Cohomologie analytique des arrangements d'hyperplans

TL;DR

This paper investigates the cohomology of analytic sheaves on certain hyperplane arrangements in projective space, revealing vanishing results and characterizing invertible functions on these rigid spaces.

Contribution

It provides new results on the cohomology of sheaves on hyperplane arrangements in projective space, especially for Drinfel'd symmetric spaces, and describes invertible functions explicitly.

Findings

Sheaf of invertible functions has no higher cohomology on these spaces.

Vanishing of the Picard group for these hyperplane arrangements.

Provides a description of global invertible functions.

Abstract

In this article, we study the cohomology of some analytic sheaves on the complementary in the projective space of a suitable infinite collection of hyperplane like the Drinfel'd symetric space. In particular, the sheaf of invertible functions on these rigid spaces has no cohomology in degree greater or equal to $1$. This proves the vanishing of the Picard goup and the methods used give a convenient description of the global invertible functions.