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Hodge Riemann Hodge-Riemann bilnear form on Symplectic symplectic manifolds.Let $\omega$ a symplectic symplectic(may be Kahler) forms on $M^{2n}$. Then we have a symmetric bilnear two form on $H^2(M,\mathbb{R})$ given by $ HR_\omega (\alpha,\beta) := < \alpha\beta[\omega]^{n-2}, [M] > $ for $\alpha, \beta \in H^2(M,\mathbb{R})$. (If $n=2$, then this form is just an intersection form.) Is there any example of $(M^{2n}, \omega, \omega')$ such that $HR_\omega$ and $HR_\omega'$ have different diagonal decomposition? (up to ordering set of eigenvalues? (with repetition) |
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Let $\omega$ a symplectic forms on $M^{2n}$. Then we have a symmetric bilnear two form on $H^2(M,\mathbb{R})$ given by $ HR_\omega (\alpha,\beta)_\omega \alpha,\beta) := < \alpha\beta[\omega]^{n-2}, [M] > $ for $\alpha, \beta \in H^2(M,\mathbb{R})$. (If $n=2$, then this form is just an intersection form.) Is there any example of $(M^{2n}, \omega, \omega')$ such that $< , >{\omega}$ HR_\omega$ and $< , >{\omega'}$ HR_\omega'$ have different diagonal decomposition? (up to ordering of eigenvalues) |
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Hodge Riemann bilnear form on Symplectic manifolds.Let $\omega$ a symplectic forms on $M^{2n}$. Then we have a symmetric bilnear two form on $H^2(M,\mathbb{R})$ given by $ (\alpha,\beta)_\omega = < \alpha\beta[\omega]^{n-2}, [M] > $ for $\alpha, \beta \in H^2(M,\mathbb{R})$. (If $n=2$, then this form is just an intersection form.) Is there any example of $(M^{2n}, \omega, \omega')$ such that $< , >{\omega}$ and $< , >{\omega'}$ have different diagonal decomposition? (up to ordering of eigenvalues)
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