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On the formality and strong Lefschetz property of symplectic manifolds

Published online by Cambridge University Press:  01 September 2008

TAEK GYU HWANG  and
JIN HONG KIM
TAEK GYU HWANG
Affiliation:
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Yusong-gu, Daejon 305-701, Republic of Korea. e-mail: hwangtaekkyu@kaist.ac.kr, jinkim11@kaist.ac.kr
JIN HONG KIM
Affiliation:
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Yusong-gu, Daejon 305-701, Republic of Korea. e-mail: hwangtaekkyu@kaist.ac.kr, jinkim11@kaist.ac.kr
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Abstract

The main aim of this paper is to give some non-trivial results that exhibit the difference and similarity between Kähler and symplectic manifolds. To be precise, it is known that simply connected symplectic manifolds of dimension greater than 8, in general, do not satisfy the formality satisfied by all Kähler manifolds. In this paper we show that such non-formality of simply connected symplectic manifolds occurs even in dimension 8. We do this by some complicated but explicit construction of a simply connected non-formal symplectic manifold of dimension 8. In this construction we essentially use a variation of the construction of a simply connected symplectic manifold by Gompf. As a consequence, we can give infinitely many simply connected non-formal symplectic manifolds of any even dimension no less than 8.

Secondly, we show that every compact symplectic manifold admitting a semi-free Hamiltonian circle action with only isolated fixed points must satisfy the strong Lefschetz property satisfied by all Kähler manifolds. This result shows that the strong Lefschetz property for the symplectic manifold admitting Hamiltonian circle actions is closely related to their fixed point set, as expected.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 2 , September 2008 , pp. 363 - 377
Copyright
Copyright © Cambridge Philosophical Society 2008

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