The quantization conjecture revisited
A strong version of the quantization conjecture of \textit{V. Guillemin} and \textit{S. Sternberg} [Invent. Math. 67, 515-538 (1982; Zbl 0503.58018)] 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.
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