Surjectivity of the hyperk\"ahler Kirwan map

Abstract

We study a class of group actions on hyperk\"ahler manifolds which we call actions of linear type. If $M$ is a hyperk\"ahler manifold possessing such a $G$-action, the hyperk\"ahler Kirwan map is surjective if and only if the natural restriction $H^\ast(M / G) \to H^\ast(M / G)$ is surjective. We prove that this restriction is an isomorphism below middle degree and an injection in middle degree. As a consequence, the hyperk\"ahler Kirwan map is surjective except possibly in middle degree, and its kernel may be determined from the kernel of the ordinary Kirwan map. These results apply in particular to hypertoric varieties, hyperpolygon spaces, and Nakajima quiver varieties.