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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.