Hard Lefschetz property of symplectic structures on compact Kähler manifolds

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, J, ω) 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 [Yan], Khesin and McDuff posed a question whether there exists a path of symplectic forms {ωt} such that the dimension h k hr (M, ω) of the space of symplectic harmonic k-forms varies along t. By [Yan] and [Ma], the hard Lefschetz property holds for (M, ω) if and only if h k hr (M, ω) 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 compact Kähler manifold whose Kähler cone is properly contained in the symplectic cone (c.f. [Dr]).Date: October 29, 2018. 2010 Mathematics Subject Classification. 53D20(primary), and 53D05(secondary).

Symplectic harmonicity and generalized coeffective cohomologies

“…Equality (ii) in the above proposition means that the behaviour of the symplectic invariantχ (8), which also gives a resolution of the same Lefschetz maps, one immediately concludes that the cohomologyȞ * (k) (M ) is isomorphic to the (k − 1)-filtered cohomology as follows:…”

“…Proof. Let us first recall that Cho proves in [8,Theorem 1.3] the existence of a compact Kähler manifold (X, ω) with dim C X = 3 such that (1) X is simply-connected,…”

“…In [28] Yan proved the existence of a 4-dimensional flexible manifold, whereas in [16] several 6-dimensional nilmanifolds satisfying such property were found. Recently, Cho has proved in [8] the existence of simply-connected flexible examples of dimension six. Our second goal in this paper is to relate the harmonic flexibility to corresponding notions of flexibility for the generalized coeffective and filtered cohomologies, as well as to construct closed manifolds which are flexible with respect to these symplectic cohomologies.…”

Hard Lefschetz property for Hamiltonian torus actions on $6$-dimensional GKM manifolds

“…It is known that the hard Lefschetz property does not hold in general. See [Cho1] or [Go] for example. However, it is not known whether (M, ω) satisfies the hard Lefschetz property when (M, ω) admits a Hamiltonian torus action with isolated fixed points.…”

Unimodality of Betti numbers for Hamiltonian circle actions with index-increasing moment Maps

“…In the symplectic category, there are a lot of examples which do not satisfy the hard Lefschetz theorem so that the unimodality of Betti numbers is not obvious in general. (See [3], [5], and [7]).…”