On Motivic Realizations of the Canonical Hermitian Variations of Hodge Structure of Calabi–Yau Type over type Domains

Abstract 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.

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