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

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**142**(2014), 2417-2428 Request permission

## 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.## References

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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
- Received by editor(s): July 22, 2012
- Published electronically: April 3, 2014
- Communicated by: Lei Ni
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**142**(2014), 2417-2428 - MSC (2010): Primary 37J05, 53D20; Secondary 37J10
- DOI: https://doi.org/10.1090/S0002-9939-2014-12014-4
- MathSciNet review: 3195764

Dedicated: This paper is dedicated to my father