Fundamental and divisor class group: finiteness and interplay

该项目的目的是调查两个不变量的代数簇。这些不变量是基本的和除数类群。他们将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。我们已经知道这些对象都有一个生成的除子群和考克斯环。我们还可以证明，对于这样的X，考克斯环构造的迭代是有限的。这个结构将发挥核心作用，以连接两个不变量。在最近的工作中，我们证明了有限的基本组的光滑轨迹的对象考虑，特别是证实了一个猜想的Kollár。从这些发现和由此产生的“代数性”的建设，本项目的目的是连接两个不变量。特别是，我们的目标是推广的概念，考克斯环和后果的本地和orbifold设置。这也将阐明局部有限性的考克斯层。第一部分将为我们的研究提供一个统一的背景，第二部分的目标是将基本类群和除数类群联系起来。目标是在X上构造一个G-（拟）扭子Z，使得Z是阶乘的，且X的正则轨迹的原像是单连通的。此外，纤维G应包括X的基本类群和约数类群。对于具有一余维环面作用的变种，我们的目标是显式构造这个准扭量，而一般来说，我们希望找到迭代次数的界限步骤和纤维G的维度。在该项目的第三部分和最后一部分，我们的目标是将我们的结果推广到更高的同伦和Chow群。特别地，我们期望除数类群的无限部分和第二同伦群之间有直接的关系。

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.