Geometry and Topology of Holomorphic Symplectic Varieties

全纯辛簇的几何和拓扑

• 批准号: 2134315
• 负责人: Junliang Shen
• 金额: $ 15.92万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2023-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2134315&HistoricalAwards=false
• 关键词: Geometry; Topology; Holomorphic; Symplectic; Varieties

Algebraic geometry is a subject studying algebraic varieties, which are spaces described by solutions of polynomial equations. This subject started with the work of the ancient Greeks about ellipses, parabolas, and hyperbolas, and now it has grown to be a modern subject sharing many rich connections with physics and computer science. The research supported by this NSF award will focus on an important class of algebraic varieties, called the holomorphic symplectic varieties, that lie at the crossroads of geometry, representation theory, and mathematical physics. A better understanding of these varieties will bring deep insights in both algebraic geometry and its related fields.In this project, the PI will build connections between compact and noncompact holomorphic symplectic varieties with a focus on the P=W conjecture, which is a central conjecture connecting Hitchin's integrable systems and Hodge theory. Compact holomorphic symplectic varieties, which are higher dimensional generalizations of K3 surfaces, have been studied intensively for decades. A systematic way to compare compact and noncompact geometries, and to introduce new techniques to Hitchin systems from compact holomorphic symplectic varieties will bring fresh perspectives and fruitful outcomes to the P=W conjecture and related questions.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.

代数几何是一门研究代数簇的学科，代数簇是用多项式方程的解来描述的空间。这门学科始于古希腊人关于椭圆、抛物线和双曲线的研究，现在它已经发展成为一门与物理学和计算机科学有许多丰富联系的现代学科。该NSF奖支持的研究将集中在一类重要的代数簇上，称为全纯辛簇，它们位于几何学、表示论和数学物理的十字路口。在这个项目中，PI将在紧全纯辛簇和非紧全纯辛簇之间建立联系，重点是P=W猜想，这是连接Hitchin可积系统和Hodge理论的一个中心猜想。紧全纯辛簇是K3曲面的高维推广，几十年来一直被广泛研究。系统地比较紧几何和非紧几何，并从紧全纯辛变体向Hitchin系统引入新技术，将为P=W猜想和相关问题带来新的视角和丰硕的结果。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。

项目成果

On intersection cohomology and Lagrangian fibrations of irreducible symplectic varieties

• DOI: 10.1090/tran/8592
• 发表时间: 2021-08
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Camilla Felisetti;Junliang Shen;Qizheng Yin

Generators for the cohomology ring of the moduli of 1-dimensional sheaves on $\mathbb{P}^2$

• DOI: 10.14231/ag-2023-017
• 发表时间: 2022-04
• 期刊: Algebraic Geometry
• 影响因子: 1.5
• 作者: Weite Pi;Junliang Shen

Perverse-Hodge complexes for Lagrangian fibrations

• DOI: 10.46298/epiga.2023.9617
• 发表时间: 2022-01
• 期刊: Épijournal de Géométrie Algébrique
• 作者: Junliang Shen;Qizheng Yin

Endoscopic decompositions and the Hausel–Thaddeus conjecture

内窥镜分解和豪塞尔·撒迪厄斯猜想

• DOI: 10.1017/fmp.2021.7
• 发表时间: 2021
• 期刊: Pi
• 作者: Maulik, Davesh;Shen, Junliang
• 通讯作者: Shen, Junliang

On the intersection cohomology of the moduli of SLn$\mathrm{SL}_n$‐Higgs bundles on a curve

曲线上 SLn$mathrm{SL}_n$−希格斯丛模的交交上同调

• DOI: 10.1112/topo.12250
• 发表时间: 2022
• 期刊: Journal of Topology
• 影响因子: 1.1
• 作者: Maulik, Davesh;Shen, Junliang
• 通讯作者: Shen, Junliang