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.

弦理论认为，小尺度上基本粒子的相互作用是由我们宇宙的隐藏维度解释的，这些维度被包裹起来形成一个小的几何空间，称为calabi-yau歧管。镜像对称现象断言，卡拉比（Calabi-Yau）歧管以镜子对和y进行了镜子对，这决定了相同的物理学，这意味着x和y的几何特性之间的惊人关系。生育的几何形状是通过切割较低尺寸并在其位置降低另一个子空间获得的空间手术的研究。空间y的模量空间参数通过变形y来获得所有可能的空间。如果x和y是镜对，则x的birational几何形状确定y的模量空间的结构，靠近一定的限制点。基于这种启发式，莫里森（Morrison）推测，卡拉比（Calabi-Yau）歧管X仅承认有限的许多可能的手术，直到X的对称性。PI的目的是表明该猜想通常是错误的，但较弱的陈述足以申请Moduli Holds，正如PI在Unlunded Calbabi上的PI型PI与Moduli Holds的申请，这是对Moduli Holds的申请。 PI还将研究称为Fano歧管和奇异性的正面弯曲空间，这些空间是通过镜像对称性的Calabi-Yau歧管的模量空间的极限点出现的。这些项目将与赠款支持的研究生一起进行。 PI还将组织研讨会和专注于研究生培训的会议。PI将研究Morrison的锥体猜想，并应用于与合作者联合的Birational几何形状和Moduli。莫里森（Morrison）猜想，卡拉比（Calabi-Yau）歧管的自态群体与理性的多面体基本域作用于其Nef锥，因此镜像歧管模量空间的尖缘邻域允许通过该数据通过looijenga的构建来确定该数据确定的紧凑型。 PI与研究生在log calabi-Yau歧管上的最新工作和奇异性的镜像变形表明，猜想总体上不存在，但是较弱的版本足以使莫迪利的应用程序保留。 PI将研究Q-Fano 3倍的镜像对称性，并应用于与研究生的K3表面模量上的分类和非弧度曲线。 Q-Fano是最小模型计划的最终产品，因此是我们对3倍的理解的基础。镜子对称启发式方法表明，Q-fano 3倍的镜像是仿射线上的K3纤维，无穷大的单体旋转，在有限的碱基变化后最大程度地单位。镜子对应于极化K3表面的模量空间上的刚性有理曲线。 计算表明，这些曲线不是shimura曲线，而是非陆上群均匀的。 PI将研究表面奇异性的Milnor纤维的镜像对称性，并应用于表面的符号构态组和模量，该项目与他人结合。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估来通过评估来获得支持的。