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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Log-concavity of the Duistermaat-Heckman measure for semifree Hamiltonian $ S^1$-actions


Author: Yunhyung Cho
Journal: Proc. Amer. Math. Soc. 142 (2014), 2417-2428
MSC (2010): Primary 37J05, 53D20; Secondary 37J10
Published electronically: April 3, 2014
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Abstract: The Ginzberg-Knutson conjecture states that for any Hamiltonian Lie group $ G$-action, the corresponding Duistermaat-Heckman measure is log-
concave. It turns out that the conjecture is not true in general, but every well-known counterexample has non-isolated fixed points. In this paper, we prove that if the Hamiltonian circle action on a compact symplectic manifold $ (M,\omega )$ is semifree and all fixed points are isolated, then the Duistermaat-Heckman measure is log-concave. With the same assumption, we also prove that $ \omega $ and every reduced symplectic form satisfy the hard Lefschetz property.


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Additional Information

Yunhyung Cho
Affiliation: School of Mathematics, Korea Institute for Advanced Study (KIAS), 87 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Republic of Korea
Email: yhcho@kias.re.kr

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-12014-4
PII: S 0002-9939(2014)12014-4
Keywords: Symplectic geometry, Duistermaat-Heckman measure
Received by editor(s): July 22, 2012
Published electronically: April 3, 2014
Dedicated: This paper is dedicated to my father
Communicated by: Lei Ni
Article copyright: © Copyright 2014 American Mathematical Society