Hodge–Riemann bilinear relations ($(-1{)}^{k(k-1)/2}{i}^{p-q}{\int }_{X}v\wedge \bar{v}\wedge {\kappa }^{d-k}>0$)

For a nonzero primitive class $v\in H^{p,q}$ of degree $k=p+q$ on a compact Kähler manifold of complex dimension $d$, the displayed expression is positive. Together with orthogonality of complementary Hodge bidegrees, this makes primitive cohomology a polarized Hodge structure when the Kähler class is rational, after clearing denominators. Positivity on the full cohomology requires the signs prescribed by its primitive decomposition.