Fano Varieties and Mirror Symmetry

代数几何是一门研究几何形状的学科，这些几何形状可以用几个变量的代数方程来定义。这些形状可以由三种类型的某些基本构建块组装而成：Fano， Calabi-Yau和general type。范诺形状是正弯曲的，就像一个球的表面。通过精细的几何论证和个案分析，它们被完全划分为一、二、三维。根据理论物理中的镜像对称现象，卡拉比-丘形状是X和Y成对出现的，这样X的几何性质就能精确地对应Y的不同几何性质，反之亦然。Fano形状也有一个镜像对称的版本：如果X是n维的Fano形状，那么它的镜像伙伴是n-1维的Calabi-Yau形状的并集。本项目将利用镜像对称来开发一种适合于计算的范诺形状分类方法，并研究由镜像对应提供的范诺形状的新观点。范诺流形是正弯曲的空间，在代数和复杂几何中起着中心作用。范诺流形在每个维上有有限多个拓扑类型。最多3维的范诺流形已经被人工分类，但由于问题的规模，需要在更高的维度上使用计算方法进行分类。有一个版本的镜像对称现象适用于Fano流形：n维的Fano流形X的镜像伙伴是n-1维的Calabi-Yau流形的一个参数族的总空间Y。该项目将使用镜像对应来描述一种计算方法来对范诺流形进行分类，并利用来自环面几何、双几何、变形理论和模、霍奇理论和枚举几何的工具，开发范诺流形几何的新视角。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。

Algebraic geometry is the study of geometric shapes which can be defined by algebraic equations in several variables. These shapes can be assembled from certain basic building blocks of three types: Fano, Calabi-Yau, and general type. Fano shapes are positively curved like the surface of a ball. They have been classified completely in dimensions one, two, and three by delicate geometric arguments and case-by-case analysis. According to the mirror symmetry phenomenon from theoretical physics, Calabi-Yau shapes come in pairs X and Y such that geometric properties of X correspond in a precise way to different geometric properties of Y, and vice versa. There is a version of mirror symmetry for Fano shapes as well: if X is a Fano shape of dimension n then its mirror partner is a union of Calabi-Yau shapes of dimension n-1. This project will use mirror symmetry to develop an approach to classification of Fano shapes which is amenable to computation, and to investigate new viewpoints on Fano shapes provided by the mirror correspondence. Fano manifolds are positively curved spaces that play a central role in algebraic and complex geometry. There are finitely many topological types of Fano manifold in each dimension. Fano manifolds of dimension at most three have been classified by hand, but due to the scale of the problem a computational approach to classification is needed in higher dimensions. There is a version of the mirror symmetry phenomenon that applies to Fano manifolds: the mirror partner of a Fano manifold X of dimension n is the total space Y of a one parameter family of Calabi-Yau manifolds of dimension n-1. This project will use the mirror correspondence to describe a computational approach to the classification of Fano manifolds and to develop new perspectives on the geometry of Fano manifolds, using tools from toric geometry, birational geometry, deformation theory and moduli, Hodge theory, and enumerative geometry.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.