Applications of Hodge-Riemann bilinear relations [closed]

Question

I am wondering if the Hodge-Riemann bilinear relations have any further applications/ developments in Kahler or algebraic geometry.

Let me briefly remind the statement.

Given a compact Kahler manifold $(M,\omega)$ of complex dimension $n$, its cohomology with complex coefficients satisfies the Hodge decomposition $H^k(M,\mathbb{C})=\oplus_{p+q=k}H^{p,q}(M)$. The hard Lefschetz theorem allows to define the primitive part $P^{p,q}\subset H^{p,q}(M)$. On $H^{p,q}$ one defines an hermitial form $$Q(\xi)=i^{p-q}(-1)^{(n-k)(n-k-1)/2}\int_M\xi\wedge \bar\xi\wedge \omega^{n-k}.$$ The Hodge-Riemann bilinear relations tell that this form $Q$ is positive definite on $P^{p,q}$.

Answer 1

Since you ask for further applications, does that mean you already know some? The Hodge-Riemann bilinear relations are used in all kinds of ways. Suppose that $M$ is a compact Riemann surface, then form the period matrix $P=(\int_{\gamma_i} \omega_j)$, where the $\gamma_i\in H_1(M,\mathbb{Z})$ and $\omega_j\in H^0(M,\Omega_M^1)$ are bases. The relations tell you that you can choose bases so that $P= (I,\Omega)$, with $\Omega^T=\Omega$, and $Im\Omega>0$. This is just the starting point for a long and beautiful story. Take a look at chapter 2 of Griffiths and Harris for more about this.