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Motivic sheaves and filtrations on Chow groups

American Mathematical Society
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  • 510 Mathematik
  • Design
  • Mathematics


Proceedings of Symposia in Pure Mathematics Volume 55 (1994), Part 1 Motivic Sheaves and Filtrations on Chow Groups U W E J A N N S E N Grothendieck's motives, as described in [Dem, K12, M a ] 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 i n a bigger category of mixed motives that allows the treatment of arbitrary varieties and an understanding of the whole Chow group o f cycles modulo rational equivalence, in fact, even of all algebraic .fiT-groups o f the varieties. In this paper we review some of these ideas and discuss some conse- quences. In particular, we show how the vast conjectural framework set up by Beilinson leads to very explicit conjectures on the existence of certain f i l i a - tions 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 o f M u m - ford 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 i n generality and sophistication. The first one does not even mention mixed motives and proposes finite filtrations F0 D F1 D • • • on rational Chow groups CHj (X)Q that are uniquely determined by their behaviour under algebraic correspon- dences. The first step is homological equivalence, but the following steps differ very much from those considered classically. F

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