Stratification of singular hyperkähler quotients

Abstract

Hyperkähler quotients by non-free actions are typically singular, but are nevertheless partitioned into smooth hyperkähler manifolds. We show that these partitions are topological stratifications, in a strong sense. We also endow the quotients with global Poisson structures which recover the hyperkähler 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 hyperkähler analogues of Sjamaar–Lerman’s theorems for singular symplectic reduction. They are based on a local normal form for the underlying complex-Hamiltonian manifold, which may be of independent interest.

Keywords

• hyperkähler quotient
• moment map
• complex symplectic reduction
• geometric invariant theory
• stratified space
• local normal form
• 53c26
• 53d20
• 57n80
• 58a35
• 32m05