Chow groups of projective varieties of very small degree
Chow groups of projective varieties of very small degree
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极小阶射影簇的 Chow 群
DOI：
10.1215/s0012-7094-97-08702-0
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发表时间：
1997
影响因子：
2.5
通讯作者：
E. Viehweg
中科院分区：
文献类型：
--
作者：
H. Esnault;M. Levine;E. Viehweg
(see[12]，[5]和那里给出的参考文献）。这些事实，连同关于代数簇的上同调和Chow群的各种说明，表明X的Chow群可能满足CHl（X）<$Q = CHl（Pk）<$Q = Q（Pk），其中l ≤ κ− 1（与注5.6和推论5.7比较）。这是由V. Srinivas和K. Paranjape在[16]，猜想1.8;推理链大致如下。假设X是光滑的。人们期望良好的过滤0 = F j+1 <$F j <$。. . <$F 0 = CH（X ×X）<$Q，其渐变片段F /F l+1由H2 j −l（X × X）控制（参见[10]）。根据Grothendieck的广义猜想[8]，群H（X）应该由余维κ子集的同调的Gysin态射下的像以及来自P的类生成。对于零圈，猜想（）由Roitman定理得出（见[17]和[18]）：
(see [12], [5] and the references given there). These facts, together with various conjectures on the cohomology and Chow groups of algebraic varieties, suggest that the Chow groups of X might satisfy CHl(X)⊗Q = CHl(Pk)⊗Q = Q (∗) for l ≤ κ− 1 (compare with Remark 5.6 and Corollary 5.7). This is explicitly formulated by V. Srinivas and K. Paranjape in [16], Conjecture 1.8; the chain of reasoning goes roughly as follows. Suppose X is smooth. One expects a good filtration 0 = F j+1 ⊂ F j ⊂ . . . ⊂ F 0 = CH(X ×X)⊗Q, whose graded pieces F /F l+1 are controlled by H2j−l(X × X) (see [10]). According to Grothendieck’s generalized conjecture [8], the groups H (X) should be generated by the image under the Gysin morphism of the homology of a codimension κ subset, together with the classes coming from P. Applying this to the diagonal in X × X should then force the triviality of the Chow groups in the desired range. For zero-cycles, the conjecture (∗) follows from Roitman’s theorem (see [17] and [18]):