  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.
