Kähler-Einstein metrics on Fano manifolds

Fano 流形上的克勒-爱因斯坦度量

• 批准号: EP/V048619/2
• 负责人: Hamid Abban
• 金额: $ 12.18万
• 依托单位: University of Nottingham
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV048619%2F2
• 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.

现代几何形状的结构是围绕导致多种多样存在规范指标的问题而设计的。一个经典的例子是Riemannian指标，在Riemann表面上具有恒定的高斯曲率。更高的维度模拟引发了希望在给定的歧管上找到“爱因斯坦公制”的希望。当有问题的流形为Kähler时，所需的度量被称为Kähler-Einstein。可以简化歧管以具有正曲率或负曲率，也可以是平坦的。当曲率为负或平坦时，Kähler-Einstein指标的存在，感谢Aubin和Yau的著名作品。在正弯曲情况下，这种度量的存在被阻塞。由于Donaldson等人的开创性工作，在这种情况下，Kähler-Einstein度量的存在取决于基础Fano品种的代数稳定性条件。但是，很难验证给定的Fano品种的这种稳定性条件。基于该领域的一些最新发展，我们旨在生产和调整一种新方法，以检查给定的Fano品种是否为K-（半）稳定。该计划是将此新方法应用于各种挑战示例。

项目成果

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

Stability of fibrations over one-dimensional bases

一维基础上纤维化的稳定性

• DOI: 10.1215/00127094-2022-0025
• 发表时间: 2022
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Abban H

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

K-stability of Fano varieties via admissible flags

• DOI: 10.1017/fmp.2022.11
• 发表时间: 2020-03
• 期刊: Forum of Mathematics, Pi
• 作者: Hamid Abban;Ziquan Zhuang

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