K-stability of birationally superrigid Fano varieties

Abstract

We prove that every birationally superrigid Fano variety whose alpha invariant is greater than (resp. no smaller than) 1 2 is K-stable (resp. K-semistable). We also prove that the alpha invariant of a birationally superrigid Fano variety of dimension n is at least 1 n+1 (under mild assumptions) and that the moduli space (if exists) of birationally superrigid Fano varieties is separated.

Birational superrigidity is not a locally closed property

“…We refer to [Tia97,Don02] for the definition of K-stability in the above statement. In our cases, K-stability is a direct consequence of the birational superrigidity by a simple application of [SZ18]. Note that Theorem 1.2 generalizes the results of [Puk00,Puk03] by allowing more general weighted complete intersections and removing the generality assumptions (albeit at the cost of increasing dimensions).…”

“…Indeed, it follows from the definition that X has terminal singularities and therefore by the seminal work of Birkar [Bir16a,Bir16b] on the Borisov-Alexeev-Borisov conjecture, birationally superrigid Fano varieties belong to a bounded family. Moreover, such moduli (if exists) satisfies the valuative criterion of separatedness by a recent result of Stibitz and the author [SZ18]. In addition, birational superrigidity is a constructible condition by [SC11, Corollary 7.8].…”

Higher codimensional alpha invariants and characterization of projective spaces

“…Recent results of [SZ18,Zhu18] can be reinterpreted from the point of view of higher codimensional alpha invariants to give the following result.…”

Algebraic hypersurfaces

“…A serious drawback is that we can prove nonrationality only for hypersurfaces X = ( I a I x I = 0) whose coefficients satisfy countably many conditions of the The recent paper of Zhuang [Zhu18] makes the final step of the Corti approach much easier in higher dimensions. The papers [SZ18,Zhu18,LZ18] contain more general results and applications.…”