Compact Moduli of Algebraic Varieties

该奖项支持代数几何的研究，这是数学的一个中心分支，旨在从实践和概念上理解多变量多项式方程组的解。代数几何在数学的其他领域也有重要的应用，例如数论、拓扑学和分析，以及物理学、生物学、密码学和工程学。包括研究生在内的年轻研究人员将参与该项目。PI将围绕代数簇的退化和其模空间的函子、几何意义的紧化等各种主题开展工作。该项目的核心部分是研究K-平凡簇的模空间的函子紧化，如K3曲面、超卡勒簇、卡-丘簇和阿贝尔簇，包括这些紧化的明确描述，以及它们的性质，如科代拉维数和正则模型。该奖项反映了NSF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。

The award supports research in algebraic geometry, a central branch of mathematics which aims to understand, both practically and conceptually, solutions of systems of polynomial equations in many variables. Algebraic geometry has important applications to other fields of mathematics, such as number theory, topology, and analysis, as well as to physics, biology, cryptography, and engineering. Young researchers including graduate students will be involved in this project.The PI will work on a variety of topics centered around degenerations of algebraic varieties and functorial, geometrically meaningful compactifications of their moduli spaces. The central part of the project is the study of functorial compactifications of moduli spaces of K-trivial varieties, such as K3 surfaces, hyperkahler varieties, Calabi-Yau varieties, and abelian varieties, including both explicit descriptions of these compactifications, as well as their properties such as Kodaira dimension and canonical models. A separate project is on the volumes of log surfaces.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.