Abstract not found

Version: 8/5/98 ABSTRACT: I prove the following strong version of the quantization conjecture of Guillemin and Sternberg: for a reductive group action on a smooth, compact, polarized variety ( X, ℓ), the cohomologies of ℓ over the GIT quotient are equal to the invariant parts of the cohomologies over X. This generalizes the theorem of [GS], which concerned the spaces of global sections, and strengthens its subsequent extensions ([M], [V]) to Riemann-Roch numbers. A remarkable by-product is the rigidity of cohomology of certain vector bundles over the GIT quotient under a small change in the defining polarization. One application is a new proof of the Borel-Weil-Bott theorem of [T] for the moduli stack M of G-bundles over a curve, and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures. Also studied are equivariant holomorphic forms and the equivariant Hodge-to-de Rham spectral sequences for X and its strata, whose collapse at E1 is shown. Collapse of the Hodge-tode Rham sequence for M is hence deduced.

These notes survey the relation between the moduli spaces of vector bundles on a curve C and the spaces of (classical) theta functions on the Jacobian J of C. The connection appears when one tries to describe the moduli space Mr of rank r

Using a global version of the equivariant Chern character, we describe an effective method for computing the complexified twisted equivariant K-theory of a space

Using a global version of the equivariant Chern character, we describe an effective method for computing the complexified twisted equivariant K-theory of a space

ABSTRACT. We determine the splitting type of the Verlinde vector bundles in higher genus in terms of simple semihomogeneous factors. In agreement with strange duality, the simple factors are interchanged by the Fourier-Mukai transform, and their spaces of sections are naturally dual.