On the splitting of the Bloch Beilinson filtration

Arnaud Beauville

Description

On the splitting of the Bloch-Beilinson filtration Arnaud BEAUVILLE Dedicated to Jacob Murre on his 75th birthday Introduction This paper deals with the Chow ring CH(X) (with rational coefficients) of a smooth projective variety X – that is, the Q-algebra of algebraic cycles on X , modulo rational equivalence. This is a basic invariant of the variety X , which may be thought of as an algebraic counterpart of the cohomology ring of a compact manifold; in fact there is a Q-algebra homomorphism cX : CH(X)? H(X,Q) , the cycle class map. But unlike the cohomology ring, the Chow ring, and in particular the kernel of cX , is poorly understood. Still some insight into the structure of this ring is provided by the deep conjectures of Bloch and Beilinson. They predict the existence of a functorial ring filtration (Fj)j≥0 of CH(X) , with CHp(X) = F0CHp(X) ? . . . ? Fp+1(X) = 0 and F1CH(X) = Ker cX . We refer to [J] for a discussion of the various candidates for such a filtration and the consequences of its existence. The existence of that filtration is not even known for an abelian vari- ety A . In that case, however, there is a canonical ring graduation given by CHp(A) =? s CHps(A) , where CHps(A) is the subspace of elements ? ? CHp(A) with

• d1 ·


• smooth projective


• splitting property


• ???y ???y


• dim x? dim


• cycle class


• bloch-beilinson filtration


• manifold

Sujets

• Beauville
• Conjugacy class
• Manifold

Informations

• Publié par: pefav
• Nombre de lectures: 8
• Langue: English

Extrait

On the splitting of the Bloch-Beilinson ¯ltration

Introduction

Arnaud B EVILLUAE

DedicatedtoJacobMurreonhis75 th birthday

This paper deals with theChow ring of a ts) rational coe±cienCH(X) (with smooth projective variety X – that is, theQ , X-algebra of algebraic cycles on modulo rational equivalence. This is a basic invariant of the variety X , which may be thought of as an algebraic counterpart of the cohomology ring of a compact manifold; in fact there is aQ-algebra homomorphismc X : CH(X)→H(XQ) , the cycle class map. But unlike the cohomology ring, the Chow ring, and in particular the kernel ofc X , is poorly understood. Still some insight into the structure of this ring is provided by the deep conjectures of Bloch and Beilinson. They predict the existence of a functorial ring ¯ltration (F j ) j¸0  CH with ,of CH(X) p (X) = F 0 CH p (X)¾. . .¾F p+1  and(X) = 0 F 1 CH(X) = Kerc X . We refer to [J] for a discussion of the various candidates for sucha¯ltrationandtheconsequencesofitsexistence. The existence of that ¯ ltration is not even known for an abelian vari- ety A . In that case, however, there is a canonicalring graduationgiven by CH p (A) =©CH s p  CH(A) , where p s (A) is the subspace of elements®∈CH p (A) with s k ¤ A ®=k 2p¡s ®for allk∈Z(k A denotes the endomorphisma7→kaof A [B2]. ) Unfortunately this does not de¯ ne the required ¯ ltration because the vanishing of the terms CH p s (A) fors <0 is not known in general – in fact, this vanishing is essentially equivalent to the existence of the Bloch-Beilinson ¯ ltration (the precise relationship is thoroughly analyzed in [Mu]). So if the Bloch-Beilinson ¯ ltration in- deed exists, itsplitsin  associated to a graduationthe sense that it is the ¯ ltration of CH(A) . In [B-V] we observed that this also happens for a K3 surface S . Here the ¯ltration is essentially trivial; the fact that it splits means that the image of the intersection product CH 1 (S)­CH 1 (S)→CH 2 (S) is always one-dimensional – an easy but somewhat surprising property.

The motivation for this paper was to understand whether the splitting of the Bloch-Beilinson ¯ltration for abelian varieties and K3 surfaces is accidental or part of a more general framework. Now asking for a conjectural splitting of a conjectural ¯ltrationmaylooklikearatheridleoccupation.Thepointwewanttomakeis that the mere existence of such a splitting has quite concrete consequences, which at least in some cases can be tested. We will restrict for simplicity to the case of

1