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，拓扑和物理学中有重要的应用。近年来，对Fano流形上Einstein度量的存在性和性质的理解有了很大的进展。特别地，这种度量的存在性是通过称为K-稳定性的代数条件来检测的。然而，在Abban和Zhuang提出一种新的方法之前，在给定的Fano上验证K-稳定性仍然是一个谜。Cheltsov及其合作者使用它检查了每个Fano 3-folds家族中通用成员的K稳定性（它们被分为105个家族）。当一个通用的元素允许爱因斯坦度量时，所有光滑元素的POLK-（多边形）稳定性在过去两年中已经有了很多活动。然而，尽管该主题非常成功，但某些技术障碍阻止了目前的全面理解。结合Abban和Cheltsov的专业知识，以及英国，美国和日本的现有活动，我们的目标是消除僵局，并在理解Fano 3-folds上爱因斯坦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.