Birational superrigidity and K-stability of projectively normal Fano manifolds of index one

Abstract

We prove that every projectively normal Fano manifold in $\mathbb{P}^{n+r}$ of index $1$, codimension $r$ and dimension $n\geq 10r$ is birationally superrigid and K-stable. This result was previously proved by Zhuang under the complete intersection assumption.