  Consider a holomorphic torus action on a possibly non-compact K\"ahler
manifold. We show that the higher cohomology groups appearing in the geometric
quantization of the symplectic quotient are isomorphic to the invariant parts
of the corresponding cohomology groups of the original manifold. For
non-Abelian group actions on compact K\"ahler manifolds, this result was proved
recently by Teleman and by Braverman. Our approach is applying the holomorphic
instanton complex to the prequantum line bundles over the symplectic cuts. We
also settle a conjecture of Zhang and the present author on the exact sequence
of higher cohomology groups in the context of symplectic cutting.
