Birational rigidity and K-stability of Fano hypersurfaces with ordinary double points

TL;DR

This paper proves that Fano hypersurfaces with ordinary double points in high dimensions are both birationally superrigid and K-stable, leading to the existence of weak Kähler–Einstein metrics.

Contribution

It extends previous results by establishing birational superrigidity and K-stability for hypersurfaces with isolated double points in dimensions five and higher.

Findings

Hypersurfaces of degree n+1 in P^{n+1} with ordinary double points are birationally superrigid.

Such hypersurfaces are K-stable.

They admit weak Kähler–Einstein metrics.

Abstract

Extending previous results, we prove that for $n \ge 5$ all hypersurfaces of degree $n+1$ in ${\mathbb P}^{n+1}$ with isolated ordinary double points are birational superrigid and K-stable, hence admit a weak K\"ahler--Einstein metric.