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I am looking for examples of closed symplectic manifolds $(M,\omega)$ whose Betti numbers do not satisfy a non-decreasing property. Meaning, it fails to satisfy $b_k(M) \leq b_{k+2}(M)$ for some $k < n=\frac{1}{2}\dim M$. (Edit: I've been told in the comments that this property for the Betti numbers is also called unimodal). It is possible there are not any known examples but I have not perused the literature enough to be sure.

If $M$ is a Kahler manifold, then a consequence of the Hard Lefschetz theorem shows that $M$ does satisfy this non-decreasing property. Outside of Kahler examples, there are symplectic manifolds which satisfy a Hard Lefschetz property. Lastly, there are also examples of symplectic manifolds which do not satisfy the Hard Lefschetz property but as far as I know, the known examples still satisfy the non-decreasing property.

This question was asked many years ago but the link in the comment leads to an unavailable page: Examples of non-Kahler symplectic manifolds.

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    $\begingroup$ As the keyword might appear in this context, it's the same as the sequence of Betti numbers not being unimodal. $\endgroup$
    – YCor
    Commented Aug 3, 2020 at 16:19
  • $\begingroup$ The broken link at the other question leads, according to the wayback machine, to Symplectic manifolds with no Kahler structure by Aleksy Tralle and John Oprea, Springer, Lecture Notes in Mathematics vol 1661. A concerning observation is that although the archived page includes the doi information for the book, the new webpage doesn't seem to prominently display this information. Is this standard for Springer now? $\endgroup$
    – Tim Campion
    Commented Aug 10, 2020 at 3:09
  • $\begingroup$ Note to self: the doi can be extracted from the modern Springer URL as the number, 10.1007 in this case, which appears before the % sign. $\endgroup$
    – Tim Campion
    Commented Aug 10, 2020 at 3:16

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