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Cohomologie analytique des arrangements d'hyperplans

Damien Junger

Vol. 17 (2023), No. 1, 1–43

Nous étudions la cohomologie de faisceaux analytiques sur le complémentaire, dans l’espace projectif, d’une collection infinie d’hyperplans bien choisie, comme l’espace symétrique de Drinfeld. En particulier, le faisceau de fonctions inversibles sur ces espaces rigides n’a pas de cohomologie de degré supérieur ou égal à 1. Ceci démontre l’annulation du groupe de Picard, et les méthodes utilisées nous donnent une description pratique des fonctions inversibles globales.

We study the cohomology of some analytic sheaves on the complement in the projective space of a suitable infinite collection of hyperplanes like the Drinfeld symmetric 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 group and the methods used give a convenient description of the global invertible functions.

rigid analytic varieties, analytic cohomology, Drinfeld symmetric spaces
Mathematical Subject Classification
Primary: 32C35, 32P05
Received: 11 November 2020
Revised: 17 January 2022
Accepted: 17 March 2022
Published: 18 February 2023
Damien Junger
Mathematisches Institut / Mathematics Münster
Fachbereich Mathematik und Informatik
Universität Münster