Canonical and log canonical thresholds of Fano complete intersections

Abstract

It is proved that the global log canonical threshold of a Zariski general Fano complete intersection of index 1 and codimension $k$ in ${\mathbb P}^{M+k}$ is equal to one, if $M\geqslant 2k+3$ and the maximum of the degrees of defining equations is at least 8. This is an essential improvements of the previous results about log canonical thresholds of Fano complete intersections. As a corollary we obtain the existence of K\" ahler-Einstein metrics on generic Fano complete intersections described above.