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Let $(M^{2n},\omega)$ be a closed connected symplectic manifold and let $HR : H^2(M;R) \times H^2(M;R) \rightarrow R$ be the Hodge-Riemann form defined by $HR(\alpha,\beta) = \int_M \alpha \beta \omega^{n-2}$.

I wonder when $HR$ is non-singular. We can easily show that $HR$ is non-singular if and only if $\omega^{n-2} : H^2(M;R) \rightarrow H^{2n-2}(M;R)$ is an isomorphism. Of course if $\omega$ is Kaehler or of Hard Lefschetz type, then it is true.

My question is, is there any other condition that makes $HR$ to be non-singular?

And if you know the examples such that $\omega^{n-2} : H^2(M;R) \rightarrow H^{2n-2}(M;R)$ is not an isomorphism, please let me know.

Thank you in advance.

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Are you assuming $M$ is compact? – S. Carnahan Dec 20 '10 at 7:43
Yes. Of course. I am sorry. I editted my question. – Yunhyung Cho Dec 20 '10 at 7:50
Ah.. There is an example in "A SIX DIMENSIONAL COMPACT SYMPLECTIC SOLVMANIFOLD WITHOUT KAHLER STRUCTURES" - 1996, Fernandez, M, de Leon, M. and Saralegui, M., Osaka J.Math – Yunhyung Cho Dec 20 '10 at 9:47

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