Symplectic cohomology and quantum cohomology of Fano manifolds

Fano流形的辛上同调和量子上同调

• 批准号: 2306204
• 负责人: Daniel Pomerleano
• 金额: $ 12.34万
• 依托单位: University of Massachusetts Boston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306204&HistoricalAwards=false
• 关键词: Symplectic; cohomology; quantum; Fano; manifolds

Symplectic geometry is the study of spaces that are locally modeled on phase spaces from classical mechanics. While symplectic manifolds have no local invariants, they do have interesting global ones. A major part of this project is concerned with “Gromov-Witten” invariants, which aim to probe a symplectic manifold by studying maps from two-dimensional surfaces into the manifold that satisfy an appropriate partial differential equation. These invariants are very powerful, but it can be quite difficult to get good control over them because of their non-local nature. One key goal of the project is to prove new fundamental facts about Gromov-Witten invariants on certain symplectic manifolds that are defined by polynomial equations. The general strategy the PI will use is to “cut open” the manifold along a divisor. The corresponding invariants of the divisor complement turn out to be very tractable and also provide a stepping stone towards understanding the Gromov-Witten invariants of the original space. The PI will continue to serve as a mentor to high school students at MIT's Research Science Institute, and will organize a workshop on homological mirror symmetry for graduate students. He will also continue to work on developing a Master’s program in mathematics at the University of Massachusetts Boston. Specifically, in the main strand of the project, the PI will prove that the quantum connection on a Fano manifold with a smooth anti-canonical divisor has a singularity of unramified exponential type. The strategy is to view the quantum cohomology of the Fano manifold as a deformation of the symplectic cohomology of the complement of the divisor. The symplectic cohomology of the complement can in turn be studied via the wrapped Fukaya category, allowing one to bring tools from noncommutative geometry to bear. In a different direction, the PI will build on his previous work to relate the symplectic cohomology of an affine log Calabi-Yau variety to certain intrinsic mirror algebras recently constructed by algebraic geometers. The PI will then go on to study homological mirror symmetry for intrinsic mirror pairs using a combination of symplectic and categorical techniques also developed in previous work.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.

辛几何是对经典力学相空间局部建模的空间的研究，虽然辛流形没有局部不变量，但它们确实有有趣的全局不变量，该项目的一个主要部分涉及“Gromov-Witten”不变量。旨在通过研究从二维表面到满足适当偏微分方程的流形的映射来探测辛流形。这些不变量非常强大，但要获得良好的效果可能相当困难。该项目的一个关键目标是证明有关由多项式方程定义的某些辛流形上的 Gromov-Witten 不变量的新基本事实。沿着除数“开”流形 除数补集的相应不变量被证明是非常容易处理的，并且也为理解原始的 Gromov-Witten 不变量提供了一个垫脚石。 PI 将继续担任麻省理工学院科学研究所高中生的导师，并将为研究生组织同调镜像对称研讨会，他还将继续致力于在大学开发数学硕士课程。具体来说，在该项目的主线中，PI 将证明具有平滑反正则除数的 Fano 流形上的量子连接具有无分支指数类型的奇点。策略是查看量子。法诺流形的上同调是除数补集的辛上同调。补集的辛上同调又可以通过包裹的 Fukaya 范畴来研究，从而允许人们将非交换几何中的工具应用到不同的领域。方向，PI 将在他之前的工作基础上将仿射对数 Calabi-Yau 簇的辛上同调与最近由然后，PI 将继续使用先前工作中开发的辛和分类技术的组合来研究本征镜像对的同调镜像对称性。该奖项反映了 NSF 的法定使命，并通过使用基金会的评估被认为值得支持。智力价值和更广泛的影响审查标准。