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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.

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Maybe you can have a look at this book: link.springer.com/book/10.1007/BFb0092608/page/1 –  Francesco Polizzi May 29 '13 at 12:18

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