Surjectivity of the hyperkähler Kirwan map

We study a class of group actions on hyperkähler manifolds which we call actions of linear type. If $M$ is a hyperkähler manifold possessing such a $G$-action, the hyperkähler 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 hyperkähler 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.

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