Birational superrigidity and K-stability

Abstract

We consider two different notions on Fano varieties: birational superrigidity, coming from the study of rationality, and K-stability, which is related to the existence of K\"ahler-Einstein metrics. In the first part, we show that birationally superrigid Fano varieties are also K-stable as long as their alpha invariants are at least $\frac{1}{2}$, partially confirming a conjecture of Odaka-Okada and Kim-Okada-Won. In the second part, we prove the folklore prediction that smooth Fano complete intersections of Fano index one are birationally superrigid and K-stable when the dimension is large. In the third part, we introduce an inductive argument to study the birational superrigidity and K-stability of singular complete intersections and in particular prove an optimal result on the birational superrigidity and K-stability of hypersurfaces of Fano index one with isolated ordinary singularities in large dimensions. Finally we provide an explicit example to show that in general birational superrigidity is not a locally closed property in families of Fano varieties, giving a negative answer to a question of Corti.