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Hard Lefschetz property of symplectic structures on compact Kähler manifolds

Author: Yunhyung Cho
Journal: Trans. Amer. Math. Soc. 368 (2016), 8223-8248
MSC (2010): Primary 53D20; Secondary 53D05
Published electronically: May 6, 2016
MathSciNet review: 3546798
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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.

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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

Keywords: Symplectic manifold, Hamiltonian action, hard Lefschetz property, non-K\"ahler manifold
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.
Dedicated: This paper is dedicated to my wife
Article copyright: © Copyright 2016 American Mathematical Society

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