On the formality and strong Lefschetz property of symplectic manifolds

Abstract

The main aim of this paper is to give some non-trivial results that exhibit the difference and similarity between Kahler 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 Kahler 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 it 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 Kahler 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.