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Derived Categories, Hodge Theory, and Birational Geometry

派生范畴、霍奇理论和双有理几何

基本信息

批准号：
2002709
负责人：
Alexander Perry
金额：
$ 11.36万
依托单位：
Institute For Advanced Study
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2021-01-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2002709&HistoricalAwards=false
关键词：
Derived Categories Hodge Theory Birational

项目摘要

Algebraic geometry is the study of the geometric objects -- called algebraic varieties -- defined by systems of polynomial equations. A fundamental problem is to classify algebraic varieties, i.e. to determine when one can be transformed into another using algebraic functions. The main theme of this project is to study the classification problem using certain algebraic invariants (derived categories and Hodge structures), which can be thought of as sophisticated "linear approximations" to algebraic varieties. These invariants have connections to many fields, ranging from number theory to symplectic geometry and high energy physics. The project has three related parts. The first is to use Bridgeland stability conditions to prove results about the geometry and period mappings of Fano varieties; this relies on a newly developed notion of stability conditions in families, and the existence of noncommutative K3 surfaces in the derived categories of certain Fano varieties. The second part is to construct more examples of noncommutative K3 surfaces, and to further develop the theory of homological projective geometry (which gives a powerful tool for studying noncommutative varieties in general). The third part is to study geometric problems suggested by the first two parts, concerning the rationality of algebraic varieties and the construction of holomorphic symplectic varieties.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.

代数几何是研究由多项式方程组定义的几何对象--代数簇。一个基本问题是分类代数簇，即确定何时可以使用代数函数将一个代数簇变换为另一个代数簇。该项目的主题是研究使用某些代数不变量（导出类别和Hodge结构）的分类问题，这些代数不变量可以被认为是代数簇的复杂“线性近似”。这些不变量与许多领域都有联系，从数论到辛几何和高能物理。该项目有三个相关部分。首先是使用Bridgeland稳定性条件来证明Fano品种的几何和周期映射的结果;这依赖于一个新发展的概念的稳定性条件的家庭，和存在的非交换K3表面的衍生类别的某些Fano品种。第二部分是构造更多的非交换K3曲面的例子，进一步发展同调射影几何理论（它为研究一般的非交换簇提供了一个强有力的工具）。第三部分是研究前两部分提出的几何问题，涉及代数簇的合理性和全纯辛簇的构造。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

期刊论文数量（4）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Stability conditions in families

DOI：
10.1007/s10240-021-00124-6
发表时间：
2019-02
期刊：
Publications mathématiques de l'IHÉS
影响因子：
0
作者：
Arend Bayer;Mart'i Lahoz;Emanuele Macrì;H. Nuer;Alexander Perry;P. Stellari
通讯作者：
Arend Bayer;Mart'i Lahoz;Emanuele Macrì;H. Nuer;Alexander Perry;P. Stellari

The integral Hodge conjecture for two-dimensional Calabi–Yau categories