Hyper-Kahler Geometry via Lagrangian Fibrations and Symplectic Resolutions

通过拉格朗日纤维和辛分辨率的超卡勒几何

• 批准号: 1801818
• 负责人: Giulia Sacca
• 金额: $ 19万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2019-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801818&HistoricalAwards=false
• 关键词: Hyper; Kahler; Geometry; via; Lagrangian

Algebraic Geometry has connections to many areas in mathematics, including topology, differential geometry, number theory, representation theory, combinatorics and the theory of differential equations. Over last 20 year important connections with string theory were discovered as well. Algebraic Geometry is the study of algebraic varieties: geometric objects that can be described as the collections of points satisfying a set of polynomial equations. One of the aims of the field is to classify algebraic varieties. This can be done by first associating discrete invariants to algebraic varieties and then studying all algebraic varieties with a given set of invariants. A basic invariant used in algebraic geometry, as well as in differential geometry, is the first Chern class. Algebraic varieties can be divided into classes according to the positivity properties (or lack thereof) of this invariant. One of the most important of these classes is that of varieties with first Chern class equal to zero. These varieties have a crucial role also in physics and in differential geometry. With this project the PI aims to advance our knowledge of hyper-K\"ahler manifolds which are, together with complex tori and Calabi-Yau manifolds, one of the building blocks of varieties with trivial first Chern class.More specifically, the PI plans to carry out the research in following directions: investigating the relation between hyper-Kahler manifolds and cubic fourfolds, improving the current knowledge of Lagrangian fibrations, using Lagrangian fibrations to expand our knowledge of the known examples, and carrying out a systematic study of symplectic resolutions. These lines of research build on past work of the PI as well as on recent progress in this field.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.

代数几何形状与数学许多领域有联系，包括拓扑，差异几何，数量理论，表示理论，组合和差分方程理论。在过去的20年中，也发现了与弦理论的重要联系。代数几何形状是代数品种的研究：可以描述为满足一组多项式方程的点的几何对象。该领域的目的之一是对代数品种进行分类。这可以通过将离散不变性与代数品种相关联，然后使用给定的一组不变式研究来完成。代数几何形状以及差异几何形状中使用的基本不变性是第一类。代数品种可以根据该不变的阳性特性（或缺乏）分为类。这些类别中最重要的类别之一是具有第一类等于零的品种。这些品种在物理和差异几何形状中也具有至关重要的作用。 With this project the PI aims to advance our knowledge of hyper-K\"ahler manifolds which are, together with complex tori and Calabi-Yau manifolds, one of the building blocks of varieties with trivial first Chern class.More specifically, the PI plans to carry out the research in following directions: investigating the relation between hyper-Kahler manifolds and cubic fourfolds, improving the current knowledge of Lagrangian fibrations, using拉格朗日纤维化以扩大我们对已知示例的了解，并对这些研究的系统进行系统的研究。