Birationally rigid Fano complete intersections. II

We prove that a generic (in the sense of Zariski topology) Fano complete intersection $V$ of the type $(d_1,\mathrm{.}\mathrm{.}\mathrm{.},d_k)$ in ${\mathbb{P}}^{M+k}$, where $d_1+\mathrm{.}\mathrm{.}\mathrm{.}+d_k=M+k$, is birationally superrigid if $M\ge 7$, $M\ge k+3$ and $\{d_i\}\ge 4$. In particular, on the variety $V$ there is exactly one structure of a Mori fibre space (or a rationally connected fibre space), the groups of birational and biregular self-maps coincide, $\mathop{\rm Bir} V= \mathop{\rm Aut} V$,andthevariety$V$ is non-rational. This fact covers a considerably larger range of complete intersections than the result of [J. reine angew. Math. {\bf 541} (2001), 55-79], which required the condition $M\ge 2k+1$. The paper is dedicated to the memory of Eckart Viehweg.