Birational superrigidity and K-stability of Fano complete intersections of index 1

Abstract

We introduce an inductive argument for proving birational superrigidity and K-stability of singular Fano complete intersections of index one, using the same types of information from lower dimensions. In particular, we prove that a hypersurface in $\mathbb{P}^{n+1}$ of degree $n+1$ with only ordinary singularities of multiplicity at most $n-5$ is birationally superrigid and K-stable if $n\gg0$. As part of the argument, we also establish an adjunction type result for local volumes of singularities.