , . . . , a n ∈ A ( k ) such that any element of g ∈ H can be written in the following form g = f ◦ t l n a n ◦ · · · ◦ t l a with f ∈ G . Choose sucha representation for each g ∈ H and denote by g ′ := f ◦ t l n − a n ◦ · · · ◦ t l − a . Also we will denoteby g the morphism g × Id b A : A × b A → A × b A . Now, for any g ∈ H we have g ∗ ( α ) = ( − g ( g ∗ ◦ [ − ∗ ◦ b F ◦ F )( α ) = ( − g ( g ∗ ◦ [ − ∗ ◦ b F )( g X q =1 η q ) = ( − g g X q =1 g ∗ ([ − ∗ ◦ b F ( η q ))This expression allows us to conclude that in order to prove the Theoreom for a cycle α it is sufficient to prove it for every [ − ∗ ◦ b F ( η q ). From now on we are going to prove theTheorem in this case.Note that for any g ∈ H we have that g ∗ ◦ [ − ∗ = [ − ∗ g ′ . Also the base change formulaapplied to the following Cartesian diagram asserts that f ∗ ◦ p ∗ = p ∗ ◦ ( f ) for every f ∈ G . A × b A f −−−−→ A × b A p y p y A f −−−−→ A
5e conclude that g ∗ ([ − ∗ ◦ b F ( η q )) = g ∗ ◦ [ − ∗ ( p ∗ ( p ∗ ( η q ) · exp( l ))) = [ − ∗ ◦ g ′∗ ( p ∗ ( p ∗ ( η q ) · exp( l ))) == [ − ∗ ◦ p ∗ ( g ′∗ ( p ∗ ( η q ) · exp( l ))) = [ − ∗ ◦ p ∗ ( p ∗ ( η q ) · exp( g ′∗ ( l ))) == [ − ∗ ◦ p ∗ ( p ∗ ( η q ) · ( ∞ X i =1 g ′∗ ( l n ))) = ∞ X i =1 ([ − ∗ ◦ p ∗ ( p ∗ ( η q ) · g ′∗ ( l n )))The sum is actually finite because A × b A is a variety of dimension 2 g , so l n = 0 for n > g .Since cup-product, pullbacks and pushforwards are linear maps, it is enough to show that theorbit of each l n under the action of elements of the form g ′∗ spans a finite-dimensional vectorspace. Moreover, since g ′∗ ( l n ) = g ′∗ ( l ) n it is enough to show it only for l . Every element ofthe form g ′ , by definition, can be written in the form g ′ = ( f × Id) ◦ ( t k n − a n × Id) ◦ · · · ◦ ( t k − a × Id) = ( f × Id) ◦ t k n ( − a n , ◦ · · · ◦ t k ( − a , Since f ∈ End ( A ), we have f × Id ∈ End ( A × b A ) . Therefore, the subset H ′ = { f ∈ End( A × b A ) |∃ g ∈ H : f = g ′ } is a subsemigroup and has property AA. Therefore, we can apply the Step 2 to H ′ and l andfinish the proof. Corollary 2.4.
Let A be an abelian variety over any field k . Fix some cycle β ∈ CH p ( A ) Q and a k -rational point x ∈ A ( k ) , then the set { α, t ∗ x ( α ) , ( t ∗ x ) ( α ) , . . . } spans finite dimensionalvector space. Proof.
Apply Theorem 2.3 to H = Z t ∗ x and α = β . Corollary 2.5.
Let A be an abelian variety over any field k . Fix some cycle β ∈ CH p ( A ) Q ,then the orbit of β under the action of the semigroup End ( A ) spans finite dimensional vectorspace. Proof.
Apply Theorem 2.3 to H = End ( A ) and α = β . Corollary 2.6.
Let A be an abelian variety over number k . Fix some cycle β ∈ CH p ( A ) Q .Then the orbit of β under the action of the group of all the endomorphism semigoup End( A )spans a finite-dimensional vector space. Proof.
According to Example 4, End( A ) has property AA. Thus we can apply Theorem 2.3to H = End( A ) and α = β . References [Bea86] A. Beauville,
Sur l’anneau de Chow d’une vari´et´e ab´elienne , Math. Ann. (1986), no. 4, 647-651.[Mil08] J. Milne (ed.),
Abelian Varieties , Lecture notes, 2008.[Ser89] J. P. Serre,
Lectures on the Mordell–Weil theorem , Aspects of Mathematics, E15. Friedr. Vieweg Sohn,Braunschweig., 1989., Aspects of Mathematics, E15. Friedr. Vieweg Sohn,Braunschweig., 1989.