Mirror Symmetry, Birational Geometry, and Moduli.

镜像对称、双有理几何和模。

• 批准号: 2200875
• 负责人: Paul Hacking
• 金额: $ 23.98万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200875&HistoricalAwards=false
• 关键词: Mirror; Symmetry; Birational; Geometry; Moduli.

String theory posits that interactions of fundamental particles at small scales are explained by hidden dimensions of our universe which are wrapped up to form a tiny geometric space called a Calabi--Yau manifold at each point. The mirror symmetry phenomenon asserts that Calabi--Yau manifolds come in mirror pairs X and Y which determine the same physics, implying surprising relations between geometric properties of X and Y. Birational geometry is the study of surgeries of spaces obtained by cutting out a subspace of lower dimension and gluing another in its place. The moduli space of a space Y parametrizes all possible spaces obtained by deforming Y. If X and Y are a mirror pair, then the birational geometry of X determines the structure of the moduli space of Y near a certain limit point. Based on this heuristic, Morrison conjectured that a Calabi--Yau manifold X admits only finitely many possible surgeries up to symmetries of X. The PI aims to show the conjecture is false in general, but a weaker statement sufficient for applications to moduli holds, as suggested by recent work of the PI with graduate students on unbounded Calabi--Yau manifolds. The PI will also study positively curved spaces called Fano manifolds and singularities that arise at limit points of the moduli space of Calabi--Yau manifolds via mirror symmetry. These projects will be pursued together with graduate students supported by the grant. The PI will also organize seminars and a conference focused on training of graduate students.The PI will study Morrison's cone conjecture and applications to birational geometry and moduli, joint with a collaborator. Morrison conjectured that the automorphism group of a Calabi--Yau manifold acts on its nef cone with rational polyhedral fundamental domain, so that a neighborhood of a cusp of the moduli space of the mirror manifold admits a compactification determined by this data via a construction of Looijenga. Recent work of the PI with graduate students on log Calabi--Yau manifolds and mirror deformations of singularities suggests that the conjecture does not hold in general, but a weaker version sufficient for applications to moduli should hold. The PI will study mirror symmetry for Q-Fano 3-folds and applications to classification and non-arithmetic curves on moduli of K3 surfaces, joint with a graduate student. Q-Fano 3-folds arise as end products of the minimal model program and so are basic to our understanding of 3-folds. Mirror symmetry heuristics suggest that the mirror of a Q-Fano 3-fold is a K3 fibration over the affine line with monodromy at infinity that is maximally unipotent after a finite base change. The mirror corresponds to a rigid rational curve on a moduli space of polarized K3 surfaces. Computations suggest that these curves are not Shimura curves but are uniformized by non-arithmetic groups. The PI will study mirror symmetry for Milnor fibers of surface singularities and applications to symplectomorphism groups and moduli of surfaces, a project that joint with others.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.

弦理论假设，在小尺度上基本粒子的相互作用可以用我们宇宙中隐藏的维度来解释，这些维度被包裹起来，形成一个微小的几何空间，在每个点上都被称为卡拉比-丘流形。镜像对称现象认为，Calabi—Yau流形是由镜像对X和Y组成的，它们决定了相同的物理性质，这意味着X和Y的几何性质之间存在着令人惊讶的关系。镜像几何是通过切割低维子空间并在其位置上粘接另一个子空间来研究空间的手术。空间Y的模空间参数化了通过Y变形得到的所有可能空间。如果X和Y是镜像对，则X的双几何决定了Y在某极限点附近模空间的结构。基于这一启发式，Morrison推测Calabi- Yau流形X只允许有限多种可能的手术达到X的对称性。PI旨在证明该猜想一般是错误的，但一个较弱的声明足以应用于模，正如PI最近与研究生一起研究无界Calabi- Yau流形所提出的那样。PI还将研究称为Fano流形的正弯曲空间，以及通过镜像对称在Calabi- Yau流形模空间的极限点上出现的奇点。这些项目将与获得资助的研究生一起进行。PI还将组织以培养研究生为重点的研讨会和会议。PI将与合作者共同研究Morrison的锥猜想及其在双几何和模中的应用。Morrison推测Calabi—Yau流形的自同构群作用于其具有有理多面体基本域的内锥上，从而使得镜像流形的模空间顶点的邻域允许由该数据通过loijenga构造确定的紧化。PI最近与研究生一起对对数Calabi- Yau流形和奇点的镜像变形的研究表明，这个猜想并不普遍成立，但一个较弱的版本足以应用于模应该成立。PI将与一名研究生合作，研究Q-Fano 3-fold的镜像对称性及其在K3曲面模上的分类和非算术曲线中的应用。Q-Fano 3-fold作为最小模型程序的最终产品出现，因此是我们对3-fold的理解的基础。镜像对称启发式表明，Q-Fano 3-fold的镜像是在无限远处具有单态的仿射线上的K3颤振，该颤振在有限碱基变化后具有最大的单性。镜面对应于偏振K3曲面模空间上的刚性有理曲线。计算表明，这些曲线不是志村曲线，而是由非算术群统一化的。PI将研究表面奇点的Milnor纤维的镜像对称性，以及在表面的辛形态群和模上的应用，这是一个与他人合作的项目。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。