CAREER: Compact Hyper-Kahler manifolds and Lagrangian fibrations

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。代数几何是研究代数的变种，用多项式方程描述的几何对象。在研究代数变量时，通常根据代数变量的几何性质将其划分为不同的类别是比较方便的。代数变种的一个基本不变量第一类陈氏不变量为零，给出了一类重要的代数变种。一个来自20世纪80年代的基本结果，称为Beauville-Bogomolov定理，指出具有零第一陈氏类的光滑紧代数变体有三种构造块：复环面、严格Calabi-Yau流形和不可约全纯simic流形。这个项目的重点是这三个构建模块中的最后一个，这是传统上研究最少的。由于微分几何中的一些基本定理，不可约全纯辛流形承认一个称为hyper-Kähler度规的特殊度规。全纯辛流形的几何不仅与代数和微分几何有关，而且与表示理论和数学物理有关。作为该项目的一部分，PI将组织活动，以加强hyper-Kähler在美国的研究社区，为来自数学领域代表性不足的少数民族的本科生举办活动，并在哥伦比亚大学周围更广泛的社区举办K-12活动。K3曲面是研究最多的代数曲面之一，不可约辛流形是它们的高维类似物。在这个类比中，紧致hyper-Kähler流形上的拉格朗日颤振是椭圆K3曲面的自然推广。与辛分辨一起，拉格朗日颤振为研究、分类和构造这类流形提供了最有力的手段。PI旨在通过系统地研究拉格朗日振动来推进（紧凑）hyper-Kähler流形的现有知识。更具体地说，PI将引入新的技术来紧化拟射影拉格朗日纤维，并将研究拉格朗日纤维紧致hyper-Kähler流形的上同调、衍生范畴和Chow群。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。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Algebraic geometry is the study of algebraic varieties, geometric objects described by polynomial equations. To study algebraic varieties, it is often convenient to divide them into different classes according to their geometric properties. An important class of algebraic varieties is given by those whose first Chern class, a basic invariant of algebraic varieties, is zero. A fundamental result from the 1980s, called the Beauville-Bogomolov theorem, states that there are exactly three kinds of building blocks for smooth compact algebraic varieties with zero first Chern class: complex tori, strict Calabi-Yau manifolds, and irreducible holomorphic symplectic manifolds. This project focuses on the last of these three building blocks, which traditionally has been the least studied. Thanks to some fundamental theorems in differential geometry, irreducible holomorphic symplectic manifolds admit a special metric called a hyper-Kähler metric. The geometry of holomorphic symplectic manifolds is relevant not only to algebraic and differential geometry, but also to representation theory and mathematical physics. As part of this project, the PI will organize activities to strengthen the hyper-Kähler research community in the United States, activities for undergraduate students from under-represented minorities in math, and K-12 activities in the broader community around Columbia University.K3 surfaces constitute one of the most studied types of algebraic surfaces, and irreducible symplectic manifolds are arguably their higher dimensional analogues. In this analogy, Lagrangian fibrations on compact hyper-Kähler manifolds are the natural generalizations of elliptic K3 surfaces. Together with symplectic resolutions, Lagrangian fibrations provide one the strongest means to study, classify, and construct this class of manifolds. The PI aims to advance the current knowledge of (compact) hyper-Kähler manifolds through the systematic study of Lagrangian fibrations. More specifically, the PI will introduce new techniques to compactify quasi-projective Lagrangian fibrations and will study the cohomology, derived categories, and Chow groups of Lagrangian fibered compact hyper-Kähler manifolds.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.