Chow groups of zero- and higher zero-cycles

在代数几何中，人们研究具有给定系数的方程的解的集合。这种系数的例子有复数、有理数或整数。这些解决方案被赋予了一定的结构，被称为品种。为了研究语簇，人们可以研究它的子空间，即子语簇，并将它们分类为等价关系。在经典的不变量理论Chow群理论中，人们将给定簇的子空间分类到有理等价。簇的两个子空间是有理等价的，如果其中一个可以变形成另一个。周群论不仅提供了一个人所研究的品种的几何信息，还提供了用来定义该品种的系数的信息。用这种理论方法找出更多关于不同系数的信息，如有理数、p进数或整数，是算术几何的一个重要目标。特别令人感兴趣的是零循环的Chow群，即直到变形的点，因为它是可计算的。许多关于零循环的结果可以归结为曲线。Chow群可以推广到更高的Chow群。这些更高的Chow群是所谓的动机上同调的模型，这是一个普遍的不变量理论。高阶Chow群作为经典理论的提炼是很有趣的，但也常常使证明经典情形中的新结果成为可能。更高的Chow组的一部分是由更高的零周期给出的。在Milnor K-理论中，这些也被称为系数为零的Chow群。就像经典的周氏零周期群一样，它们在许多情况下可能会被计算出来，但还没有得到如此深入的研究。此外，它们还涉及到其他重要的理论，如高K理论、类场理论和加藤猜想。在我们的项目中，我们想要研究不同系数上的零循环的高阶Chow群，如p-进数、有理数和复数，并探索这些群与其他理论的关系。我们还想把一些已知的光滑簇的结果推广到出现奇点的情况。为此，我们发展了高圈的变形技巧，高阶Chow群的对角线在复数上的分解，Levine-Webel Chow群的推广，以及与Kato猜想的新关系。在最后一种情况下，我们对p部分特别感兴趣。此外，我们还想研究方案的局部类场理论中的一些混合特征现象。总而言之，该项目将大大扩大(动机)不变量的图景，我们希望这与我们打算在逗留期间学习的新技术相结合，导致新的猜想和结果。

In algebraic geometry one studies sets of solutions of equations with given coefficients. Examples of such coefficients are the complex numbers, the rational numbers or the integers. Endowed with a certain structure, these sets of solutions are called varieties. In order to study a variety one can study its subspaces, i.e. subvarieties, and classify them up to an equivalence relation. In the theory of Chow groups, a classical theory of invariants, one classifies subspaces of a given variety up to rational equivalence. Two subspaces of a variety are rationally equivalent if one can be deformed into the other. The theory of Chow groups does not just give information on the geometry of the variety one studies but also on the coefficients which are used to define it. Using this theoretical approach to find out more about different coefficients such as the rational numbers, the p-adic numbers or the integers is an important goal of arithmetic geometry. Of particular interest is the Chow group of zero-cycles, i. e. points up to deformation, because of its computability. Many results about zero-cycles may be reduced to curves.Chow groups may be generalised to higher Chow groups. These higher Chow groups are a model for so called motivic cohomology which is a universal theory of invariants. Higher Chow groups are interesting as a refinement of the classical theory but also often make it possible to prove new results in the classical case. One part of higher Chow groups is given by higher zero-cycles. These are also called Chow groups of zero-cycles with coefficients in Milnor K-theory. Just like classical Chow groups of zero-cycles they may in many cases be computed but have not been studied in such depth. Furthermore they are related to other important theories like higher K-theory, class field theory and Kato conjectures. In our project we would like to study higher Chow groups of zero-cycles over different coefficients such as the p-adic, rational and complex numbers and explore the mentioned relations with other theories. We would also like to extend some results which are known for smooth varieties to cases in which singularities appear. In order to do so we develop deformation techniques for higher cycles, a decomposition of the diagonal for higher Chow groups over the complex numbers, a generalisation of the Levine-Weibel Chow group and new relations with Kato conjectures. In the last case we are particularly interested in the p-part. Furthermore we would like to study some mixed-characteristic phenomena in the local class field theory of schemes. In sum the project would substantially enlarge the picture of (motivic) invariants and we hope that this - in combination with new techniques we intend to learn during the stay - leads to new conjectures and results.