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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)=Mαβωn2.

I wonder when HR is non-singular. We can easily show that HR is non-singular if and only if ωn2:H2(M;R)H2n2(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 ωn2:H2(M;R)H2n2(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
    Commented Dec 20, 2010 at 7:43
  • Yes. Of course. I am sorry. I editted my question. Commented Dec 20, 2010 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 Commented Dec 20, 2010 at 9:47

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