= Solution
Fix a finite-rank <free abelian group> $H_{\mathbb Z}$. A pure integral <Hodge structure> of weight $w$ is a decomposition
$$
H_{\mathbb C}=H_{\mathbb Z}\otimes\mathbb C
=\bigoplus_{p+q=w}H^{p,q},
\qquad \overline{H^{p,q}}=H^{q,p}.
$$
Its <Hodge numbers> are $h^{p,q}=\dim_{\mathbb C}H^{p,q}$, and its <Hodge filtration> is $F^p=\bigoplus_{r\ge p}H^{r,w-r}$. Equivalently, the filtration has these dimensions and satisfies
$$
H_{\mathbb C}=F^p\oplus\overline{F^{w-p+1}}.
$$
The 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 $Q(u,v)=(-1)^wQ(v,u)$, extended complex-bilinearly, with
$$
Q(H^{p,q},H^{r,s})=0\quad\text{unless }(r,s)=(q,p).
$$
We choose the geometric sign convention in which the positivity condition is
$$
\boxed{(-1)^{w(w-1)/2}i^{p-q}Q(v,\bar v)>0
\quad(0\ne v\in H^{p,q}).}
$$
The 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 variety>
$$
\check D=\left\{
F^\bullet:
\dim F^p=\sum_{r\ge p}h^{r,w-r},\
Q(F^p,F^{w-p+1})=0
\right\}.
$$
The 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 \b[$\check D$ retains the complex flag conditions, whereas $D$ retains genuine <polarized Hodge structures>].
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