Hodge Realizations of Motivic Cohomology

* 动机上同调的Hodge实现 * 让我们暂时想象一下，你正在进行一次星星旅行探险 去发现太阳系不同行星上的文明 你首先在一个星球上遇到一个文明， 暗示，以及你随后遇到的文明， 其他行星表明，有一种深刻的 “似曾相识”的感觉。在这一点上， 想象一下，所有这些行星文明仅仅是“化身” 一个“太阳系的先进宇宙文明”，你可以 既看不见也摸不着，但你本能地知道存在。在数学中， 这就是当前的代数循环和动机上同调世界。 太阳系先进的宇宙文明的类比是 动机上同调（一种普遍的上同调理论），这是一个反射 所有的上同调理论（类似于行星文明） 共同点所有已知的动机上同调的候选者都使用以下数据： 代数循环作为积木。调节器（或实现）是类似的 一个特定的行星文明更准确地说， 一种实现是从动机上同调到更“通俗”的上同调理论的映射。 霍奇实现只是实现的一个特殊类别。 我们理解动机上同调复杂性的唯一途径是通过它的实现。

***Hodge Realizations of Motivic Cohomology*** Let us imagine for the moment that you are on a star trek expedition to discover civilizations on different planets in our solar system. You first encounter a civilization on one planet that has ``earthly'' overtones, and as your subsequent encounters of civilizations on other planets indicate, there is a sort of profound ``deja vu'' feeling about your discoveries. One might at this point, imagine that all such planetary civilizations are mere ``incarnations'' of an ``advanced universal civilization of the solar system'', which you can neither see nor touch, but you instinctively know exists. In mathematics, this is the current world of algebraic cycles and motivic cohomology. The analog of the advanced universal civilization of the solar system is precisely motivic cohomology (a universal cohomology theory), which is a reflection of what all cohomology theories (analogous to planetary civilizations) have in common. All known candidates of motivic cohomology use the datum of algebraic cycles as building blocks. A regulator (or realization) is analogous to one particular planetary civilization. In more precise terms, a realization is a map from motivic cohomology to a more ``earthly'' cohomology theory. Hodge realizations are simply a particular class of realizations. Our only way to understand the complexity of motivic cohomology is via its realizations.