[Paper Review] Local structure of singular hyperkahler quotients

Maxence Mayrand|arXiv (Cornell University)|Jul 16, 2018

Geometry and complex manifolds13 references7 citations

TL;DR

This paper establishes a hyperkähler analogue of the Sjamaar-Lerman stratification for singular hyperkähler quotients, proving they decompose into smooth symplectic strata and are locally biholomorphic to affine complex-symplectic GIT quotients, with holomorphic Poisson compatibility. The work extends the Marsden-Weinstein reduction framework to singular, hyperkähler settings using hyperkähler quotient constructions and complex-symplectic geometry.

ABSTRACT

When a compact Lie group acts freely and in a Hamiltonian way on a symplectic manifold, the Marsden-Weinstein theorem says that the reduced space is a smooth symplectic manifold. If we drop the freeness assumption, the reduced space might be singular, but Sjamaar-Lerman (1991) showed that it can still be partitioned into smooth symplectic manifolds which fit together nicely in the sense that they form a stratification. In this paper, we prove a hyperkahler analogue of this statement, using the hyperkahler quotient construction. We also show that singular hyperkahler quotients are complex spaces which are locally biholomorphic to affine complex-symplectic GIT quotients with biholomorphisms that are compatible with natural holomorphic Poisson brackets on both sides.

Motivation & Objective

• To extend the Sjamaar-Lerman stratification theorem from symplectic to hyperkähler geometry.
• To analyze the local structure of singular hyperkähler quotients under non-free group actions.
• To establish that these quotients are complex spaces locally isomorphic to affine complex-symplectic GIT quotients.
• To prove that the local biholomorphisms preserve natural holomorphic Poisson brackets.
• To provide a hyperkähler analogue of the Marsden-Weinstein reduction in the singular case.

Proposed method

• Use the hyperkähler quotient construction to define reduced spaces under Hamiltonian actions of compact Lie groups.
• Apply the stratification framework of Sjamaar-Lerman to the hyperkähler setting, showing strata are smooth symplectic manifolds.
• Analyze the local structure of singular hyperkähler quotients using complex-symplectic geometry.
• Establish local biholomorphisms between singular hyperkähler quotients and affine complex-symplectic GIT quotients.
• Verify that these biholomorphisms are compatible with the natural holomorphic Poisson brackets on both sides.
• Leverage the hyperkähler structure to relate the symplectic and complex structures in the quotient.

Experimental results

Research questions

• RQ1 How can the Sjamaar-Lerman stratification be generalized to the hyperkähler setting?
• RQ2 What is the local complex-analytic structure of a singular hyperkähler quotient?
• RQ3 Are singular hyperkähler quotients locally biholomorphic to affine complex-symplectic GIT quotients?
• RQ4 Do the local biholomorphisms between singular hyperkähler quotients and complex-symplectic GIT quotients preserve the holomorphic Poisson bracket?
• RQ5 What is the role of the hyperkähler quotient construction in resolving singularities of Hamiltonian quotients?

Key findings

• Singular hyperkähler quotients admit a stratification into smooth symplectic manifolds, analogous to the Sjamaar-Lerman theorem in the symplectic case.
• The singular hyperkähler quotient is a complex space, locally biholomorphic to an affine complex-symplectic GIT quotient.
• The local biholomorphisms between the quotient and the GIT quotient are compatible with the natural holomorphic Poisson brackets on both spaces.
• The hyperkähler quotient construction provides a natural framework for resolving singularities in Hamiltonian group actions.
• The local structure of the quotient is governed by complex-symplectic geometry, with the hyperkähler structure ensuring compatibility across complex and symplectic components.
• The results establish a hyperkähler analogue of the Marsden-Weinstein theorem in the singular case, extending the scope of symplectic reduction to non-free actions.