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Annals of Mathematics, II. Series Vol. 152, No. 1, pp. 143 (2000) 

The quantization conjecture revisitedConstantin TelemanReview from Zentralblatt MATH: A strong version of the quantization conjecture of Guillemin and Sternberg (GS) is proved. One considers the linear action of a reductive group G on a projectively embedded complex manifold X and its associated Ginvariant 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 GS theorem on global sections and shows its extensions to RiemannRoch 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 Hodgetode Rham spectral sequences for X and its strata are also studied and their collapse is proven. A new proof of the BorelWeilBott theorem for the moduli stack of Gbundles 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 Keywords: geometric quantization; reductive group; polarized variety; global section; cohomology of vector bundles; polarization change; moduli stack; parabolic structure; spectral sequence collapse Classification (MSC2000): 37XX 5399 Full text of the article:
Electronic fulltext finalized on: 8 Sep 2001. This page was last modified: 22 Jan 2002.
© 2001 Johns Hopkins University Press
