Chow motives and cycle modules

Chow 动机和循环模块

• 批准号: RGPIN-2017-04174
• 负责人: Gille, Stefan
• 金额: $ 1.17万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635444
• 关键词: Chow; motives; cycle; modules

Chow motives have been introduced by Alexander Grothendieck (1928-2014), winner of the fields Medal (analog of Noble Price) in 1966. These objects and their generalizations as for instance Vladimir Voevodsky's triangulated categories of motives play a significant role in algebra and algebraic geometry. Starting with Vladimir Voevodsky's proof of the Milnor conjecture which uses Markus Rost's computation of the Chow motive of a norm quadric, motives and the closely related algebraic cycles invaded first the theory of quadratic forms and later the theory of algebraic groups. The so called Rost nilpotence principle which has been verified for projective homogeneous varieties as for instance quadrics plays here an important role, as one of the main tools to decompose motives. It is conjectured that this principle is true for all smooth projective varieties, and has been proven by myself for surfaces and certain threefolds over fields of characteristic 0. In these proofs the cycle modules of Markus Rost, which in particular have been introduced as a tool to compute Chow groups, are indispensable.

Chow 动机是由 1966 年领域奖章（类似于 Noble Price）获得者 Alexander Grothendieck（1928-2014）提出的。这些对象及其概括，例如 Vladimir Voevodsky 的动机三角范畴，在代数和代数几何中发挥着重要作用。从 Vladimir Voevodsky 对米尔诺猜想的证明开始，该猜想使用了 Markus Rost 对范数二次函数的 Chow 动机的计算，动机和密切相关的代数循环首先侵入了二次形式的理论，后来侵入了代数群的理论。所谓的罗斯特幂零原理，已在投影齐次簇（例如二次方程）中得到验证，在这里发挥着重要作用，作为分解动机的主要工具之一。据推测，这一原理对于所有光滑射影簇都是正确的，并且我自己已经证明了曲面和特征 0 域上的某些三倍。在这些证明中，Markus Rost 的循环模块（特别是作为计算 Chow 群的工具而引入）是必不可少的。