The quantization conjecture revisited
type: Item
instance of: scholarly article
title: The quantization conjecture revisited
zbMATH Open document ID: 0980.53102
DOI: 10.2307/2661378
author: Constantin Teleman
published in: Annals of Mathematics. Second Series
publication date: 2001-05-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9808029
full work available at URL: https://eudml.org/doc/121618
full work available at URL: http://citeseerx.ist.psu.edu/viewdoc/summary
full work available at URL: 152 1.html
review text: A strong version of the quantization conjecture of \textit{V. Guillemin} and \textit{S. Sternberg} [Invent. Math. 67, 515-538 (1982; Zbl 0503.58018)] is proved. One considers the linear action of a reductive group \(G\) on a projectively embedded complex manifold \(X\) and its associated \(G\)-invariant stratification by locally closed, smooth subvarieties. It is shown that, for a reductive action of \(G\) on a smooth, compact, polarized variety \((X,L)\), the cohomologies of \(L\) over the quotient \(X//G\) (in geometric invariant theory) equal the invariant part of the cohomologies over \(X\). This result generalizes the G-S theorem on global sections and shows its extensions to Riemann-Roch numbers. The invariance of cohomology of vector bundles over \(X//G\) under a small change in the defining polarization or under desingularization, as well as a new proof of Boutot's theorem, are obtained as important consequences. The equivariant Hodge-to-de Rham spectral sequences for \(X\) and its strata are also studied and their collapse is proven. A new proof of the Borel-Weil-Bott theorem for the moduli stack of \(G\)-bundles over a curve is given as an application. Analogous results are obtained for the moduli stacks and spaces of bundles with parabolic structures.
reviewed by: Gheorghe Zet
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 81S10
zbMATH DE Number: 1545109
zbMATH Keywords: geometric quantization
zbMATH Keywords: reductive group
zbMATH Keywords: polarized variety
zbMATH Keywords: global section
zbMATH Keywords: cohomology of vector bundles
zbMATH Keywords: polarization change
zbMATH Keywords: moduli stack
zbMATH Keywords: parabolic structure
zbMATH Keywords: spectral sequence collapse
MaRDI profile type: Publication
Wikidata QID: Q123206171
OpenAlex ID: W2160494180
arXiv ID: math/9808029
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description: scientific article