Hyper-Kahler Geometry via Lagrangian Fibrations and Symplectic Resolutions

代数几何与数学的许多领域都有联系，包括拓扑学、微分几何、数论、表示论、组合学和微分方程理论。在过去的20年里，也发现了与弦理论的重要联系。代数几何是对代数变量的研究：几何对象可以被描述为满足一组多项式方程的点的集合。该领域的目标之一是对代数变量进行分类。这可以通过首先将离散不变量与代数变量关联起来，然后用给定的不变量集研究所有代数变量来实现。在代数几何和微分几何中使用的一个基本不变量是第一个陈氏类。代数变量可以根据这个不变量的正性质（或不正性质）划分为若干类。这些类中最重要的一类是第一陈氏类等于零的变种。这些变体在物理学和微分几何中也起着至关重要的作用。通过这个项目，PI的目标是提高我们对hyper-K\ ahler流形的认识，这些流形与复环面和Calabi-Yau流形一起，是具有平凡第一陈氏类的变种的基石之一。更具体地说，PI计划在以下方向开展研究：研究超kahler流形与三次四倍流形之间的关系，改进拉格朗日纤维的现有知识，利用拉格朗日纤维扩展我们对已知例子的知识，并对辛分辨率进行系统的研究。这些研究路线建立在PI过去的工作以及该领域的最新进展之上。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。

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.