CAREER: Compact Hyper-Kahler manifolds and Lagrangian fibrations

职业：紧凑超卡勒流形和拉格朗日纤维

• 批准号: 2144483
• 负责人: Giulia Sacca
• 金额: $ 40万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2144483&HistoricalAwards=false
• 关键词: CAREER; Compact; Hyper; Kahler; manifolds

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.

该奖项是根据2021年《美国救援计划法》（公法117-2）全部或部分资助的。代数几何形状是对多项式平衡描述的代数品种的研究。为了研究代数品种，通常可以根据其几何特性将它们分为不同的类别是方便的。一类重要的代数品种由那些Chern类（代数品种的基本不变）为零的人给出。 1980年代的基本结果称为Beauville-Bogomolov定理，指出，恰好有三种构建块，用于平滑紧凑的代数品种，具有零的第一chern类：复杂的Tori，严格的Calabi-yau歧管，以及不可估的Holomorphic Symmetric歧管。该项目着重于这三个构建基块中的最后一个，传统上是研究最少的。感谢您在差异几何形状方面的一些基本定理，不可还原的塑形象征性歧管承认一个特殊的指标，称为Hyper-Kähler指标。全体形态符号歧管的几何形状不仅与代数和差异几何形状有关，还与表示理论和数学物理学有关。作为该项目的一部分，PI将组织活动，以加强美国的超级Kähler研究社区，来自代表性不足的数学少数群体的本科生的活动以及哥伦比亚大学更广泛社区的K-12活动。K3表面表面构成了最典型的综合表面，并且是较高的综合表面，并且构成了较高的综合表面，并且构成了较高的表现。在这种类比中，紧凑的超喀勒歧管上的拉格朗日纤维是椭圆形K3表面的自然概括。 Lagrangian纤维与对称分辨率一起提供了一种强大的手段，可以进行研究，分类和构建这类歧管。 PI旨在通过系统的Lagrangian纤维化研究来促进当前（紧凑）Hyper-Kähler歧管的知识。更具体地说，PI将引入新技术，以压缩准标准的拉格朗日纤维纤维，并将研究共同体，派生类别和组成的Lagrangian纤维紧凑型Hyper-Kähler歧管组的食物组。该奖项通过评估了NSF的法规诚实的支持，对NSF的诚实进行了评估，该奖项已被评估诚实地构成了基础。