Birationally rigid Fano hypersurfaces

We prove that a smooth Fano hypersurface $V=V_M\subset\mathbb P^M$, $M\geqslant 6$, is birationally superrigid. In particular, it cannot be fibred into uniruled varieties by a non-trivial rational map, and every birational map of $V$ onto a minimal Fano variety of the same dimension is a biregular isomorphism. The proof is based on the method of maximal singularities combined with the connectedness principle of Shokurov and Kollar.