Cohomology of the Mumford Quotient

الأصل · EN

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 the Witten deformation of the Dolbeault complex of L 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 for symplectic reduction are, in fact, equalities.