论文标题
霍奇表示
Hodge Representations
论文作者
论文摘要
Green-Griffiths-Kerr引入了Hodge表示，以对极化霍奇结构的Hodge组进行分类，以及一个时期域的相应的Mumford-Tate子域。本文的目的是提供有关如何给定固定周期域$ \ MATHCAL {D} $的说明，以枚举与Mumford-Tate子域相对应的HODGE表示。 After reviewing the well-known classical cases that $\mathcal{D}$ is Hermitian symmetric (weight $n=1$, and weight $n=2$ with $p_g = h^{2,0}=1$), we illustrate this in the case that $\mathcal{D}$ is the period domain parameterizing polarized Hodge structures of (effective) weight two Hodge structures with first Hodge number $ p_g = h^{2,0} = 2 $。我们还对Calabi-yau类型的hodge表示进行了分类，并列举了CY 3倍类型的水平表示。 （“水平”表示，具有对应域$ d \ subset \ mathcal {d} $的属性的人满足了无穷小时的关系，又称格里菲斯的横向性，因此是hermitian。）
Hodge representations were introduced by Green-Griffiths-Kerr to classify the Hodge groups of polarized Hodge structures, and the corresponding Mumford-Tate subdomains of a period domain. The purpose of this article is to provide an exposition of how, given a fixed period domain $\mathcal{D}$, to enumerate the Hodge representations corresponding to Mumford-Tate subdomains $D \subset \mathcal{D}$. After reviewing the well-known classical cases that $\mathcal{D}$ is Hermitian symmetric (weight $n=1$, and weight $n=2$ with $p_g = h^{2,0}=1$), we illustrate this in the case that $\mathcal{D}$ is the period domain parameterizing polarized Hodge structures of (effective) weight two Hodge structures with first Hodge number $p_g = h^{2,0} = 2$. We also classify the Hodge representations of Calabi-Yau type, and enumerate the horizontal representations of CY 3-fold type. (The "horizontal" representations those with the property that corresponding domain $D \subset \mathcal{D}$ satisfies the infinitesimal period relation, a.k.a. Griffiths' transversality, and is therefore Hermitian.)
