Classical Period Domains
Classical Period Domains
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古典时期领域
DOI：
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Zhenghe Zhang
中科院分区：
文献类型：
--
作者：
R. Laza;Zhenghe Zhang
本文综述了厄米对称域在研究Hodge结构变化中所起的作用。这些是基于第一作者在温哥华“霍奇理论的进展”学校（2013年6月）所做讲座的扩展笔记。引言有两个经典的情况下，期间地图起着至关重要的作用，研究模空间，即模的主要极化阿贝尔品种和极化K3表面的模。这两种情况的共同点是周期域实际上是埃尔米特对称域。众所周知，周期域是Hermitian对称的唯一情况是h2，0 = 1的权1 Hodge结构和权2 Hodge结构.一般来说，通过周期映射来研究模空间是很困难的。这个方向的一个主要困难来自格里菲斯的横截关系。典型地，周期域D中的周期映射的像Z将是高条件的超越解析子簇。唯一的情况下，当Z可以代数描述是当Z是一个埃尔米特对称子域的D与全测地嵌入（并满足水平关系）。这是密切相关的几何方面的理论志村品种德利涅。在[GGK12]的意义下，这也是无约束周期子域的情况。我们称这种情况为经典的，在“非经典”的情况下，格里菲斯的横截关系是非平凡的。本文综述了厄米对称域在研究Hodge结构变化中的作用。让我们简要概述一下文件的内容。在第一节中，我们回顾了埃尔米特对称域的基本定义和性质（第1.1节）以及它们的分类（第1.2节）。分类是通过从相关的（半简单的）Shimura数据中重建Hermitian对称域来完成的，这也便于在Hermitian对称域上构造Hodge结构的变体（1.3节）。作为题外话，我们还讨论了如果周期子域上的泛族Hodge结构满足Griffiths横截性，则该子域必须是Hermitian对称的（即无约束的Hermitian对称）。
We survey the role played by Hermitian symmetric domains in the study of variations of Hodge Structure. These are extended notes based on the lectures given by the first author in Vancouver at the “Advances in Hodge Theory” school (June 2013). Introduction There are two classical situations where the period map plays an essential role for the study of moduli spaces, namely the moduli of principally polarized abelian varieties and the moduli of polarized K3 surfaces. What is common for these two situations is the fact that the period domain is in fact a Hermitian symmetric domain. It is well known that the only cases when a period domain is Hermitian symmetric are weight 1 Hodge structures and weight 2 Hodge structures with h 2,0 = 1. In general, it is difficult to study moduli spaces via period maps. A major difficulty in this direction comes from the Griffiths’ transversality relations. Typically, the image Z of the period map in a period domain D will be a transcendental analytic subvariety of high condimension. The only cases when Z can be described algebraically are when Z is a Hermitian symmetric subdomain of D with a totally geodesic embedding (and satisfying the horizontality relation). This is closely related to the geometric aspect of the theory of Shimura varieties of Deligne. It is also the case of unconstrained period subdomains in the sense of [GGK12]. We call this case classical, in contrast to the “non-classical” case when the Griffiths’ transversality relations are non-trivial. The purpose of this survey is to review the role of Hermitian symmetric domains in the study of variations of Hodge structure. Let us give a brief overview of the content of the paper. In Section 1, we review the basic definitions and properties of Hermitian symmetric domains (Section 1.1) and their classification (Section 1.2) following [Mil04]. The classification is done by reconstructing Hermitian symmetric domains from the associated (semisimple) Shimura data, which are also convenient for the purpose of constructing variations of Hodge structure over Hermitian symmetric domains (Section 1.3). As a digression, we also include the discussion that if the universal family of Hodge structures over a period subdomain satisfies Griffiths transversality then the subdomain must be Hermitian symmetric (i.e. unconstrained ⇒ Hermitian symmetric).
DOI：
10.1007/978-3-642-00639-5
发表时间：
2007-11
期刊：
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影响因子：
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作者：
J. Rohde
通讯作者：
J. Rohde
将 Prym 映射扩展到阿贝尔簇模空间的环形紧化
DOI：
10.4171/jems/678
发表时间：
2017
期刊：
arXiv: Algebraic Geometry
影响因子：
--
作者：
S. Casalaina-Martin;S. Grushevsky;K. Hulek;R. Laza;Dutour Sikiric
通讯作者：
Dutour Sikiric
Calabi-Yau 流形的最大自同构与最大单能单向
影响因子：
0.6
作者：
J.-C. Rohde
通讯作者：
J.-C. Rohde