2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/77443Let $X$ 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.A mistake in the proof of Theorem 3.1.b is corrected. The definition of the integration map is slightly changed. To appear in "Quantization of singular symplectic quotients"Symplectic GeometryAlgebraic GeometryDifferential Geometrytext