Let ${\mathcal{D}}$ be the irreducible Hermitian symmetric domain of type $D_{2n}^{\mathbb{H}}$. There exists a canonical Hermitian variation of real Hodge structure ${\mathcal{V}}_{\mathbb{R}}$ of Calabi–Yau type over ${\mathcal{D}}$. This short note concerns the problem of giving motivic realizations for ${\mathcal{V}}_{\mathbb{R}}$. Namely, we specify a descent of ${\mathcal{V}}_{\mathbb{R}}$ from $\mathbb{R}$ to $\mathbb{Q}$ and ask whether the $\mathbb{Q}$-descent of ${\mathcal{V}}_{\mathbb{R}}$ can be realized as sub-variation of rational Hodge structure of those coming from families of algebraic varieties. When $n=2$, we give a motivic realization for ${\mathcal{V}}_{\mathbb{R}}$. When $n\geqslant 3$, we show that the unique irreducible factor of Calabi–Yau type in $\text{Sym}^{2}{\mathcal{V}}_{\mathbb{R}}$ can be realized motivically.

This paper contains two results on Hodge loci in $\mathsf{M}_{g}$. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibers and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fiber is contained in a proper Hodge locus. The second result deals with divisors in $\mathsf{M}_{g}$. It is proved that the image under the period map of a divisor in $\mathsf{M}_{g}$ is not contained in a proper totally geodesic subvariety of $\mathsf{A}_{g}$. It follows that a Hodge locus in $\mathsf{M}_{g}$ has codimension at least 2.