Birational Superrigidity and K-Stability of Singular Fano Complete Intersections

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 $n\gg 0$. As part of the argument, we also establish an adjunction-type result for local volumes of singularities.