Hi.

I want to find an example of symplectic (smooth compact) manifold $(M,\omega)$ whose Betti numbers are not unimodal, i.e.

$b_i(M) \leq b_{i+2}(M)$ fails for some $i < n$. ($ \dim{M} = 2n$.)

(Every Kahler manifold satisfies the hard Lefschetz property so that the manifold that I want to find is non-Kahler.)

Thank you in advance.