Kähler-Einstein metrics on Fano manifolds

Fano 流形上的克勒-爱因斯坦度量

• 批准号: EP/V048619/1
• 负责人: Hamid Abban
• 金额: $ 25.77万
• 依托单位: Loughborough University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV048619%2F1
• 关键词: hler; Einstein; metrics; Fano; manifolds

The fabric of modern geometry is designed around questions that lead to the existence of canonical metrics on manifolds. A classical example is the Riemannian metrics with constant Gauss curvature on Riemann surfaces. The higher dimensional analogue sparks the hope of finding an "Einstein metric" on a given manifold. When the manifold in question is Kähler, then the desired metric is called Kähler-Einstein. Manifolds can be simplified to have positive or negative curvature, or be flat. The existence of a Kähler-Einstein metric when the curvature is negative or flat is known, thanks to the celebrated work of Aubin and Yau. The existence of such metric is obstructed in the positive curvature case. Due to the pioneering work of Donaldson et al, the existence of a Kähler-Einstein metric in this case is determined by an algebraic stability condition on the underlying Fano variety. However, it is difficult to verify such stability condition for a given Fano variety.Based on some recent developments in the field, we aim to produce and fine-tune a new method to check whether a given Fano variety is K-(semi)stable or not. The plan is to apply this new method to various challenging examples.

现代几何的结构是围绕着导致流形上存在正则度量的问题而设计的。一个经典的例子是黎曼曲面上具有常高斯曲率的黎曼度量。更高维度的类似物激发了在给定流形上找到“爱因斯坦度量”的希望。当所讨论的流形是Kähler时，则所需的度规称为Kähler-Einstein。流形可以被简化为具有正曲率或负曲率，或者是平坦的。由于奥宾和丘的著名工作，当曲率为负或平坦时，凯勒-爱因斯坦度规的存在是已知的。在正曲率的情形下，这种度量的存在性受到阻碍。由于唐纳森等人的开创性工作，在这种情况下凯勒-爱因斯坦度量的存在性是由基础法诺簇上的代数稳定性条件确定的。然而，这是很难验证这样的稳定性条件，对于一个给定的Fano品种的基础上，在该领域的一些最新进展，我们的目标是生产和微调一个新的方法来检查是否一个给定的Fano品种是K-（半）稳定或不。我们的计划是将这种新方法应用于各种具有挑战性的例子。

项目成果

Explicit Birational Geometry of Fano threefold complete intersections

Fano三重完全交集的显式双有理几何

• DOI: 10.48550/arxiv.2301.06481
• 发表时间: 2023
• 作者: Guerreiro T

K-Stability of Log Del Pezzo Hypersurfaces with Index 2

• DOI: 10.1142/s0129167x22500707
• 发表时间: 2022-02
• 期刊: International Journal of Mathematics
• 影响因子: 0.6
• 作者: In-Kyun Kim;Nived Viswanathan;Joonyeong Won

Classification of higher mobility closed-loop linkages

高机动性闭环连杆机构的分类

• DOI: 10.1007/s10231-022-01258-y
• 发表时间: 2022
• 期刊: Annali di Matematica Pura ed Applicata (1923 -)
• 作者: Duarte Guerreiro T

Sarkisov links from toric weighted blowups of $\mathbb{P}^4$ at a point

Sarkisov 从 $mathbb{P}^4$ 的环面加权放大链接到一点

• DOI: 10.4171/pm/2087
• 发表时间: 2022
• 期刊: Portugaliae Mathematica
• 影响因子: 0.8
• 作者: Duarte Guerreiro T

Seshadri constants and K-stability of Fano manifolds

• DOI: 10.1215/00127094-2022-0026
• 发表时间: 2021-01
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Hamid Abban;Ziquan Zhuang