Birational superrigidity and K-stability of Fano complete intersections of index 1

Abstract

We prove that every smooth Fano complete intersection of index $1$ and codimension $r$ in ${\mathbb{P}}^{n+r}$ is birationally superrigid and K-stable if $n\ge 10r$. We also propose a generalization of Tian’s criterion of K-stability and, as an application, prove the K-stability of the complete intersection of a quadric and a cubic in ${\mathbb{P}}^5$. In the appendix (written jointly with C. Stibitz), we prove the conditional birational superrigidity of Fano complete intersections of higher index in large dimension.