Stratification of singular hyperkahler quotients

TL;DR

This paper demonstrates that singular hyperkahler quotients can be partitioned into smooth hyperkahler manifolds, with a detailed topological and Poisson structure analysis, extending classical symplectic reduction results.

Contribution

It establishes that hyperkahler quotients are topologically stratified and introduces a local model linking them to linear complex-symplectic reductions, generalizing Sjamaar-Lerman's theorems.

Findings

Hyperkahler quotients are topologically stratified into smooth manifolds.

Global Poisson structures induce hyperkahler structures on strata.

Local models show quotients are locally isomorphic to linear complex-symplectic reductions.

Abstract

Hyperkahler quotients by non-free actions are typically highly singular, but are remarkably still partitioned into smooth hyperkahler manifolds. We show that these partitions are topological stratifications, in a strong sense. We also endow the quotients with global Poisson structures which induce the hyperkahler structures on the strata. Finally, we give a local model which shows that these quotients are locally isomorphic to linear complex-symplectic reductions in the GIT sense. These results can be thought of as the hyperkahler analogues of Sjamaar-Lerman's theorems for symplectic reduction. They are based on a local normal form for the underlying complex-Hamiltonian manifold, which may be of independent interest.