Cohomology of the Mumford quotient

Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use a Witten type deformation of the Dolbeault complex of L, introduced by Tian and Zhang, to show, that the cohomology of the sheaf of holomorphic sections of the induced bundle on the Mumford quotient of (X,L) is equal to the G-invariant part on the cohomology of the sheaf of holomorphic sections of L. This result, which was recently proven by C. Teleman by a completely different method, generalizes a theorem of Guillemin and Sternberg, which addressed the global sections. It also shows, that the Morse-type inequalities of Tian and Zhang [11] for symplectic reduction are, in fact, equalities. 1