Compact Moduli of Algebraic Varieties

代数簇的紧模

• 批准号: 1902157
• 负责人: Valery Alexeev
• 金额: $ 34.97万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902157&HistoricalAwards=false
• 关键词: Compact; Moduli; Algebraic; Varieties

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.

该奖项支持代数几何的研究，代数几何是数学的一个核心分支，旨在从实践和概念上理解多变量多项式方程组的解。代数几何在数论、拓扑和分析等其他数学领域以及物理学、生物学、密码学和工程学中都有重要的应用。包括研究生在内的年轻研究人员将参与该项目。 PI 将致力于围绕代数簇的退化及其模空间的函数、几何上有意义的紧缩等各种主题开展工作。该项目的核心部分是研究 K-平凡簇的模空间函子紧致化，例如 K3 表面、超卡勒簇、Calabi-Yau 簇和阿贝尔簇，包括这些紧致化的明确描述，以及它们的诸如 Kodaira 维度和规范模型等属性。一个单独的项目是关于原木表面的体积。该奖项反映了 NSF 的法定使命，并且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

Compactifications of moduli of elliptic K3 surfaces: Stable pair and toroidal

椭圆 K3 曲面模量的紧化：稳定副和环形

• DOI: 10.2140/gt.2022.26.3525
• 发表时间: 2022
• 期刊: Geometry & Topology
• 影响因子: 2
• 作者: Alexeev, Valery;Brunyate, Adrian;Engel, Philip
• 通讯作者: Engel, Philip

Hodge classes on the moduli space of $W(E_6)$-covers and the geometry of $\mathcal{A}_6$

$W(E_6)$ 模空间上的 Hodge 类覆盖和 $mathcal{A}_6$ 几何

• DOI: 10.4310/pamq.2022.v18.n4.a1
• 发表时间: 2022
• 期刊: Pure and Applied Mathematics Quarterly
• 影响因子: 0.7
• 作者: Alexeev, Valery;Donagi, Ron;Farkas, Gavril;Izadi, Elham;Ortega, Angela
• 通讯作者: Ortega, Angela

ADE surfaces and their moduli

ADE 表面及其模量

• DOI: 10.1090/jag/762
• 发表时间: 2021
• 期刊: Journal of Algebraic Geometry
• 影响因子: 1.8
• 作者: Alexeev, Valery;Thompson, Alan
• 通讯作者: Thompson, Alan

The Flex Divisor of a K3 Surface

K3 Surface 的 Flex 除数

• DOI: 10.1093/imrn/rnac089
• 发表时间: 2022
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Alexeev, Valery;Engel, Philip
• 通讯作者: Engel, Philip