Birational rigidity and alpha invariants of Fano varieties

TL;DR

This paper constructs examples of Fano varieties with nearly maximal alpha invariants and demonstrates the existence of G-birationally super-rigid Fano varieties with arbitrarily small alpha invariants, advancing understanding of their birational properties.

Contribution

It establishes the existence of Fano varieties with alpha invariants arbitrarily close to 1/2 and G-birationally super-rigid Fano varieties with arbitrarily small alpha invariants.

Findings

Existence of Fano varieties with alpha invariants approaching 1/2.

Existence of G-birationally super-rigid Fano varieties with arbitrarily small alpha invariants.

Demonstrates the range of alpha invariants in relation to birational rigidity.

Abstract

We prove that for every $\epsilon>0$, there is a birationally super-rigid Fano variety $X$ such that $\frac{1}{2}\leqslant\alpha(X)\leqslant \frac{1}{2}+\epsilon$. Also we show that for every $\epsilon>0$, there is a Fano variety $X$ and a finite subgroup $G\subset\mathrm{Aut}(X)$ such that $X$ is $G$-birationally super-rigid, and $\alpha_G(X)<\epsilon$.