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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.

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.