Derived Categories, Hodge Theory, and Birational Geometry

代数几何是研究由多项式方程系统定义的几何对象（称为代数变量）的学科。一个基本问题是对代数变量进行分类，即确定何时可以用代数函数将一个变量变换为另一个变量。该项目的主要主题是研究使用某些代数不变量（衍生类别和Hodge结构）的分类问题，这些不变量可以被认为是代数变量的复杂“线性近似”。这些不变量与许多领域有联系，从数论到辛几何和高能物理。该项目有三个相关部分。首先利用桥地稳定性条件证明了Fano变量的几何映射和周期映射；这依赖于一个新发展的族稳定条件的概念，以及在某些Fano变体的派生范畴中存在非交换K3曲面。第二部分是构造更多的非交换K3曲面的例子，并进一步发展同调射影几何理论（它为研究一般的非交换变量提供了一个有力的工具）。第三部分是研究前两部分提出的几何问题，涉及代数变分的合理性和全纯辛变分的构造。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 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.