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

TL;DR

This paper proves that certain high-dimensional Fano manifolds are both birationally superrigid and K-stable, extending previous results beyond complete intersections.

Contribution

It establishes birational superrigidity and K-stability for projectively normal Fano manifolds of index one without the complete intersection assumption.

Findings

Fano manifolds of dimension at least 10 times their codimension are birationally superrigid.

Such manifolds are also K-stable.

The results generalize previous work by Zhuang.

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.