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组的解。这样的系数的例子是复数、有理数或整数。这些解的集合被赋予一定的结构，称为变种。为了研究一个簇，人们可以研究它的子空间，即子簇，并将它们分类为等价关系。在Chow群的理论中，一个经典的不变量理论，人们将给定簇的子空间分类为有理等价。一个簇的两个子空间是有理等价的，如果其中一个可以变形为另一个。Chow群的理论不仅提供了几何的各种研究的信息，而且还提供了用于定义它的系数的信息。使用这种理论方法来找出更多关于不同系数的信息，例如有理数，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圈的Chow群，即。e.因为它的可计算性，它指向变形。许多关于零圈的结果可以化为曲线，Chow群可以推广到更高Chow群。这些更高的Chow群是所谓的动机上同调的模型，动机上同调是不变量的普遍理论。高阶Chow群作为经典理论的一种改进是很有趣的，但也常常使得在经典情况下证明新的结果成为可能。高周群的一部分由高零圈给出。在Milnor K-理论中，这些群也被称为具有系数的零圈Chow群。就像经典的Chow群的零圈，他们可以在许多情况下计算，但没有被深入研究。此外，它们还与其他重要的理论，如高等K理论，类场理论和加藤定理有关。在我们的项目中，我们想研究不同系数的零圈的高阶Chow群，如p-adic，有理数和复数，并探索与其他理论的关系。我们还想推广一些结果是已知的光滑品种的情况下，奇点出现。为了做到这一点，我们开发的变形技术更高的周期，分解的对角线更高的周组的复数，一个概括的莱文-魏贝尔周组和新的关系与加藤代数。在最后一种情况下，我们对p部分特别感兴趣。此外，我们还想研究局部类场论中的一些混合特征线现象。总之，该项目将大大扩大（动机）不变量的图片，我们希望这-结合我们打算在逗留期间学习的新技术-导致新的成果和结果。

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.