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 $\hat{I}¦:M\to {\mathfrak{k}}^{\ast }$. 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 $M_0:=\hat{I}{¦}^{-1}(0)/K$ 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 $M_0$. As an immediate consequence, this result provides an answer to the question posed by Weitsman.

17 pages, comments welcome!