Primitive cohomology ($P^k=\ker ([\kappa {]}^{d-k+1}:H^k\to H^{2d-k+2})$)

For a compact Kähler manifold of complex dimension $d$ and $k\le d$, this is the cohomology killed by the indicated power of a Kähler class. The Hodge–Riemann bilinear relations polarize its Hodge structure. On a K3 surface, primitive degree-two cohomology is the orthogonal complement of the chosen positive class.

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.