Surjectivity of the hyperk\"ahler Kirwan map

TL;DR

This paper investigates the surjectivity of the hyperk"ahler Kirwan map for certain group actions on hyperk"ahler manifolds, establishing conditions under which it is surjective and relating it to the classical Kirwan map.

Contribution

It provides criteria for the surjectivity of the hyperk"ahler Kirwan map and relates it to the ordinary Kirwan map, especially for actions of linear type.

Findings

The hyperk"ahler Kirwan map is surjective except possibly in middle degree.

The restriction map on cohomology is an isomorphism below middle degree.

The kernel of the hyperk"ahler Kirwan map can be derived from the kernel of the ordinary 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.