Kirwan surjectivity for the equivariant Dolbeault cohomology

Abstract

Consider the holomorphic Hamiltonian action of a compact Lie group K on a compact Kähler manifold M with a moment map Φ:M→k∗. Assume that 0 is a regular value of the moment map. Weitsman raised the question of what we can say about the cohomology of the Kähler quotient M0≔Φ−1(0)∕K if all the ordinary cohomology of M is of type (p,p).In this paper, using the Cartan–Chern–Weil theory we show that in the above context there is a natural surjective Kirwan map from an equivariant version of the Dolbeault cohomology of M onto the Dolbeault cohomology of the Kähler quotient M0. As an immediate consequence, this result provides an answer to the question posed by Weitsman.