Fundamental and divisor class group: finiteness and interplay

• Applicant: Dr. Lukas Braun
• Subject Area: Mathematics
• Term: from 2020 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 452847893

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.

• DFG Programme: WBP Position