Fibrés vectoriels semi-stables sur une courbe de Mumford. (Semistable vector fibers on a Mumford curve)
type: Item
instance of: scholarly article
title: Fibrés vectoriels semi-stables sur une courbe de Mumford. (Semistable vector fibers on a Mumford curve)
zbMATH Open document ID: 0566.14008
DOI: 10.1007/BF01472132
author: M. van der Put
author: Marc Reversat
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164074
review text: Semi-stable vector bundles of degree zero on a compact Riemann surface of genus \(\geq 2\) correspond to unitary representations of the fundamental group. This theorem of M. Narasimhan and C. S. Seshadri has a rigid analytic version recently proved by G. Faltings. In this version the curve is a Mumford curve and the unitary representation is replaced by a \(\Phi\)-bounded representation of a Schottky group. In the present paper this theorem of Faltings is proved for arbitrary non-archimedean valued fields. New ideas and simplifications are introduced. In particular sheaves of normed vector spaces and their cohomology are introduced. The cohomology groups of a sheaf of normed vector spaces corresponding to a vector-bundle on the Mumford curve turn out to be free modules of finite rank over a certain non commutative ring. Using these cohomology groups one finds in an explicit way the r-dimensional \(\Phi\)-bounded representations which correspond to a given semi-stable vector bundle on the Mumford curve.
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 3902805
zbMATH Keywords: Mumford curve
zbMATH Keywords: Schottky group
zbMATH Keywords: non-archimedean valued fields
zbMATH Keywords: sheaves of normed vector spaces
zbMATH Keywords: semi-stable vector bundle
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3902805