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Hard Lefschetz property of symplectic structures on compact Kähler manifolds

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Title
Hard Lefschetz property of symplectic structures on compact Kähler manifolds
Author(s)
Yunhyung Cho
Publication Date
2016-11
Journal
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v.368, no.11, pp.8223 - 8248
Publisher
AMER MATHEMATICAL SOC
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, ω, J)and a symplectic form σ on M which does not satisfy the hard Lefschetz property, but is symplectically deformation equivalent to the Kähler form ω. As a consequence, we can give an answer to the question posed by Khesin and Mc-Duff as follows. According to symplectic Hodge theory, any symplectic form ω on a smooth manifold M defines symplectic harmonic forms on M. In a paperby D. Yan (1996), Khesin and McDuff posed a question whether there exists a path of symplectic forms {ωt} such that the dimension hk hr(M,ω) of the spaceof symplectic harmonic k-forms varies along t. By Yan and O. Mathieu, the hard Lefschetz property holds for (M,ω) if and only if hk hr(M, ω) is equal to the Betti number bk(M) for all k > 0. Thus our result gives an answer tothe question. Also, our construction provides an example of a compact Kähler manifold whose Kähler cone is properly contained in the symplectic cone. © 2016 American Mathematical Society
URI
https://pr.ibs.re.kr/handle/8788114/3038
ISSN
0002-9947
Appears in Collections:
Center for Geometry and Physics(기하학 수리물리 연구단) > Journal Papers (저널논문)
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