Differential operators on a Riemann surface with projective structure
type: Item
instance of: scholarly article
title: Differential operators on a Riemann surface with projective structure
zbMATH Open document ID: 1085.14030
author: Indranil Biswas
published in: Journal of Geometry and Physics
publication date: 2005-06-27T00:00:00Z
review text: The author considers arbitrary Riemann surfaces (not necessarily compact or of finite type) equipped with a projective structure and a theta characteristic, i.e. a line bundle which is a square root of the canonical bundle. The complement of the zero section in the total space of the line bundle has a natural symplectic structure, and using the projective structure, this symplectic structure has a canonical quantization by a star product (i.e. a deformation quantization). Using the quantization (via the symbol map) a canonical isomorphism between the space of holomorphic differential operators on certain tensor powers of the theta characteristics \(L\) and the space of holomorphic sections over the Riemann surface \(X\) of certain tensor powers of \(L\) and the canonical bundle \(\Omega\) is constructed. In detail: \[ H^0(X, \text{Diff}_X^k(L^{\otimes j},L^{\otimes (i+j+2k)})) \cong \bigoplus_{l=0}^{k} H^0(X, L^{\otimes i} \otimes \Omega_X^{\otimes l}), \] for \(i,j\in\mathbb Z, n\geq 0\) and \(i\notin [-2(k-1),0]\).
reviewed by: M. Schlichenmaier
Mathematics Subject Classification ID: 14H60
Mathematics Subject Classification ID: 53B10
Mathematics Subject Classification ID: 53D55
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 81S10
zbMATH DE Number: 2183099
zbMATH Keywords: differential operator
zbMATH Keywords: projective structure
zbMATH Keywords: quantization
Wikidata QID: Q115353545
MaRDI profile type: Publication
full work available at URL: j.geomphys.2003.09.002
OpenAlex ID: W2010252467
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DOI: 10.1016/J.GEOMPHYS.2003.09.002
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description: scientific article; zbMATH DE number 2183099