Cohomology of complements of toric arrangements associated with root systems

Olof Bergvall

doi:10.1007/s40687-021-00305-z

Abstract

We develop an algorithm for computing the cohomology of complements of toric arrangements. In the case a finite group Gamma is acting on the arrangement, the algorithm determines the cohomology groups as representations of Gamma . As an important application, we determine the cohomology groups of the complements of the toric arrangements associated with root systems of exceptional type as representations of the corresponding Weyl groups.

Highlights

An arrangement is a finite set of closed subvarieties of a variety
Arrangements are of interest to a wide range of areas of mathematics such as algebraic geometry, topology, combinatorics, Lie theory and singularity theory
We show that the cohomology of the complement of a toric arrangement can be computed using only the arithmetic of Zmodules

Summary

Introduction

An arrangement is a finite set of closed subvarieties of a variety. Despite their simple definition, arrangements are of interest to a wide range of areas of mathematics such as algebraic geometry, topology, combinatorics, Lie theory and singularity theory. The results of the present paper have been essential ingredients in determining the cohomology of many of the moduli spaces mentioned above (others, in particular the case of Del Pezzo surfaces of degree 1 is work in progress). We refer to Remark 4.2 for a more thorough comparison with their work but mention already that while the hyperplane case is entirely determined by the intersection poset, see Definition 2.2, we need to take “local” topological and arithmetic data into account in the toric setting. 4, we use these algorithms to compute the cohomology groups of the complements of the toric arrangements associated with exceptional root systems as representations of the corresponding Weyl groups. See Example 2.6 for an explicit arrangement together with its intersection poset and Möbius function

Group actions on arrangements

Group actions on toric arrangements

Findings

H2 H3 H4 H5 H6 H7 H8