Counterexamples to hyperkahler Kirwan surjectivity

Abstract

Suppose that M is a complete hyperkahler manifold with a compact Lie group K acting via hyperkahler isometries and with hyperkahler moment map $({\mu }_{\mathbb{C}},{\mu }_{\mathbb{R}}):M\to {\mathfrak{k}}^{\ast }\otimes \mathrm{Im}(\mathbb{H})$. It is a long-standing problem to determine when the hyperkahler Kirwan map $H_K^{\ast }(M,\mathbb{Q})⟶H^{\ast }(M//K,\mathbb{Q})$ is surjective. We show that for each $n\ge 2$, the natural $U(n)$-action on $M=T^{\ast }(S{L}_n\times {\mathbb{C}}^n)$ 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.

References in corpus (1)

• A hyperkahler structure on the cotangent bundle of a complex Lie group

Cited by in corpus (1)

• Bow varieties---geometry, combinatorics, characteristic classes