# Equivariant Morse theory for the norm-square of a moment map on a variety

## TL;DR

This paper extends Morse theory to singular spaces, specifically for the norm-square of a moment map on affine varieties, establishing homotopy equivalences that respect group actions.

## Contribution

It generalizes Morse theory to a class of singular spaces and verifies conditions for the norm-square of a moment map on affine varieties, including quiver representation spaces.

## Key findings

- Morse theory applies to certain singular spaces with specific conditions.
- Homotopy equivalence is equivariant under Hamiltonian group actions.
- Main theorem holds for the norm-square of a moment map on quiver representation spaces.

## Abstract

We show that the main theorem of Morse theory holds for a large class of functions on singular spaces. The function must satisfy certain conditions extending the usual requirements on a manifold that Condition C holds and the gradient flow around the critical sets is well-behaved, and the singular space must satisfy a local deformation retract condition. We then show that these conditions are satisfied when the function is the norm-square of a moment map on an affine variety, and that the homotopy equivalence from this theorem is equivariant with respect to the associated Hamiltonian group action. An important special case of these results is that the main theorem of Morse theory holds for the norm square of a moment map on the space of representations of a finite quiver with relations.