K-Trivial Varieties - Degenerations, Automorphisms, and Periods

代数几何研究的是代数变量，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，并且与各种现实世界的应用相关，从密码学到计算生物学，再到物理学中的宇宙模型。事实上，Calabi-Yau三折，一类特殊的代数变体，是弦理论中宇宙形状的抽象表示。关于Calabi-Yau三倍体，一个非常重要的、广泛开放的问题是，这种物体是否存在有限多种类型。这个提议是关于Calabi-Yau三倍和一个相关的，更广泛的代数变种，所谓的k平凡变种的研究。一系列问题，从上述的有限性问题到更具体的问题，将被调查。这项研究将涉及一些PI的研究生和博士后。还计划了与该主题有关的一些其他研究和外联活动。k -平凡变异体，即具有平凡正则类的代数变异体，是代数几何中的一个中心课题。提议的项目将集中在两类k平凡的变种：超kaehler流形和Calabi-Yau三倍。驱动本研究的动机目标是此类对象的变形类型的有限性，以及构建新变形类的补充问题（特别是在hyper-Kaehler案例中）。实现这些挑战性目标的中间步骤包括关于自同构群的问题，即这些对象的对称性；变形和退化，将k平凡变量分解成更简单，更易于管理的部分；以及k平凡变体的振动（特别是超凯勒流形的拉格朗日振动），即从低维对象构造这些变体。这项研究的一个主要工具是霍奇理论和相关的时期图。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 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.