Hard Lefschetz property of symplectic structures on compact Kähler manifolds
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Abstract:
In this paper, we give a new method to construct a compact symplectic manifold which does not satisfy the hard Lefschetz property. Using our method, we construct a simply connected compact Kähler manifold $(M,\omega ,J)$ and a symplectic form $\sigma$ on $M$ which does not satisfy the hard Lefschetz property, but is symplectically deformation equivalent to the Kähler form $\omega$. As a consequence, we can give an answer to the question posed by Khesin and McDuff as follows. According to symplectic Hodge theory, any symplectic form $\omega$ on a smooth manifold $M$ defines symplectic harmonic forms on $M$. In a paper by D. Yan (1996), Khesin and McDuff posed a question whether there exists a path of symplectic forms $\{ \omega _t \}$ such that the dimension $h^k_{hr}(M,\omega )$ of the space of symplectic harmonic $k$-forms varies along $t$. By Yan and O. Mathieu, the hard Lefschetz property holds for $(M,\omega )$ if and only if $h^k_{hr}(M,\omega )$ is equal to the Betti number $b_k(M)$ for all $k>0$. Thus our result gives an answer to the question. Also, our construction provides an example of a compact Kähler manifold whose Kähler cone is properly contained in the symplectic cone.References
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Additional Information
- Yunhyung Cho
- Affiliation: Departamento de Matemática, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos-LARSYS, Instituto Superior Técnico, Av. Rovisco Pais 1049-001 Lisbon, Portugal
- Address at time of publication: Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, Republic of Korea 37673
- Email: yhcho@ibs.re.kr
- Received by editor(s): March 1, 2015
- Received by editor(s) in revised form: August 22, 2015, and November 10, 2015
- Published electronically: May 6, 2016
- Additional Notes: The author was supported by IBS-R003-D1.
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 8223-8248
- MSC (2010): Primary 53D20; Secondary 53D05
- DOI: https://doi.org/10.1090/tran/6894
- MathSciNet review: 3546798
Dedicated: This paper is dedicated to my wife