Fundamental and divisor class group: finiteness and interplay

该项目的目的是研究代数簇的两个不变量。这些不变量是基本的和除数类群。它们将被研究两个不同的代数几何对象X：弱Fano对和Kawamata对数终端奇点。已知这些对象有有限生成的除数类群和Cox环。在最近的工作中，我们证明了所考虑的对象的光滑轨迹的基本群是有限的，特别是证实了Kollár的一个猜想。从这些发现和由此得到的结构的‘代数性’出发，本项目的目的是将这两个不变量联系起来。项目的第一部分提供了必要的基础。特别地，我们的目的是将COX环的概念和结果推广到局部集和ORBORBOLD集上。这也将有助于揭示COX层的局部有限性。这一部分为我们的研究提供了一个统一的环境。在第二部分中，我们的目标是将基本类群和除数类群联系起来。我们的目的是在X上构造一个G-(拟)扭子Z，使得Z是阶乘的，且X的正则轨迹的原象是单连通的。对于具有一维环面作用的簇，我们的目标是显式地构造这个拟扭量，而一般而言，我们期望找到关于纤维G的迭代步数和维度的界。在项目的第三部分，也是最后一部分，我们的目的是将我们的结果推广到高次同伦和Chow群。特别地，我们期望除子类群的无限部分与第二同伦群之间存在直接关系。

The aim of the project is the investigation of two invariants of algebraic varieties. These invariants are the fundamental and the divisor class group. They will be investigated for two different algebro-geometric objects X: weakly Fano pairs and Kawamata log-terminal singularities. It is already known that these objects have finitely generated divisor class group and Cox ring. We could also prove that an iteration of the Cox ring construction is finite for such X. This construction will play a central role in order to link both invariants.In recent work, we proved finiteness of the fundamental group of the smooth loci of the objects in consideration, in particular confirming a conjecture of Kollár.Starting from these findings and the resulting 'algebraicity' of the constructions, the aim of the present project is to link both invariants.The first part of the project provides the necessary groundwork. In particular, we aim to generalize the concept of Cox rings and consequences to the local and the orbifold setting. This will shed light also on local finiteness of Cox sheaves. This preliminary part will provide us with a unified setting for our investigations.In the second part, our objective is to relate fundamental and divisor class group with each other. The goal is to construct a G-(quasi-)torsor Z over X, such that Z is factorial and the preimage of the regular locus of X is simply connected. Moreover, the fiber G shall comprise both the fundamental and the divisor class group of X.For varieties with a one-codimensional torus action, we aim to explicitly construct this quasi-torsor, while in general, we expect to find bounds on both the number of iteration steps and the dimension of the fiber G.In the third and last part of the project, we aim to generalize our results to higher homotopy and Chow groups. In particular, we expect a direct relation between the infinite part of the divisor class group and the second homotopy group.