Symplectic actions on compact manifolds
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- by Sol Schwartzman
- Proc. Amer. Math. Soc. 143 (2015), 259-263
- DOI: https://doi.org/10.1090/S0002-9939-2014-12410-5
- Published electronically: August 25, 2014
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Abstract:
Let $M^{2n}$ be a compact connected Kähler manifold. We prove two theorems that in this case imply the following:
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If we have a symplectic action of a compact connected Lie group $G$ on $M^{2n}$ and there exists an orbit under the action of $G$ that is contained in a subset $A$ of $M^{2n}$ whose first Betti number equals zero, then our action is Hamiltonian.
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If we have a continuous symplectic flow on $M^{2n}$ that preserves some invariant metric, then exactly one of the following statements is true:
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Our flow is Hamiltonian.
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Our flow has a continuous eigenfunction which is not invariant under the flow.
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Bibliographic Information
- Sol Schwartzman
- Affiliation: Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881
- Email: solschwartzman@gmail.com
- Received by editor(s): November 13, 2012
- Received by editor(s) in revised form: March 26, 2013
- Published electronically: August 25, 2014
- Communicated by: Yingfei Yi
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 259-263
- MSC (2010): Primary 37Jxx, 37J10
- DOI: https://doi.org/10.1090/S0002-9939-2014-12410-5
- MathSciNet review: 3272751
Dedicated: Dedicated to the Memory of Lew Pakula