Hodge Type Realizations of Algebraic Cycles

我的研究的一个简短的标题描述是动机的监管者。动机的世界是由a·格罗滕迪克（a . Grothendieck）发明的，他是菲尔兹奖得主，可以说是20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，产生于非常不同的起源（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，叫做动机范畴，它将以某种普遍的方式统一和编码数学中所有深奥的思想。这个动机的世界，目前仍以推测的形式存在，一直是20世纪和21世纪顶尖数学头脑关注的一个来源。一个恰当的例子是2004年菲尔兹奖得主v·沃沃茨基（V. Voevodsky）发明的具有“大多数”期望属性的动机候选版本。监管机构是对动机的臆测世界的实现，这些动机被定义为更“世俗”的类别。所以从某种意义上说，监管机构给了我们一个快照，让我们看到了我们试图证明存在的东西！当然，当我们构建一个监管机构时，我们脑海中有一个候选动机类别。问题在于，除了格罗滕迪克最初提出的具体建议（这需要重大的推测才能启动）之外，没有其他具有预期属性的候选方案是很容易描述的，而这正是监管机构发挥作用的地方。尽管文献中有大量非常抽象的监管机构定义，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，我自己和不同的合作者在第一篇论文中提供了这样的描述

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.