K-stable Fano 3-folds

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.