arXiv 2003.001372020-03-05math.AGmath.RT

霍奇表示

Hodge Representations

Xiayimei Han, Colleen Robles

发表信息：: Exp Results 1 (2020) e51
DOI：: 10.1017/exp.2020.55

摘要：

Green-Griffiths-Kerr引入了Hodge表示来分类极化Hodge结构的Hodge群，以及周期域的相应Mumford-Tate子域。本文的目的是提供一个博览会，给定一个固定的周期域$\mathcal{D}$，枚举Mumford-Tate子域$D \subset \mathcal{D}$相应的霍奇表示。在回顾了著名的经典情况下，$\mathcal{D}$是厄米对称（重量$n=1$，重量$n=2$与$p_g = h^{2,0}=1$），我们说明了这一点的情况下，$\mathcal{D}$是周期域参数化极化霍奇结构的（有效）重量两个霍奇结构与第一霍奇数$p_g = h^{2,0} = 2$。我们还对Calabi-Yau型的Hodge表示进行了分类，并列举了CY 3-fold型的水平表示。(The“水平”表示具有相应域$D \subset \mathcal{D}$满足无穷小周期关系的性质的表示，也称为“水平”表示。Griffiths' transversality，and therefore is Hermitian.）

英文摘要：

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.)

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