Abstract
A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety $(X,\mathcal{L})$, the cohomologies of $\mathcal{L}$ over the GIT quotient $X//G$ equal the invariant part of the cohomologies over $X$. This generalizes the theorem of [GS] on global sections, and strengthens its subsequent extensions ([JK], [M]) 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 equivarient 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 Hodge-to-de Rham sequences for these stacks is also shown.
