Let (M2n,ω) be a closed connected symplectic manifold and let HR:H2(M;R)×H2(M;R)→R be the Hodge-Riemann form defined by HR(α,β)=∫Mαβωn−2.
I wonder when HR is non-singular. We can easily show that HR is non-singular if and only if ωn−2:H2(M;R)→H2n−2(M;R) is an isomorphism. Of course if ω 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 ωn−2:H2(M;R)→H2n−2(M;R) is not an isomorphism, please let me know.
Thank you in advance.