Hodge theory

Hodge theory connects de Rham cohomology with harmonic differential forms through the Hodge Laplacian on a compact Riemannian manifold. The Hodge decomposition theorem supplies a unique harmonic representative of each cohomology class. On a compact Kähler manifold, the decomposition additionally respects the complex bidegrees of differential forms.

A pure integral Hodge structure is a finite-rank free abelian group with a complex decomposition satisfying $\overline{H^{p,q}}=H^{q,p}$ and $p+q=w$. The integer $w$ is its weight. Its equivalent Hodge filtration is $F^p={⨁}_{r\ge p}{H}^{r,w-r}$.

A pure integral variation consists of a local system of free abelian groups, holomorphic filtration subbundles of its associated flat bundle, and fibrewise Hodge structures of fixed weight and Hodge numbers. Its flat connection satisfies Griffiths transversality. A polarized variation additionally has a flat integral bilinear form polarizing every fibre.

In a variation of Hodge structure, differentiation lowers the filtration index by at most one. In the relative de Rham construction, an absolute form of degree at least $p$ contributes, after extracting one base differential, a relative form of degree at least $p-1$. This filtration shift proves the condition for geometric variations.

The flat connection on the complexified fibre-cohomology local system differentiates cohomology classes using topological parallel transport. For a smooth proper holomorphic family, relative de Rham cohomology identifies it with the connecting morphism obtained by retaining one base differential in the absolute de Rham complex.

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.

A decreasing filtration defines a Hodge structure when $H_{\mathbb{C}}=F^p\oplus \overline{F^{w-p+1}}$ for every $p$. The decomposition is recovered as $H^{p,q}=F^p\cap \overline{F^q}$. This filtration is the holomorphic datum that varies in a variation of Hodge structure.