K-Trivial Varieties - Degenerations, Automorphisms, and Periods

代数几何学研究的是代数簇，即由多项式方程定义的几何对象。这样的对象在数学中无处不在，并且与各种真实的世界应用相关，从密码学到计算生物学，再到物理学中的宇宙模型。事实上，卡-丘三重（Calabi-Yau threefolds）是一类特殊的代数簇，是弦理论中宇宙形状的抽象表示。关于卡-丘三重性，一个非常重要的、开放性很强的问题是，存在着许多类型的这样的物体。这个建议是关于研究卡-丘三重和一个相关的，更广泛的一类代数簇，所谓的K-平凡簇。从上述有限性问题到更具体的问题，将研究一些问题。这项研究将涉及一些PI的研究生和博士后。还计划开展一些与这一主题有关的其他研究和外联活动。K-平凡簇，即具有平凡标准类的代数簇，是代数几何中的一个中心课题。拟议的项目将集中在两个主要类别的K-平凡品种：超凯勒流形和卡拉比-丘三倍。驱动这项研究的动机目标是有限的变形类型，这样的对象，和建设新的变形类（特别是在超凯勒的情况下）的补充问题。 实现这些具有挑战性的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，即将K-平凡簇分解成更简单，更易于管理的片段;以及K-平凡簇的纤维化（特别是超Kaehler流形的拉格朗日纤维化），即从低维对象构建这些簇。 该奖项反映了NSF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.