In the realm of almostKahler geometry , to what extent , the hard Lefschetz property is still holds?
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Since on any symplectic manifold you can easily find an almostKahler metric, this question can be reformulated as follows: "In the realm of symplectic geometry , to what extent , the hard Lefschetz property is still holds?" The answer is: It does not hold to a very large extent. There are a lot of examples of symplectic manifolds, where this property doesn't hold. In fact, checking that the hard Lefschetz property does not hold for a given symplectic manifold is one of the most favourite tools to establish that a given symplectic manifold is not Kahler. You can check, for example this page: http://en.wikipedia.org/wiki/Lefschetz_manifold 

