Compact Moduli of Algebraic Varieties

该奖项支持代数几何的研究，代数几何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。代数几何在其他数学领域也有重要的应用，如数论、拓扑学和分析，以及物理、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这个项目。PI将围绕代数簇的退化和模空间的函数化、几何意义紧化等一系列主题展开工作。该项目的中心部分是研究K-平凡簇，如K3曲面，Hyperkahler簇，Calabi-Yau簇和Abel簇的模空间的函数紧化，包括这些紧化的显式描述，以及它们的性质，如Kodaira维度和规范模型。另一个单独的项目是关于原木表面的体积。这个奖项反映了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。

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.