Motivic sheaves and filtrations on Chow groups

Abstract

Grothendieck's motives, as described in [Dem, K12,Ma] are designed as a tool to understand the cohomology of smooth projective varieties and the algebraic cycles modulo homological and numerical equivalence on them. According to Beilinson and Deligne, Grothendieck's category of pure motives should embed in a bigger category of mixed motives that allows the treatment of arbitrary varieties and an understanding of the whole Chow group of cycles modulo rational equivalence, in fact, even of all algebraic .fiT-groups of the varieties.In this paper we review some of these ideas and discuss some consequences. In particular, we show how the vast conjectural framework set up by Beilinson leads to very explicit conjectures on the existence of certain filiations on Chow groups of smooth projective varieties. These filtrations would offer an understanding of several phenomena and counterexamples that for some time have led people to believe that the behaviour of the algebraic cycles is absolute chaos for codimension bigger than one.In § 1 we review some basic facts on Chow groups, correspondences, and cycle maps into cohomology theories. We recall a counterexample of Mumford implying that in general Chow groups are not representable and the Abel-Jacobi map has a huge kernel and some investigations of Bloch on this topic.In § §2 and 4 we state altogether four versions of Beilinson's conjectures on mixed motives and filtrations on Chow groups, increasing in generality and sophistication. The first one does not even mention mixed motives and proposes finite filtrations (X) Q that are uniquely determined by their behaviour under algebraic correspondences. The first step is homological equivalence, but the following steps differ very much from those considered classically. For example, algebraic equivalence does not appear, and the second step is something like the kernel 1991 Mathematics Subject Classification. Primary 14C15; Secondary 14A20, 14C25. This paper is in final form and no version of it will be submitted for publication elsewhere.

On Finite-dimensional Motives and Murre's Conjecture

“…It is known [Ja2] that this conjecture, taken for all smooth projective varieties, is equivalent to the conjecture of Bloch-Beilinson on a certain functorial filtration on Chow groups, and that this Bloch-Beilinson filtration would be equal to the filtration F · defined above. The advantage of Murre's conjecture is that it can be formulated and proved for specific varieties, and that will be used below.…”

“…On the other hand it is known that the Bloch-Beilinson-Murre conjecture would imply the conjecture (2.4) of Kimura-O'Sullivan ([AK], [An1]). Let us note here that, more precisely, Murre's conjecture for X, X × X and X × X × X implies N (X) ([Ja2] pp. 294, 295), so that Murre's conjecture for all sufficient high powers X N implies finite-dimensionality of X (3.9; note that we have assumed C(X), hence S(X)).…”

“…Conjecture N (X) would follow from the existence of the Bloch-Beilinson filtration [Ja2], or Murre's conjecture [Mu1], or the following conjecture of Voevodsky:…”

“…Finally, by 3.1 every other idempotent lifting π X i is of the form (1 + a)π i (1 + a) −1 with a ∈ J(X), cf. [Ja2] 5.4. But every endomorphism of h rat (X) commutes with F (cf.…”

On algebraic 1-motives related to Hodge cycles

“…Following the general framework of mixed motives (e.g., see [11], [25] and [23] for a full overview) we may expect the following picture for non-singular algebraic varieties over a field k (algebraically closed of characteristic zero for simplicity).…”