K-stable Fano 3-folds

爱因斯坦度量是复流形上的一种特殊类型的度量，在几何、拓扑和物理中有重要的应用。近年来，在理解Fano流形上爱因斯坦度量的存在性和性质方面取得了显著的进展。特别地，这种度量的存在是由一个称为K-稳定性的代数条件来检测的。然而，验证给定Fano上的K-稳定性一直是一个谜，直到阿班和庄提出了一种新的方法。利用它，切尔佐夫和他的合作者检验了每个Fano 3-折叠家族(它们被分成105个家族)中普通成员的K-稳定性。在过去的两年里，验证所有光滑元素的K-(聚)稳定性，当一个普通的元素承认爱因斯坦度量时，已经有了很多的活动。然而，尽管这一主题取得了令人难以置信的成功，但某些技术障碍目前阻碍了人们对这一问题的充分理解。结合阿班和切尔佐夫的专业知识，以及英国、美国和日本的现有活动，我们的目标是打破僵局，在理解爱因斯坦关于Fano 3折的度量的存在和性质方面取得飞跃。

Einstein metrics are a special type of metric on complex manifolds that have important applications in geometry, topology, and physics. In recent years, there has been significant progress in understanding the existence and properties of Einstein metrics on Fano manifolds. Particularly, the existence of such metric is detected by an algebraic condition known as K-stabiliyty. However, verifying K-stability on a given Fano had remained a mystery until a new methodology was proposed by Abban and Zhuang. Using it, Cheltsov and collaborators have examined K-stability for generic members in each family of Fano 3-folds (they are classified into 105 families). Verifying K-(poly)stability for all smooth elements, when a generic one admits an Einstein metric, has seen much activity in the past two years. However, despite the subject being incredibly successful, certain technical obstacles prevent a full understanding at the moment. Combining expertise of Abban and Cheltsov, together with existing activities in the UK, USA, and Japan, we aim to remove the deadlocks and make a leap in understanding of the existence and properties of Einstein metrics on Fano 3-folds.