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.

Cohomogeneity-One Lagrangian Mean Curvature Flow

“…The following argument is mostly taken from [39] and [33], where more details may be found. We consider the set M H := {w ∈ M : Stab(w) = H}, which may be shown to be a Kähler submanifold of M .…”