On K-stability of certain Fano 3-folds

The existence of Kähler-Einstein (KE) metrics on Fano manifolds is a long-standing problem in algebraic geometry. Since the solution to the Yau-Tian-Donaldson conjecture, the existence of a KE metric on a Fano manifold $X$ has been proven to be equivalent to the K-polystabilty of $X$. In particular, the problem of the existence of a KE metric has been translated in algebro-geometric terms. In my talk, I will explain how the Abban-Zhuang theory can be used to prove K-stability of Fano varieties and apply it to prove the K-stability of certain Fano 3-folds obtained as blow-up of $\mathbb P^3$ in a curve. Everything is based on a joint work with Tiago Duarte Guerreiro and Nivedita Viswanathan.