Counterexamples to hyperkahler Kirwan surjectivity

TL;DR

This paper constructs counterexamples showing that the hyperkahler Kirwan map is not always surjective, using specific hyperkahler quotients involving cotangent bundles of groups and representations.

Contribution

It provides the first known counterexamples to hyperkahler Kirwan surjectivity for certain group actions, expanding understanding of hyperkahler quotient topology.

Findings

Counterexamples for $U(n)$-actions on specific hyperkahler manifolds

Failure of hyperkahler Kirwan map surjectivity in these cases

Establishment of a 'Kahler = GIT quotient' correspondence for certain cotangent bundles

Abstract

Suppose that M is a complete hyperkahler manifold with a compact Lie group K acting via hyperkahler isometries and with hyperkahler moment map $(\mu_{\mathbb{C}}, \mu_{\mathbb{R}}): M\rightarrow \mathfrak{k}^*\otimes\operatorname{Im}(\mathbb{H})$. It is a long-standing problem to determine when the hyperkahler Kirwan map $H^*_K(M,\mathbb{Q}) \longrightarrow H^*(M//K, \mathbb{Q})$ is surjective. We show that for each $n\geq 2$, the natural $U(n)$-action on $M = T^*(SL_n\times\mathbb{C}^n)$ admits a hyperkahler quotient for which the hyperkahler Kirwan map fails to be surjective. As a tool, we establish a ``Kahler $=$ GIT quotient'' assertion for products of cotangent bundles of reductive groups, equipped with the Kronheimer metric, and representations.