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Geometric and Arithmetic Hyperbolicity in Moduli Spaces

模空间中的几何和算术双曲性

基本信息

批准号：
1702149
负责人：
Benjamin Bakker
金额：
$ 13.77万
依托单位：
University of Georgia Research Foundation Inc
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1702149&HistoricalAwards=false
关键词：
Geometric Arithmetic Hyperbolicity Moduli Spaces

项目摘要

Algebraic geometry is concerned with geometrically understanding algebraic varieties -- the spaces of solutions to polynomial equations -- in order to solve algebraic equations. Abelian varieties are especially interesting because these spaces possess the structure of an Abelian group, that is, the points of the space can be added to each other to produce other points. Abelian varieties appear ubiquitously in mathematics as important invariants of more complicated varieties and are therefore a crucial tool in algebraic geometry, number theory, representation theory, and complex analysis. This research project aims to understand how algebraic varieties vary in families by studying the geometry of the moduli spaces that parametrize abelian varieties and related objects.More specifically, the project is concerned with hyperbolicity phenomena in locally symmetric varieties. Some such varieties can be interpreted as moduli spaces of Hodge structures, and in those cases hyperboliciity relates to the fact that the existence of variations of those Hodge structures (for example the periods of abelian varieties or hyperkähler varieties) over a base B imposes strong conditions on the birational geometry of B. This is closely related to conjectural uniform boundedness properties of such variations over varying bases B, which in turn strongly parallel conjectural boundedness properties exhibited by Galois representations coming from geometry over arithmetic bases. The project aims to expand upon new techniques recently developed by the investigator and collaborators to explore such phenomena, both in the geometric and arithmetic contexts.

代数几何涉及从几何角度理解代数簇（多项式方程解的空间），以便求解代数方程。阿贝尔簇特别有趣，因为这些空间具有阿贝尔群的结构，即空间中的点可以相互相加以产生其他点。阿贝尔簇作为更复杂簇的重要不变量在数学中无处不在，因此是代数几何、数论、表示论和复分析中的重要工具。该研究项目旨在通过研究参数化阿贝尔簇和相关对象的模空间的几何形状来了解代数簇在族中的变化。更具体地说，该项目涉及局部对称簇中的双曲现象。一些此类簇可以解释为 Hodge 结构的模空间，在这些情况下，双曲性与以下事实相关：这些 Hodge 结构在基 B 上的变体（例如阿贝尔簇或超凯勒簇）的存在对 B 的双有理几何施加了强条件。这与这些变体在不同基上的猜想一致有界性质密切相关。 B，这反过来又与来自算术基础上的几何的伽罗瓦表示所展示的猜想有界性性质强烈平行。该项目旨在扩展研究者和合作者最近开发的新技术，以在几何和算术背景下探索此类现象。