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On motivic realizations of the canonical Hermitian variations of Hodge structure of CalabiYau type over type $D^{\mathbb H}$ domains


Zheng Zhang,
Department of Mathematics, Texas A&M University, College Station, TX 778433368, USA
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 CalabiYau 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 subvariation
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 $n \geq 3$, we show that
the unique irreducible factor of CalabiYau type in $\mathrm{Sym}^2
\mathcal{V}_{\mathbb{R}}$ can be realized motivically.
MSC Classifications: 
14D07, 32G20, 32M15 show english descriptions
Variation of Hodge structures [See also 32G20] Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30] Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15]
14D07  Variation of Hodge structures [See also 32G20] 32G20  Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30] 32M15  Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15]
