On the global log canonical threshold of Fano complete intersections

We show that the global log canonical threshold of generic Fano complete intersections of index 1 and codimension $k$ in ${\mathbb P}^{M+k}$ is equal to 1 if $M\geqslant 3k+4$ and the highest degree of defining equations is at least 8. This improves the earlier result where the inequality $M\geqslant 4k+1$ was required, so the class of Fano complete intersections covered by our theorem is considerably larger. The theorem implies, in particular, that the Fano complete intersections satisfying our assumptions admit a K\" ahler-Einstein metric. We also show the existence of K\" ahler-Einstein metrics for a new finite set of families of Fano complete intersections.