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On a compact Kähler manifold, complex conjugation commutes with the real Hodge Laplacian and exchanges harmonic forms of bidegrees $(p,q)$ and $(q,p)$. The identification of Dolbeault cohomology with harmonic forms therefore gives the symmetry of Hodge numbers.

Fix a finite-rank free abelian group $H_{\mathbb{Z}}$. A pure integral Hodge structure of weight $w$ is a decompositionIts Hodge numbers are $h^{p,q}={\dim }_{\mathbb{C}}{H}^{p,q}$, and its Hodge filtration is $F^p={⨁}_{r\ge p}{H}^{r,w-r}$. Equivalently, the filtration has these dimensions and satisfiesThe decomposition is recovered as $H^{p,q}=F^p\cap \overline{F^q}$.

A polarized Hodge structure additionally has a nondegenerate integral bilinear form $Q$ satisfying $Q(u,v)=(-1{)}^{w}Q(v,u)$, extended complex-bilinearly, withWe choose the geometric sign convention in which the positivity condition isThe expression is real by the symmetry and conjugation conditions. In particular it is $iQ(v,\bar{v})>0$ on a weight-one $(1,0)$ summand. Other conventions absorb the constant factor $(-1{)}^{w(w-1)/2}$ into $Q$; a consistent choice matters in weight two.

Fix $H_{\mathbb{Z}}$, $Q$ and the Hodge numbers. The compact dual of a period domain is the projective flag varietyThe orthogonality conditions are algebraic. The Griffiths domain $D$ is the subset whose filtrations also satisfy real opposition and the displayed positivity. It is open in the usual complex topology of $\check{D}$; the inequalities are not additional complex algebraic equations. Thus $\check{D}$ retains the complex flag conditions, whereas $D$ retains genuine polarized Hodge structures.

In weight one, the Hodge numbers are $h^{1,0}=h^{0,1}=r$, and the Hodge filtration is determined by the $r$-plane $H=F^1$. The bilinear orthogonality condition is $\psi (H,H)=0$. An isotropic $r$-plane in a symplectic $2r$-space is a Lagrangian subspace, soThe ordinary Grassmannian has dimension $r^2$. Restriction of the alternating form to an $r$-plane imposes $r(r-1)/2$ independent conditions. Equivalently, near a fixed Lagrangian plane, transverse Lagrangians are graphs of symmetric matrices, with $r(r+1)/2$ independent entries. Therefore

Write ${\mathbb{C}}^{2r}={\mathbb{C}}^r\oplus {\mathbb{C}}^r$ in the block order of the given matrix. On $D$, positivity is $i\psi (v,\bar{v})>0$ for every nonzero $v\in H$. The plane $H$ cannot contain $(x,0)\ne 0$, because that vector has $\psi (v,\bar{v})=0$. Projection onto the second coordinate factor is therefore injective on $H$ and, by equal dimensions, an isomorphism. There is a unique matrix $Z$ such thatFor $v=(Zu,u)$ and $v^{\prime }=(Zw,w)$, direct evaluation givesThus isotropy is exactly $Z=Z^{\mathsf{T}}$. For such $Z$,Hence positivity is precisely positive definiteness of $\mathrm{Im}Z$. Conversely, symmetry and this inequality define a positive Lagrangian plane; it is opposed to its conjugate because $(Z-\bar{Z})u=0$ forces $u=0$. We have explicitly identifiedthe Siegel upper half-space. The choice of the graph $(Zu,u)$, rather than $(u,Zu)$, is what matches the sign of the given symplectic form.

Let an integral element of the symplectic group have block matrixActing on graph vectors givesThe image remains a positive Lagrangian plane, so its second projection is invertible. Thus $CZ+D_0$ is invertible and the induced action isThe symplectic identities give symmetry of this matrix, and preservation of $i\psi (v,\bar{v})$ givesThis is a Hermitian congruence by an invertible matrix, so it proves preservation of the positive domain as well as describing the action. These are coordinates on marked Hodge structures; passing to the integral-group quotient forgets the symplectic marking.

Let $\Lambda =H^2(X,\mathbb{Z})$ with the cup-product pairing $q$ of signature $(3,19)$. For a K3 surface, the Hodge numbers are $(h^{2,0},h^{1,1},h^{0,2})=(1,20,1)$. A nonzero holomorphic two-form $\omega$ spans $F^2$, and its wedge products giveThe remaining filtration is forced by orthogonality:Thus the usual K3 period domain and its quadric compactification areThe quadric is smooth and has complex dimension $21-1=20$. Writing $\omega =x+iy$, its equations say that $x,y$ are orthogonal positive vectors of equal square. A period therefore determines an oriented positive two-plane in ${\Lambda }_{\mathbb{R}}$.

There is a polarization qualification: the full cup-product form is not a polarization in the strict sense of part (a). Its real $(1,1)$ space has signature $(1,19)$, so it cannot satisfy the negative-definiteness requirement there. Fixing an ample integral class $h$ of square $2d>0$ and requiring it to stay of type $(1,1)$ gives the primitive cohomology latticeIts genuine polarized Griffiths domain isIts compact dual is the corresponding smooth quadric of complex dimension nineteen, and $D_h$ has two components, distinguished by the orientation of the positive two-plane. For ample-polarized geometric K3 moduli one also removes the loci $(\omega ,\delta )=0$ with $\delta \in {\Lambda }_h$, ${\delta }^2=-2$, since an ample class cannot be orthogonal to an effective rational curve. Those exclusions concern the ample moduli locus; they are not part of the definition of the polarized Hodge domain.

If one wants a strict polarization on the full $H^2$ of a fixed projective surface, reverse the sign on its rational Kähler line and retain $q$ on the orthogonal complement. For example the integral formhas the required positivity and signature $(2,20)$. It is a different form from the cup product. The twenty-dimensional full period quadric and the nineteen-dimensional primitive polarized domain should therefore be distinguished. The positivity and signature calculations above explain why these two domains describe different data.

A pure integral variation of Hodge structure of weight $w$ on a complex manifold $S$ consists of a local system ${\mathbb{H}}_{\mathbb{Z}}$ of finite-rank free abelian groups and a decreasing filtration by holomorphic subbundlesEach fibre, with this filtration and the conjugation from its integral lattice, must be a Hodge structure of weight $w$ with locally constant Hodge numbers. The local system gives a flat connection $\nabla$: in a locally constant frame it differentiates the coefficient functions. The filtration must satisfy Griffiths transversality,A polarized variation additionally has a locally constant integral bilinear form $Q$ which polarizes every fibre. Flatness means $\nabla Q=0$. A collection of fibrewise Hodge decompositions without a holomorphic filtration or the transversality condition is not yet a variation of Hodge structure.

Here a smooth family is understood as a smooth proper family of compact complex manifolds: $f:\mathcal{X}\to S$ is a proper holomorphic submersion, with compact Kähler fibres. Properness and compactness are essential for the usual statement. Indeed, the projection $({\mathbb{C}}_t\times {\mathbb{C}}_z)\setminus \{(0,0)\}\to {\mathbb{C}}_t$ is a holomorphic submersion with Kähler fibres, but its fibres are $\mathbb{C}$ for $t\ne 0$ and ${\mathbb{C}}^{\ast }$ for $t=0$. Their first Betti numbers are zero and one, so their cohomology cannot give a fixed-rank variation. This supplies a counterexample if “smooth family” is read without the usual proper compact-fibre hypothesis. The Ehresmann fibration theorem gives local differentiable trivializations, sois a local system with fibre $H^k(X_s,\mathbb{Z})/\text{torsion}$. Its associated flat bundle has the Gauss–Manin connection, given by differentiating cohomology classes in topologically constant frames.

The Kähler Hodge decomposition givesThe Betti numbers are locally constant by differentiable triviality. The dimensions of $H^q(X_s,{\Omega }_{X_s}^p)$ are upper semicontinuous, and their sums for fixed $p+q=k$ equal the fixed Betti number. Near any fixed fibre, every summand can only decrease, while their sum cannot change; hence none decreases. Thus the Hodge numbers are locally constant. The relative de Rham complex and the Grauert base change theorem then makethe fibres of holomorphic subbundles $F^p\mathcal{H}$. More intrinsically these are the images of the hypercohomology of the truncated relative de Rham complex ${\Omega }_{\mathcal{X}/S}^{\ge p}$ in the full relative de Rham cohomology. Kähler degeneration of the spectral sequence ensures that these images have the required ranks.

For the transversality proof, let $A^2$ denote the ideal in the absolute holomorphic de Rham complex generated by two pulled-back base one-forms. Keeping zero or one base differential gives a short exact sequence of cochain complexes, with the usual shifted differential in the left-hand complexThe connecting morphism in degree $k$ is the Gauss–Manin connectionTo compute it, lift a relative de Rham cocycle to the middle complex, apply the absolute differential, and retain the terms with one base differential. A cocycle in filtration degree $p$ can be represented using absolute forms of degree at least $p$. In a term containing one base differential, removing that factor leaves a relative form of degree at least $p-1$. The connecting construction therefore lands in $F^{p-1}$:This is the filtration shift behind Griffiths transversality. It explains the inequality rather than merely naming a theorem. The unfiltered construction agrees with topological parallel transport by the de Rham comparison, and its square is zero, as required for the flat connection.

Consequently the fibre cohomology gives a pure variation of Hodge structure. To claim a polarized integral variation one needs the additional compatible flat polarization. In a smooth projective family, a relative ample line bundle supplies a flat integral Kähler class; its primitive pieces are polarized by the Hodge–Riemann bilinear relations, and the full cohomology is polarized using the appropriate primitive-piece signs. For a general compact Kähler family, the unpolarized variation above is automatic, but a flat integral polarization is not supplied merely by knowing that each fibre is Kähler. This distinction is particularly relevant to unrestricted K3 families.

The Hodge decomposition theorem for compact Kähler manifolds stateswith each summand represented uniquely by harmonic forms of type $(p,q)$. Part b shows that these are precisely the constant-coefficient forms $d{z}_I\wedge d{\overline{z}}_J$. Therefore the Hodge numbers of a complex $n$-torus are