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We prove that every smooth Fano complete intersection of index $1$ andcodimension $r$ in $\mathbb{P}^{n+r}$ is birationally superrigid and K-stableif $n\ge 10r$. We also propose a generalization of Tian's criterion ofK-stability and, as an application, prove the K-stability of the completeintersection of a quadric and a cubic in $\mathbb{P}^5$. In the appendix(written jointly with C. Stibitz), we prove the conditional birationalsuperrigidity of Fano complete intersections of higher index in largedimension.

Birational superrigidity and K-stability of Fano complete intersections of index one (with an appendix written jointly with Charlie Stibitz)