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 P M+k is equal to one, if M 2k + 3 and the maximum of the degrees of defining equations is at least 8. This is an essential improvement of the previous results about log canonical thresholds of Fano complete intersections. As a corollary we obtain the existence of Kähler-Einstein metrics on generic Fano complete intersections described above.

On the sharpness of Tian's criterion for K-stability

“…Recall that a smooth P-regular complete intersection X ⊆ P n+r of dimension n ≥ 2r +3 is automatically r-strongly P-regular, we obtain the following immediate corollary in the spirit of [Puk18].…”

“…Proof. This should be well known to experts and is essentially a direct consequence of the work of Pukhlikov [Puk18,§4]. For reader's convenience we sketch the proof.…”

“…For this we recall the regularity condition introduced by Pukhlikov (see e.g. [Puk18] or [Puk13, §3.2]). Let x ∈ X and let f i be the defining equation of F i .…”

“…We also provide a generalization of [Jia17, Theorem 1.2] in Theorem 3.9. In Section 4 we estimate the global log canonical thresholds of general hypersurfaces or complete intersections based on Pukhlikov's work [Puk05,Puk13,Puk18]. Finally in Section 5 we present our main constructions to prove Theorem 1.3.…”

Remarks on Galois rational covers

“…For the double covers of the space P M in [4] a similar improvement was shown: for M 10 the double space, branched over a Zariski general hypersurface of degree 2M with at worst quadratic singularities of rank 4, is divisorially canonical, and the branch hypersurfaces, for which the corresponding double cover is not divisorially canonical, form a set of codimension 1 2 (M − 4)(M − 1) in P(H 0 (P M , O P M (2M))). For a Zariski general non-singular complete intersection of type holds, the divisorial canonicity was shown in [5]. Before that paper, in [6] and [7] the divisorial canonicity was shown for smaller classes of complete intersections of index 1.…”