Chow-Künneth decomposition for 3- and 4-folds fibred by varieties with trivial Chow group of zero-cycles
Chow-Künneth decomposition for 3- and 4-folds fibred by varieties with trivial Chow group of zero-cycles
复制标题
由具有零周期 Chow 群的品种进行 3 倍和 4 倍纤维化的 Chow-Künneth 分解
DOI：
10.1090/s1056-3911-2014-00616-0
复制
发表时间：
2014
影响因子：
1.8
通讯作者：
Vial C
中科院分区：
文献类型：
--
作者：
Vial C
令 k 为一个域，并令 Ω 为 k 上的通用域。令 f: X→ S 为在 k 上定义的显性态射，从平滑射影簇 X 到维数≤ 2 的平滑射影簇 S，使得 fΩ 的一般纤维具有零循环的平凡 Chow 群。例如，X 可以是其一般成员有理连接的二维变族族的总空间。假设X的维度≤4。那么我们证明X具有自对偶Murre分解，即X具有自对偶Chow-Künneth分解，满足Murre猜想(B)和(D)。此外，我们证明动机莱夫谢茨猜想对于 X 成立，因此莱夫谢茨标准猜想也成立。我们还给出了 Kimura 有限维一般类型三重的新例子、具有自对偶 Murre 分解的一般类型四重的新例子，以及具有有限度三无分支上同调的簇的新例子。
Let k be a field and let Ω be a universal domain over k. Let f: X→ S be a dominant morphism defined over k from a smooth projective variety X to a smooth projective variety S of dimension≤ 2 such that the general fibre of fΩ has trivial Chow group of zero-cycles. For example, X could be the total space of a two-dimensional family of varieties whose general member is rationally connected. Suppose that X has dimension≤ 4. Then we prove that X has a self-dual Murre decomposition, ie that X has a self-dual Chow–Künneth decomposition which satisfies Murre’s conjectures (B) and (D). Moreover we prove that the motivic Lefschetz conjecture holds for X and hence so does the Lefschetz standard conjecture. We also give new examples of threefolds of general type which are Kimura finite-dimensional, new examples of fourfolds of general type having a self-dual Murre decomposition, as well as new examples of varieties with finite degree three unramified cohomology.