Geometric quantization of symplectic manifolds with respect to reducible non-negative polarizations
type: Item
instance of: scholarly article
title: Geometric quantization of symplectic manifolds with respect to reducible non-negative polarizations
zbMATH Open document ID: 0876.58016
DOI: 10.1007/BF02506413
author: J. E. Andersen
published in: Communications in Mathematical Physics
publication date: 1997-11-25T00:00:00Z
review text: Geometric quantization requires to construct a (``prequantum'') line bundle \(L\) over a symplectic manifold \((M, \omega)\) and to choose a polarization \(P\) on \(M\). An important detail is to study the cohomology groups of the leafwise complex with respect to \(P\) with values in \(L\). If \(P\) is a Kähler polarization, index and Lefschetz theorems are applicable. The author initiates a program to present similar theorems for more general polarizations. To that end he considers polarizations that satisfy certain non-negativity and reducibility conditions. For compact symplectic manifolds he can then show that the above-mentioned complex has finitedimensional cohomology groups. Moreover, a Lefschetz theorem and a fixed point theorem can be proven for this case.
Mathematics Subject Classification ID: 53D50
zbMATH DE Number: 1016974
zbMATH Keywords: reducible non-negative polarizations
zbMATH Keywords: geometric quantization
zbMATH Keywords: non-negativity
zbMATH Keywords: reducibility
zbMATH Keywords: compact symplectic manifolds
zbMATH Keywords: Lefschetz theorem
zbMATH Keywords: fixed point theorem
reviewed by: Volker Perlick
MaRDI profile type: Publication
cites work: Q4195061
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description: scientific article; zbMATH DE number 1016974