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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Hard Lefschetz property of symplectic structures on compact Kähler manifolds
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by Yunhyung Cho PDF
Trans. Amer. Math. Soc. 368 (2016), 8223-8248 Request permission

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
  • 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.

  • Dedicated: This paper is dedicated to my wife
  • © 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