Hodge Type Realizations of Algebraic Cycles

代数环的 Hodge 型实现

• 批准号: RGPIN-2018-04344
• 负责人: Lewis, James
• 金额: $ 3.35万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758352
• 关键词: Hodge; Type; Realizations; Algebraic; Cycles

An abridged title description of my research is Regulators of Motives. The world of motives was the invented by A. Grothendieck, a Fields medalist who is arguably the most influential mathematician in the 20th century. Grothendieck observed that many different mathematical objects, arising from very different origins (e.g. algebra, analysis, or geometry) had more in common than what was formerly understood. He conjectured that there was an entirely complete mathematical world in itself, called the category of motives, that would unify and encode all of the deep ideas in mathematics, and in a somewhat universal way. This world of motives, which is still currently in conjectural form has been a source of preoccupation among the leading mathematical minds of the 20th and 21st centuries. A case at point is a candidate version of motives possessing ``most'' of the desired properties, invented by 2004 Fields medalist V. Voevodsky. Regulators are realizations of the conjectural world of motives into categories that are more ``earthly'' defined. So in a sense, regulators gives us a snapshot of something we are trying to show exists! Of course when we construct a regulator, we have a candidate category of motives in mind. The trouble is that outside of Grothendieck's original concrete proposal (which requires major conjectures to get off the ground), no other candidate possessing the expected properties is very easy to describe by itself, and that is precisely where regulators come into play. Although there are plenty of very abstract definitions of regulators in the literature, the absence of a very explicit description was a big obstacle. Over the course of the last 10 years, I took it upon myself with various collaborators to provide that description in a first paper(with Kerr and M\"uller-Stach, Compositio Math 142).A second paper (with Kerr, Inventiones 170), which was much more involved, was a tour de force, ``take no prisoners'' approach to regulators involving the very general arena of ``mixed motives'', where applications to number theory and physics are apparent. The next paper in this direction which is currently in progress, and using a larger cast of collaborators, will be a explicit and decisive description of regulators using Bloch's simplicial higher Chow groups as well as using the Voevodsky machinery. Finally, Kerr and Lewis have reworked a simplicial version of the Bloch regulator, and in the process, have discovered a serious error in the simplicial realregulator provided by Goncharov, which has been extensively used over the past decade. The results will appear in the Journal of Algebraic Geometry.The corollaries to all this work will hopefully influence a new generation of algebraic geometers for many years to come.My current preoccupation is in the direction of the Beilinson-Hodge conjecture, and its connections to theBloch-Kato theorem, as well as work on height pairings.

我研究的删节标题描述是动机的监管者。 动机的世界是由奖牌获得者A. Grothendieck发明的，他可以说是20世纪最具影响力的数学家。 Grothendieck观察到，许多不同的数学对象是由截然不同的起源（例如代数，分析或几何形状）产生的，其共同点比以前理解的更多。他猜想，存在一个完全完整的数学世界，即动机类别，可以统一并以一种普遍的方式统一和编码数学中的所有深刻思想。目前仍处于猜想的形式的这个动机世界一直是20世纪和21世纪领先的数学思想的关注源。一个案例是由2004年Fields Medalist V. Voevodsky发明的所需属性的动机的候选版本。监管机构是将动机的猜想世界的认识为更``地球''所定义的类别的。因此，从某种意义上说，监管机构为我们提供了我们试图展示的东西的快照！当然，当我们构建监管机构时，我们会考虑一个候选动机类别。问题在于，除了Grothendieck的原始具体提议之外（这需要重大猜想才能脱颖而出），没有其他拥有预期属性的候选人本身很容易描述，这正是监管机构开始发挥作用的地方。尽管文献中对监管因子的定义非常抽象，但缺乏非常明确的描述是一个很大的障碍。在过去的十年中，我与各种合作者在第一篇论文中提供了描述（Kerr and M \“ Uller-Stach，Compositio Math 142）。第二篇论文（与Kerr，Kerr，Invention 170）相比，这是一项旅游局长，对囚犯的态度不适用，涉及一般的``'''''''''''''''''''''complate''物理学是明显的。在过去的十年中，该结果将出现在代数几何学杂志上。