Birational superrigidity and K-stability of singular Fano complete intersections

TL;DR

This paper develops an inductive approach to prove birational superrigidity and K-stability for certain singular Fano complete intersections, especially hypersurfaces with controlled singularities, extending known results to higher dimensions.

Contribution

It introduces an inductive method leveraging lower-dimensional data to establish birational superrigidity and K-stability for singular Fano varieties, including hypersurfaces with ordinary singularities.

Findings

Hypersurfaces of degree n+1 in P^{n+1} with mild singularities are birationally superrigid and K-stable for large n.

Established an adjunction-type result for local volumes of singularities.

Provided conditions under which singular Fano complete intersections exhibit stability properties.

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.