The quantization conjecture revisited

Constantin Teleman

Review from Zentralblatt MATH:

A strong version of the quantization conjecture of Guillemin and Sternberg (G-S) 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

Keywords: geometric quantization; reductive group; polarized variety; global section; cohomology of vector bundles; polarization change; moduli stack; parabolic structure; spectral sequence collapse

Classification (MSC2000): 37-XX 53-99

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