Geometric and Arithmetic Hyperbolicity in Moduli Spaces

代数几何关注的是从几何上理解代数变量——多项式方程的解的空间——以便解代数方程。阿贝尔变体特别有趣，因为这些空间具有阿贝尔群的结构，即空间的点可以相互相加产生其他点。阿贝尔变量作为复杂变量的重要不变量在数学中无处不在，因此是代数几何、数论、表示理论和复分析的重要工具。本研究项目旨在通过研究参数化阿贝尔变量和相关对象的模空间的几何来了解代数变量如何在族中变化。更具体地说，该项目是有关双曲现象的局部对称变种。一些这样的变化可以被解释为Hodge结构的模空间，在这些情况下，双曲性与这些Hodge结构的变化（例如阿贝尔变化或hyperkähler变化的周期）在基B上的存在性有关，这对B的双几何形状提出了很强的条件。这与这些变化在不同基B上的一致有界性的推测密切相关。这反过来又与基于算术基的几何伽罗瓦表示所表现出的猜想有界性强烈平行。该项目旨在扩展研究者和合作者最近开发的新技术，以探索几何和算术背景下的这种现象。

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.