AbstractWe obtain estimates on the character of the cohomology of anS1-equivariant holomorphic vector bundle over a Kähler manifoldMin terms of the cohomology of the Lerman symplectic cuts and the symplectic reduction ofM. In particular, we prove and extend inequalities conjectured by Wu and Zhang.The proof is based on constructing a flat family of complex spacesMt(t∈ ℂ) such thatMtis isomorphic toMfort≠ 0, whileM0is a singular reducible complex space, whose irreducible components are the Lerman symplectic cuts.