The quantization conjecture revisited

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 // G equal the invariant part of the cohomologies over X. This generalizes the theorem of [Invent. Math. 67 (1982), 515-538] on global sections, and strengthens its subsequent extensions 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 [Invent. Math. 134 (1998), 1-57] 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 Hodge-to-de Rham sequences for these stacks is also shown.