The quantization conjecture revisited

• 成果类型: Article
• 署名作者: Teleman, C
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/2661378
• 发表日期: 2000
• 页码: 1-43
• 关键词: generalized theta-functions; yang-mills equations; parabolic g-bundles; invariant-theory; vector-bundles; geometric-quantization; symplectic reduction; riemann surfaces; MODULI SPACES; picard group

摘要

A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety (X, L), the cohomologies of L over the GIT quotient X parallel toG equal the invariant part of the cohomologies over X. This generalizes the theorem of CGS] on global sections, and strengthens its subsequent extensions ([JK], [MI) to Riemann-Roch numbers. Remarkable by-products are the invariance of cohomology of vector bundles over X//G under a small change in the defining polarization or under shift desingularization, as well as a new proof of Boutot's theorem. Also studied are equivariant holomorphic forms and the equivariant Hodge-to-de Rham spectral sequences for X and its strata, whose collapse is shown. One application is a new proof of the Borel-Weil-Bott theorem of [T1] for the moduli stack of G-bundles over a curve, and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures. Collapse of the Hedge-to-de Rham sequences for these stacks is also shown.