K-stability of cubic threefolds

Abstract

We prove the K-moduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler-Einstein metric as well as provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension three of the volumes of kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of three dimensional canonical and terminal singularities, which was established during the study of the explicit three dimensional minimal model program.

Interaction Between Singularity Theory and the Minimal Model Program

“…For instance, by revisiting the classification results of three dimensional singularities, we show that vol(x, X) ≤ 16 if x ∈ X is singular and the equality holds if and only if x ∈ X is a rational double point (see [LX17b]). As a consequence, we could solve the question on the existence of Kähler-Einstein metrics for cubic threefolds.…”

“…Conjecture 4.10 is verified in [LX17a] for x ∈ X ∞ where X ∞ is a Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds. Since any point will have its volume less 16 or equal to n n (see [LX17b,Appendix]), Conjecture 4.10 implies that for a klt singularity x ∈ (X, ∆), vol(x, X) ≤ n n /|π loc 1 (x, X)|,…”

“…A degeneration argument in [Liu17] implies that the volume function should be lower semi-continuous. A special case we know is that the volume of any ndimensional klt non-smooth point is always less than n n (see [LX17b,Appendix]).…”

Big and Nef Classes, Futaki Invariant and Resolutions of Cubic Threefolds

“…Here we do not need to recall the full definition of K-stability. Instead it is enough to recall that it is a GIT stability notion for polarized varieties (when no polarization is specified, it is assumed to be the anticanonical bundle), and that the Hilbert-Mumford criterion for K-stability implies the following elementary To begin with observe that by results of Allcock [1] and Liu-Xu [24] we have the following clear picture of K-stability of cubic threfolds. • X is K-stable if and only if it is smooth or it has isolated singularities of type A k with k ≤ 4.…”

“…In particular the case of cubic threefolds is discussed in Section 4. Thanks to Allcock [1] and Liu-Xu [24], as recalled in Theorem 4.2, Kpolystable cubic threefolds are now classified, and for example among them it appears the zero locus X of…”