K-Trivial Varieties - Degenerations, Automorphisms, and Periods

K-平凡簇 - 简并、自同构和周期

基本信息

批准号：
2101640
负责人：
Radu Laza
金额：
$ 32万
依托单位：
SUNY at Stony Brook
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101640&HistoricalAwards=false
关键词：
Trivial Varieties Degenerations Automorphisms Periods

项目摘要

Algebraic geometry is concerned with the study of algebraic varieties, that is geometric objects defined by polynomial equations. Such objects are ubiquitous in mathematics, and are relevant to a variety of real world applications ranging from cryptography, to computational biology, to models of the universe in physics. Indeed, the Calabi-Yau threefolds, a special class of algebraic varieties, are abstract representations of the shape of the universe in string theory. A very consequential, wide-open question regarding the Calabi-Yau threefolds is the existence of finitely many types of such objects. This proposal is concerned with the study of Calabi-Yau threefolds and of a related, wider class of algebraic varieties, the so-called K-trivial varieties. A number of questions ranging from the above-mentioned finiteness question to more tangible questions will be investigated. This study will involve a number of the PI’s graduate students and postdocs. Some additional research and outreach activities related to the subject are also planned. The study of K-trivial varieties, that is algebraic varieties with trivial canonical class, is a central subject in algebraic geometry. The proposed projects will focus on two main classes of K-trivial varieties: hyper-Kaehler manifolds and Calabi-Yau threefolds. The motivational goals driving this study are the finiteness of deformation types for such objects, and the complementary question of constructing new deformation classes (especially in the hyper-Kaehler case). Intermediate steps towards these challenging objectives include questions regarding the automorphism groups, that is the symmetries of such objects; the deformations and degenerations, that is breaking up the K-trivial varieties into simpler, more manageable pieces; and the fibrations of K-trivial varieties (especially Lagrangian fibrations for hyper-Kaehler manifolds), that is constructing such varieties from lower dimensional objects. A main tool in this investigation is Hodge theory, and the associated period maps.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.

代数几何涉及代数簇的研究，即由多项式方程定义的几何对象，这些对象在数学中无处不在，并且与从密码学到计算生物学到宇宙模型的各种现实世界应用相关。事实上，卡拉比-丘三重是一类特殊的代数簇，是弦理论中宇宙形状的抽象表示，是一个关于宇宙的非常重要的、广泛开放的问题。卡拉比-丘三重是此类对象的有限多种类型的存在，该提议涉及卡拉比-丘三重和一类相关的、更广泛的代数簇，即所谓的 K-平凡簇的研究。从上述有限性问题到更具体的问题，本研究将涉及一些与该主题相关的额外研究和外展活动。 K-平凡簇的研究，即具有平凡正则类的代数簇，是代数几何的核心课题。拟议的项目将集中于两个主要类别的 K-平凡簇：超凯勒流形和卡拉比-丘。推动这项研究的动机目标是此类对象的变形类型的有限性，以及构建新变形类（尤其是在超凯勒变形类中）的补充问题。实现这些具有挑战性的目标的中间步骤包括有关自同构群的问题，即此类对象的对称性；将 K-平凡簇分解为更简单、更易于管理的部分以及 K 的纤维； - 平凡簇（尤其是超凯勒流形的拉格朗日纤维），即从低维对象构造此类簇。这项研究的主要工具是霍奇理论以及相关的。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。