The Morse–Bott–Kirwan condition is local

Tara Holm; Yael Karshon

doi:https://doi.org/10.1186/s40687-016-0066-9

The Morse–Bott–Kirwan condition is local

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Article number	25
Journal	Research in Mathematical Sciences
Volume	3
Issue number	1
DOIs	https://doi.org/10.1186/s40687-016-0066-9
State	Published - 1 Dec 2016
Externally published	Yes

Keywords

Hamiltonian group action
Kirwan surjectivity
Momentum map
Morse theory
Morse–Bott

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Mathematics (miscellaneous)
Computational Mathematics
Applied Mathematics

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Cite this

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title = "The Morse–Bott–Kirwan condition is local",

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{\textquoteright}s result that the norm square of a momentum map satisfies Kirwan{\textquoteright}s condition.",

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N2 - 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.

AB - 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.

KW - Hamiltonian group action

KW - Kirwan surjectivity

KW - Momentum map

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The Morse–Bott–Kirwan condition is local

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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