Hodge Type Realizations of Algebraic Cycles

我的研究有一个简短的标题描述：动机的调节器。动机的世界是由A·格罗森迪克发明的，他是菲尔兹奖获得者，可以说是20世纪最有影响力的数学家。Grothendieck观察到，许多不同的数学对象，来自非常不同的起源(例如，代数、分析或几何)，比以前所理解的有更多的共同点。他推测，有一个完全完整的数学世界本身，被称为动机范畴，它将统一和编码数学中的所有深层思想，并以某种普遍的方式。这个动机的世界，目前仍处于猜想的形式，已经成为20世纪和21世纪领先的数学思想家关注的来源。一个典型的例子是由2004年菲尔兹奖牌得主沃沃茨基发明的动机的候选版本，该版本拥有“大多数”想要的属性。监管者是对动机的猜想世界的实现，将其划分为更“世俗”定义的类别。因此，在某种意义上，监管机构给了我们一个快照，让我们看到我们试图展示的东西的存在！当然，当我们构建一个监管机构时，我们心中有一个候选的动机类别。问题是，除了Grothendieck最初的具体提议(需要进行重大猜测)之外，没有其他拥有预期房产的候选人本身就很容易描述，而这正是监管机构发挥作用的地方。尽管文献中对监管机构有很多非常抽象的定义，但缺乏非常明确的描述是一大障碍。在过去的10年里，我与不同的合作者一起，在第一篇论文(与科尔和M\“Uller-Stach，Compostio Math 142)中提供了这种描述。第二篇论文(与科尔，Invenones 170)涉及更多内容，是对监管机构的一种环球式的方法，涉及到非常广泛的”混合动机“领域，在这些领域，数论和物理的应用是显而易见的。这一方向的下一篇论文目前正在进行中，并使用了更多的合作者，将明确而果断地描述使用Bloch简单的Higher Chow小组以及使用Voevodsky机制的监管机构。最后，克尔和刘易斯修改了一个简单版本的布洛赫调节器，在这个过程中，他们发现贡查洛夫提供的简单真实调节器中存在一个严重错误，该调节器在过去十年中得到了广泛使用。这一结果将发表在《代数几何》杂志上。所有这些工作的推论有望在未来许多年里影响新一代代数几何学家。我目前关注的是贝林森-霍奇猜想及其与布洛赫-加藤定理的联系，以及关于高度配对的工作。

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 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.