Hyperkähler Manifolds, Moduli Spaces, and Fano Varieties

这个项目的重点是代数几何的问题。这个数学领域的重点是几何空间，称为代数变量，它可以局部描述为多项式方程组的几个变量的解的集合。因此，该领域适合于计算和信息以及计算机科学和物理学的应用。代数几何的主要目的之一是对代数变量进行分类。一个可以用来区分不同类别的离散不变量是曲率：曲率为正的变种称为Fano，曲率为零的变种称为Calabi-Yau，曲率为负的变种称为general type。这个项目的重点是一类特殊的Calabi-Yau品种，它们被称为hyperkähler流形。本项目利用模理论探讨hyperkähler流形与Fano型、Calabi-Yau型、general type型的关系。该项目还支持PI继续努力和开展活动，扩大代表性不足群体的参与。该项目围绕三大目标展开。(1) PI旨在通过形式化将Fano变量与K3型hyperkähler流形相关联的几何结构来加强hyperkähler流形与Fano变量之间的联系。(2)另一个目标是通过研究哪些阿贝尔变体可能作为拉格朗日纤维hyperkähler流形的光滑纤维出现来推进hyperkähler流形的拉格朗日纤维的研究。(3) PI将通过研究hyperkähler流形的模空间的几何来扩展其模理论，特别是描述hyperkähler流形的模空间是一般型的情况。此外，PI将共同举办各种活动，以教育和培训下一代，并扩大数学科学的广泛参与。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is focused on questions in algebraic geometry. This field of mathematics focuses on geometric spaces, called algebraic varieties, which can locally be described as the set of solutions of a system of polynomial equations in several variables. As such, the field lends itself to applications in computing and information as well as computer science and physics. One of the main aims of algebraic geometry is to classify algebraic varieties. One discrete invariant that can be used to distinguish distinct classes is curvature: varieties with positive curvature are called Fano, while varieties with zero curvature are called Calabi-Yau, and those with negative curvature are called general type. This project is focused on a particular class of Calabi-Yau varieties, which are called hyperkähler manifolds. The project uses moduli theory to explore relationships between hyperkähler manifolds and varieties that are Fano, varieties that are Calabi-Yau, and varieties that are general type. The project also supports the PI’s continued efforts and activities towards broadening participation among underrepresented groups.The project centers around three broad goals. (1) The PI aims to strengthen connections between hyperkähler manifolds and Fano varieties by formalizing geometric constructions associating a Fano variety to a hyperkähler manifold of K3 type. (2) Another goal is to advance the study of Lagrangian fibrations of hyperkähler manifolds by investigating which abelian varieties may arise as smooth fibers of a Lagrangian-fibered hyperkähler manifold. (3) The PI will expand the theory of moduli of hyperkähler manifolds by studying the geometry of such moduli spaces, in particular describing when moduli spaces of hyperkähler manifolds are of general type. In addition, the PI will co-organize various events with a view towards educating and training the next generation and growing broad participation in the mathematical sciences.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.