The Morse-Bott-Kirwan condition is local

TL;DR

This paper proves that the Kirwan condition, initially defined globally for smooth functions, can be verified locally, simplifying the application of Morse-Bott theory, with an application to momentum maps.

Contribution

It establishes that the Kirwan condition is local, enabling easier verification and application in Morse-Bott theory, including for momentum maps.

Findings

The Kirwan condition is equivalent locally and globally.

Local normal form theorem can be used to verify the condition.

Application to momentum maps confirms the condition holds for norm-square functions.

Abstract

Kirwan identified a condition on a smooth function under which the usual techniques of Morse-Bott theory can be applied to this function. We prove that if a function satisfies this condition locally then it also satisfies the condition globally. As an application, we use the local normal form theorem to recover Kirwan's result that the norm-square of a momentum map satisfies Kirwan's condition.