Morse theory with the norm-square of a hyperkahler moment map

TL;DR

This paper establishes a gradient inequality for the norm-square of hyperkahler moment maps, enabling Morse-theoretic computations of topological invariants of toric hyperkahler orbifolds without the properness assumption.

Contribution

It extends gradient inequalities to hyperkahler moment maps and applies Morse theory to compute topological invariants of toric hyperkahler orbifolds.

Findings

Bounded the gradient flow of hyperkahler moment maps

Computed Betti numbers of toric hyperkahler orbifolds

Determined cohomology rings of these orbifolds

Abstract

We prove that the norm-square of a moment map associated to a linear action of a compact group on an affine variety satisfies a certain gradient inequality. This allows us to bound the gradient flow, even if we do not assume that the moment map is proper. We describe how this inequality can be extended to hyperkahler moment maps in some cases, and use Morse theory with the norm-squares of hyperkahler moment maps to compute the Betti numbers and cohomology rings of all toric hyperkahler orbifolds.