On the global log canonical threshold of Fano complete intersections

TL;DR

This paper proves that generic Fano complete intersections with certain index, codimension, and degree conditions have a global log canonical threshold of 1, implying they admit Kähler-Einstein metrics, and extends previous results to a larger class.

Contribution

It improves the bounds on the dimension for which Fano complete intersections have a log canonical threshold of 1, expanding the class of varieties known to admit Kähler-Einstein metrics.

Findings

Global log canonical threshold equals 1 under relaxed conditions.

Fano complete intersections admit Kähler-Einstein metrics.

Existence of Kähler-Einstein metrics for new families of Fano complete intersections.

Abstract

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.