Projective structures on a Riemann surface. III
type: Item
instance of: scholarly article
title: Projective structures on a Riemann surface. III
zbMATH Open document ID: 1031.14015
DOI: 10.1016/S0926-2245(01)00061-4
author: R. Smith
published in: Differential Geometry and its Applications
publication date: 2002-08-14T00:00:00Z
review text: For part I and II see Int. Math. Res. Not. 1996, 753-768 (1996; Zbl 0888.14012) and ibid. 1999, 685-716 (1999; Zbl 0939.14014). A projective structure on a Riemann surface \(X\) is a covering of \(X\) by holomorphic coordinate charts whose transition functions are Möbius transformations. Let \(\Delta\) denote the diagonal divisor of \(X\times X\) and let \(p_i: X\times X\to X\), \(i=1,2\), be the projection onto the \(i\)-th factor. Let \(K_X\) be the holomorphic cotangnet bundle of \(X\), \({\mathcal M}=p_1^*K_X\otimes p^*_2 K_X\otimes {\mathcal O}_{X\times X} (2\Delta)\), and denote by \(\Delta_n\) the \(n\)-th order infinitesimal neighborhood of \(\Delta\) in \(X\times X\). The authors introduce the notion of generalized projective structure on \(X\) as the choice of an extension of the trivialization of the line bundle \({\mathcal M}\) on \(\Delta_1\) to a trivialization on \(\Delta_n\) for \(n\geq 2\), and they provide several canonically equivalent descriptions of the space \({\mathcal S}_n(X)\) of generalized projective structures on \(X\). Moreover, they construct a canonical involution on \({\mathcal S}_n(X)\), and interpret it in the isomorphic spaces introduced above. The paper under review represents a rather interesting progress of the previous work (part I and II, loc. cit.) by both authors.
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 14H30
zbMATH DE Number: 1780854
zbMATH Keywords: covering of Riemann surface
zbMATH Keywords: generalized projective structures
Wikidata QID: Q115459051
reviewed by: Jose Manuel Gamboa
MaRDI profile type: Publication
cites work: Complex Analytic Connections in Fibre Bundles
cites work: Differential operators on complex manifolds with a flat projective structure
cites work: Invariants for a class of equivariant immersions of the universal cover of a compact Riemann surface into a projective space
cites work: Projective structures on a Riemann surface
cites work: Q4719524
cites work: Equations différentielles à points singuliers réguliers
cites work: Q5565313
cites work: Q5843703
cites work: Curves in Grassmannians
cites work: Q4154999
full work available at URL: s0926-2245(01)00061-4
OpenAlex ID: W2795644542
description: scientific article; zbMATH DE number 1780854