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--称为代数簇。一个基本的问题是对代数族进行分类，即确定一个代数族何时可以使用代数函数转换为另一个代数族。这个项目的主要主题是研究使用某些代数不变量(派生范畴和Hodge结构)的分类问题，这些代数不变量可以被认为是对代数簇的复杂的“线性逼近”。这些不变量与许多领域有关，从数论到辛几何和高能物理。该项目由三个相关部分组成。首先利用Bridgeland稳定性条件证明了Fano簇的几何映射和周期映射的结果；这依赖于新发展的族稳定性条件的概念，以及在某些Fano簇的派生范畴中非交换K3曲面的存在性。第二部分是构造更多的非对易K3曲面的例子，并进一步发展同调射影几何理论(这为研究非对易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，涉及代数变体的合理性和全纯辛变体的构造。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.