This question may be quite vague and weird, but let me try to make the motivation clear.
Let $X$ be a compact Kähler manifold of dimension $n$. The Hodge index theorem says that
The signature of the intersection form on $H^n(X,\mathbb R)$ is $\sum_{a,b}(-1)^ah^{a,b}(X)$.
And the Hodge--Riemann relations imply that
Let $\omega$ be a Kähler class. Then for any even $k\leq n$, the signature of the bilinear form $$ H^k(X,\mathbb R)\times H^k(X,\mathbb R)\to \mathbb R$$ given by $(\alpha, \beta)\mapsto \int_X\alpha\wedge\beta\wedge\omega^{n-k}$ is totally determined by the Hodge structure of $H^*(X,\mathbb R)$.
Proofs of both results can be found in Voisin's book Hodge Theory and Complex Algebraic Geometry Section 6.3.
Let $X$ be a complex projective manifold with an ample line bundle $L$. Let $\omega=c_1(L)$ be the first Chern class of $L$. Let $k\leq n$ be an even number. In an attempt to understand the symmetric bilinear form $$H^k(X,\mathbb Q)\times H^k(X,\mathbb Q)\to \mathbb Q$$ given by $(\alpha, \beta)\mapsto \int_X\alpha\wedge\beta\wedge\omega^{n-k}$, we may pass to coefficients $\mathbb R$ or $\mathbb Q_p$. The Hodge--Riemann relations roughly say that with coefficients $\mathbb R$, the nondegenerate bilinear form is determined by the Hodge structure of $H^*(X,\mathbb Q)$, since a real nondegerate symmetric bilinear form is essentially determined by its signature. Knowing information about this bilinear form with $\mathbb Q_p$-coefficients would be helpful for the understanding of the bilinear form with $\mathbb Q$-coefficients itself.
Can we say something about this bilinear form with $\mathbb Q_p$-coefficients (e.g. the discriminant, the $\epsilon$-invariant)?