Birational rigidity and alpha invariants of Fano varieties

Abstract

We prove that for every $\epsilon >0$, there is a birationally super-rigid Fano variety $X$ such that $\frac{1}{2}⩽\alpha (X)⩽\frac{1}{2}+\epsilon$. 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 ${\alpha }_G(X)<\epsilon$.

23 pages. This paper is submitted for publication in a volume on the occasion of the 70th anniversary of Slava Shokurov