Hodge Type Realizations of Algebraic Cycles

* 我的研究的一个简短的标题描述是动机的调节器。 动机的世界是由A.格罗滕迪克是菲尔兹奖得主，可以说是世纪最有影响力的数学家。格罗滕迪克观察到，许多不同的数学对象，从非常不同的起源（例如代数，分析或几何）产生了比以前理解的更多的共同点。他断言，有一个完全完整的数学世界本身，所谓的类别的动机，将统一和编码的所有深刻的想法在数学，并在某种程度上普遍的方式。这个动机的世界，目前仍处于理论的形式，一直是20世纪和21世纪数学界的主要思想家们关注的问题。一个很好的例子是2004年菲尔兹奖获得者V. Voevodsky发明的一个候选版本的动机，它拥有“大多数”期望的属性。调节器是将动机的自然世界划分为更“世俗”定义的类别的实现。因此，从某种意义上说，监管机构给了我们一个快照，我们试图表明存在的东西！当然，当我们构建一个调节器时，我们心中有一个候选动机类别。问题是，除了格罗滕迪克最初的具体提议（需要重大的假设才能启动），没有其他拥有预期属性的候选人是很容易描述的，而这正是监管机构发挥作用的地方。尽管文献中有很多关于监管者的非常抽象的定义，但缺乏非常明确的描述是一个很大的障碍。在过去的10年里，我与各种合作者一起在第一篇论文中提供了这种描述（与Kerr和M\“uller-Stach，Compositio Math 142）。第二篇论文（与Kerr合著，Inventiones 170）涉及的内容要多得多，是一种对监管者的“不抓囚犯”的方法，涉及非常普遍的"混合动机“竞技场，在那里数论和物理学的应用是显而易见的。下一个文件在这个方向上，目前正在进行中，并使用更大的演员的合作者，将是一个明确的和决定性的描述监管机构使用布洛赫的单纯高周组以及使用Voevodsky机器。最后，Kerr和刘易斯重新设计了Bloch调节器的一个简单版本，并在此过程中发现了Goncharov提供的简单真实的 * 调节器中的一个严重错误，该调节器在过去十年中被广泛使用。结果将发表在代数几何杂志上。*所有这些工作的推论将有望在未来许多年内影响新一代的代数几何学家。我目前的主要工作是在贝林森-霍奇猜想的方向，以及它与 * 布洛赫-加藤定理的联系，以及高度配对的工作。

***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 real***regulator 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 the***Bloch-Kato theorem, as well as work on height pairings.