Motivic sheaves and filtrations on Chow groups
Motivic sheaves and filtrations on Chow groups
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Chow 组的动力滑轮和过滤装置
DOI：
10.1090/pspum/055.1/1265533
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发表时间：
1994
影响因子：
2.5
通讯作者：
U. Jannsen
中科院分区：
文献类型：
--
作者：
U. Jannsen
Grothendieck的动机，如[Dem， K12， M a]中所描述的，被设计为一种工具，用于理解光滑投影变量的上同调及其上的代数循环模同调和数值等价。根据Beilinson和Deligne的观点，Grothendieck的纯动机范畴应该把它嵌入到一个更大的混合动机范畴中，这个范畴允许处理任意的变量，并且可以理解整个周氏循环群的模有理等价，事实上，甚至可以理解所有的代数组的变量。在本文中，我们回顾了其中的一些观点并讨论了一些结果。特别地，我们展示了Beilinson建立的庞大的猜想框架是如何导致对光滑射影变的Chow群上的某些函数的存在性的非常明确的猜想的。这些过滤将提供对一些现象和反例的理解，这些现象和反例在一段时间内使人们相信余维大于1的代数循环的行为是绝对混沌的。在§1中，我们回顾了关于Chow群、对应和循环映射到上同调理论的一些基本事实。我们回顾了M M M ford的一个反例，它暗示在一般情况下，Chow群是不可表示的，Abel-Jacobi映射有一个巨大的核，以及Bloch对这个主题的一些研究。在§2和§4中，我们总共陈述了四个版本的Beilinson关于Chow群体的混合动机和过滤的猜想，增加了它的普遍性和复杂性。第一个甚至没有提到混合动机，并提出了在理性Chow群CH (X)Q上的有限过滤F D F D•••，这些过滤是由它们在代数对应下的行为唯一决定的。第一步是同源等价，但接下来的步骤与经典的考虑有很大的不同。例如，代数等价没有出现，第二步类似于核
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 consequences. 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 F D F D • • • on rational Chow groups CH (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