Polarized Hodge structure ($(-1{)}^{w(w-1)/2}{i}^{p-q}Q(v,\bar{v})>0$)

A polarization is a nondegenerate integral bilinear form of symmetry $(-1{)}^w$, orthogonal on Hodge summands except complementary bidegrees, satisfying the displayed positivity for every nonzero $v\in H^{p,q}$. This convention takes the cup-product pairing as the primitive geometric polarization. Omitting the constant sign factor and rescaling $Q$ is an equivalent convention. The full intersection pairing of a K3 surface is not itself a polarization: its real $(1,1)$ space has one positive direction as well as nineteen negative directions.

Fix a lattice, a bilinear polarization and Hodge numbers. The period domain consists of the Hodge filtrations with these dimensions satisfying orthogonality, opposition and positivity. It is an open subset of its compact dual of a period domain, described algebraically by the dimensions and orthogonality conditions alone.

On the K3 intersection lattice, this open quadric has complex dimension twenty and records the period line $H^{2,0}$. Fixing a positive class $h$ of type $(1,1)$ adds $q(h,\omega )=0$ and yields a nineteen-dimensional domain for primitive polarized cohomology of signature $(2,19)$. The distinction is necessary: the unrestricted full lattice has signature $(3,19)$ and does not satisfy polarization positivity on all of $H^{1,1}$.

Positive complex Lagrangian subspaces for the integral symplectic form $\left(\begin{array}{cc}0& -I\\ I& 0\end{array}\right)$ are graphs $H_Z=\{(Zu,u):u\in {\mathbb{C}}^r\}$. Isotropy is the symmetry of $Z$, and $iQ(v,\bar{v})=2u^{\mathsf{T}}(\mathrm{Im}Z)\bar{u}$ gives positivity. For $g=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$ in the symplectic group, the action is $Z\mapsto (AZ+B)(CZ+D{)}^{-1}$.

The compact dual is the complex projective flag variety imposing prescribed filtration dimensions and $Q(F^p,F^{w-p+1})=0$, but neither real opposition nor positivity. In weight one it is the complex Lagrangian Grassmannian. In polarized K3 weight two it is an isotropic-line quadric.