Kirwan Surjectivity for the Equivariant Dolbeault cohomology

TL;DR

This paper proves a surjective Kirwan map from equivariant Dolbeault cohomology of a Kähler manifold to the Dolbeault cohomology of its quotient, answering a question about cohomology types in Hamiltonian actions.

Contribution

It establishes a natural surjective Kirwan map for equivariant Dolbeault cohomology in the context of Hamiltonian Kähler actions, extending classical results.

Findings

Surjective Kirwan map from equivariant Dolbeault to quotient cohomology

Answer to Weitsman's question on cohomology types

Application of Cartan-Chern-Weil theory in this context

Abstract

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.