Novel Approaches to Geometry of Moduli Spaces

模空间几何的新方法

• 批准号: 2401387
• 负责人: Evgueni Tevelev
• 金额: $ 27.5万
• 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: 2024
• 资助国家: 美国
• 起止时间: 2024-06-01 至 2027-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401387&HistoricalAwards=false
• 关键词: Novel; Approaches; Geometry; Moduli; Spaces

Algebraic geometry has long occupied a central role in mathematics, providing a sophisticated language to describe geometric shapes known as algebraic varieties - with applications ranging from configuration spaces in physics to parametric models in statistics. This versatile language is used throughout algebra and has fueled multiple recent advances, not only in algebraic geometry itself but also in representation theory, number theory, symplectic geometry and other fields. Algebraic varieties are typically endowed with additional structures, such as vector bundles. Local sections of vector bundles are mathematical abstractions of fields in physics, making algebraic geometry indispensable for the study of physical phenomena like mirror symmetry and other dualities. A recurring theme in moduli theory is the interplay between moduli spaces of vector bundles, which parametrize them geometrically and can be studied analytically, and the derived categories of algebraic varieties, which encode algebraic and homological properties of vector bundles. Derived categories provide a bridge from algebraic geometry to the emerging field of non-commutative geometry. Indeed, functors and equivalences between derived categories are deeply related to the birational (local) geometry of algebraic varieties. This project will further the study of derived categories. The PI will deliver graduate-level mini-courses and lectures at conferences, professional development events, and summer schools for graduate students. Many sub-projects are suitable as thesis topics for graduate students. Furthermore, several problems are designed specifically for undergraduate participants in the research and training program in algebraic geometry organized by the PI.In more detail, the proposed reserch is centered around two main themes. The first is the study of derived categories of moduli spaces and Fano varieties more broadly. The derived categories of Fano varieties, unlike Calabi-Yau or most canonically polarized varieties, admit semi-orthogonal decompositions; from the perspective of non-commutative geometry, Fano varieties are built from more elementary blocks. A beautiful picture emerges, where the decompositions of various Fano varieties, related by birational transformations, undergo rearrangements, which we call weaving patterns. Their construction is motivated by ideas of mirror symmetry, quantum cohomology, vanishing theorems of the minimal model program, and quantization. The PI will advance this program for a wide variety of spaces: moduli spaces of vector bundles, parabolic bundles and Higgs bundles on curves, toric varieties, flag varieties, moduli of sheaves with one-dimensional support on K3 surfaces, and fixed-point loci of anti-symplectic involutions on projective hyperkahler varieties. The second theme is to continue the study of the categorical Milnor fiber for deformations of singular algebraic varieties, describe its mirror symmetry interpretation, and find applications to moduli of algebraic surfaces of geometric genus zero, including Dolgachev surfaces and fake del Pezzo surfaces.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.

代数几何长期以来一直在数学中占据着核心地位，它提供了一种复杂的语言来描述被称为代数簇的几何形状-其应用范围从物理学中的配置空间到统计学中的参数模型。这种通用的语言在整个代数中使用，并推动了多个最近的进展，不仅在代数几何本身，而且在表示论，数论，辛几何和其他领域。代数簇通常被赋予额外的结构，如向量丛。向量丛的局部截面是物理学领域的数学抽象，使得代数几何对于研究镜像对称和其他对偶等物理现象不可或缺。模理论中一个反复出现的主题是向量丛的模空间之间的相互作用，它将向量丛几何参数化，可以解析地研究，而代数簇的派生范畴编码了向量丛的代数和同调性质。导出范畴提供了一个桥梁，从代数几何的新兴领域的非交换几何。事实上，函子和导出范畴之间的等价关系与代数簇的双有理（局部）几何有很深的关系。这个项目将进一步研究派生类别。PI将在会议，专业发展活动和研究生暑期学校提供研究生水平的迷你课程和讲座。许多子项目适合作为研究生的论文题目。此外，几个问题是专门为大学生参与的研究和培训计划，在代数几何组织的PI。更详细地说，拟议的研究是围绕两个主题。第一个是研究更广泛的模空间和Fano簇的导出范畴。与卡-丘或大多数正则极化簇不同，法诺簇的派生范畴允许半正交分解;从非交换几何的角度来看，法诺簇是从更多的基本块构建的。一幅美丽的画面出现了，在那里，各种Fano变种的分解，通过双有理变换，经历了重排，我们称之为编织模式。他们的建设是出于镜像对称，量子上同调，消失定理的最小模型程序，和量化的想法。PI将在各种各样的空间中推进这个程序：向量丛的模空间，抛物线丛和曲线上的希格斯丛，复曲面簇，旗簇，K3曲面上一维支撑层的模，以及投影超卡勒簇上反辛对合的不动点轨迹。第二个主题是继续研究奇异代数簇变形的范畴Milnor纤维，描述其镜像对称解释，并找到几何亏格为零的代数曲面的模的应用，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进行评估，被认为值得支持审查标准。