The monodromy representations of projective structures
type: Item
instance of: scholarly article
title: The monodromy representations of projective structures
zbMATH Open document ID: 0656.32006
DOI: 10.1007/BF01194420
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: Let M be a complex manifold of complex dimension n, identified with \(\tilde M/\pi_ 1(M)\), where \(\tilde M\) is the universal covering of M. Thena (flat) projective structure on M is a locally biholomorphic map \(f: \tilde M\to P^ n({\mathbb{C}})\) for which there is a homomorphism \(h: \pi_ 1(M)\to PGL(n+1,{\mathbb{C}})\) such that \(f\circ \gamma =h(\gamma)\circ f\) for all \(\gamma\) in \(\pi_ 1(M)\). Here, h (if it exists) is uniquely determined by f, and is called the monodromy representation of the projective structure. In this note, it is shown that the obstruction to lifting the monodromy representation to one of \(\pi_ 1(M)\) into \(GL(n+1,{\mathbb{C}})\) is the same for all projective structures (if any) on M, namely the image of the first Chern class of M in \(H^ 2(M,{\mathbb{Z}}/(n+1))\). Corollary: this obstruction vanishes in the case of a Riemann surface (a classical but often-proved result). The proof involves only linear-algebraic considerations on projective space.
Mathematics Subject Classification ID: 32Q99
zbMATH DE Number: 4071325
zbMATH Keywords: monodromy representation
zbMATH Keywords: projective structure
author: R. R. Simha
reviewed by: R. R. Simha
MaRDI profile type: Publication
cites work: Special coordinate coverings of Riemann surfaces
cites work: Q5825483
description: scientific article; zbMATH DE number 4071325