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On Motivic Realizations of the Canonical Hermitian Variations of Hodge Structure of Calabi–Yau Type over type $D^{\mathbb{H}}$ Domains

  • Zheng Zhang (a1)

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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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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