Let $(M^{2n},\omega)$ be a 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.

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.

Let $(M^{2n},\omega)$ be a 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 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.

Let $(M^{2n},\omega)$ be a 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.