Desingularisation of orbifolds obtained from symplectic reduction at generic coadjoint orbits

TL;DR

This paper presents a method to resolve singularities in symplectic orbifolds derived from symplectic reduction at generic coadjoint orbits, enabling desingularisation and smoothing of these geometric structures.

Contribution

It introduces a construction for resolving symplectic orbifolds from presymplectic quotients, extending to desingularise generic symplectic quotients and smoothing symplectic cuts.

Findings

Resolution of symplectic orbifolds via torus actions

Desingularisation of generic symplectic quotients

Modification of symplectic cuts to smooth manifolds

Abstract

We show how to construct a resolution of symplectic orbifolds obtained as quotients of presymplectic manifolds with a torus action. As a corollary, this allows us to desingularise generic symplectic quotients. Given a manifold with a Hamiltonian action of a compact Lie group, symplectic reduction at a coadjoint orbit which is transverse to the moment map produces a symplectic orbifold. If moreover the points of this coadjoint orbit are regular elements of the Lie coalgebra, that is, their stabiliser is a maximal torus, the result for torus quotients may be applied to obtain a desingularisation of these symplectic orbifolds. Regular elements of the Lie coalgebra are generic in the sense that the singular strata have codimension at least three. Additionally, we show that even though the result of a symplectic cut is an orbifold, it can be modified in an arbitrarily small neighbourhood of the cut hypersurface to obtain a smooth symplectic manifold.