  Consider the holomorphic Hamiltonian action of a compact Lie group $K$ on a
compact K\"ahler manifold $M$ with a moment map $\Phi: M\rightarrow
\mathfrak{k}^*$. 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 K\"ahler
quotient $M_0:=\Phi^{-1}(0)/K$ 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 K\"ahler
quotient $M_0$. As an immediate consequence, this result provides an answer to
the question posed by Weitsman.
