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Symplectic actions on compact manifolds

Author: Sol Schwartzman
Journal: Proc. Amer. Math. Soc. 143 (2015), 259-263
MSC (2010): Primary 37Jxx, 37J10
Published electronically: August 25, 2014
MathSciNet review: 3272751
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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:

  1. 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.
  2. If we have a continuous symplectic flow on $ M^{2n}$ that preserves some invariant metric, then exactly one of the following statements is true:
    1. Our flow is Hamiltonian.
    2. Our flow has a continuous eigenfunction which is not invariant under the flow.

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Additional Information

Sol Schwartzman
Affiliation: Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881

Received by editor(s): November 13, 2012
Received by editor(s) in revised form: March 26, 2013
Published electronically: August 25, 2014
Dedicated: Dedicated to the Memory of Lew Pakula
Communicated by: Yingfei Yi
Article copyright: © Copyright 2014 American Mathematical Society