On the global log canonical threshold of Fano complete intersections (Q253170)

The log canonical threshold of a variety is of basic important in the study of the birational geometry of uniruled varieties. For example, acconding to [\textit{J.P. Demailly} and \textit{J. Kollár}, Ann. Sci. Éc. Norm. Supér. (4) 34, No. 4, 525--556 (2001; Zbl 0994.32021); \textit{A.M. Nadel}, Ann. Math. (2) 132, No. 3, 549--596 (1990; Zbl 0731.53063); \textit{G. Tian}, Invent. Math. 89, 225--246 (1987; Zbl 0599.53046)], if the log canonical threshold of \(X\) is greater than \(\frac{\dim X}{\dim X+1}\), there exists a Kähler--Einstein metric on \(X\). The paper under consideration improves the bounds on global log canonical thresholds of generic Fano complete intersections, and thereby establiishes as a corollary the existence of Kähler-Einstein metrics on several families of complete intersection Fanos in \(\mathbb P^{24}\); for example, it is established that a \((2,5,5,5,7)\) complete intetrsection in \(\mathbb P^{24}\) admits a Kähler-Einstein metric. This extends results from [\textit{A. Pukhlikov}, Manuscr. Math. 121, No. 4, 491--526 (2006, Zbl 1135.14032); \textit{A. Pukhlikov}, Math. Notes 88, No. 4, 552--558 (2010, Zbl 1244.14029)]. The main result is that a general Fano complete intersection of index \(1\) and codimension \(k\) in \(\mathbb P^{M+k}\) is equal to \(1\) if \(M \geq 3k+4\) one of the defining equations has degree at least \(8\).