K-stability of cubic threefolds

By Yuchen Liu. Duke mathematical journal, 2017

86 citations recorded by the source corpus.

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 K\"ahler-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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c1daad4af858749af32404c0a5fd8734f1279ec7

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In Applied Mathematics

1. Chi Li
2. Chenyang Xu
3. H. Blum
4. Yuchen Liu
5. Mattias Jonsson
6. Ziquan Zhuang
7. Xiaowei Wang
8. S. Boucksom

In Geometry and Topology

1. Chenyang Xu
2. Chi Li
3. H. Blum
4. Yuchen Liu
5. Mattias Jonsson
6. Ziquan Zhuang
7. S. Boucksom
8. Kento Fujita

Citations in the corpus

• Thresholds, valuations, and K-stability (2017, background): “ a Q-Fano variety and L= K X, Theorem D generalizes the volume bounds found in [Fuj18] and [Liu18], in which Xis assumed K-semistable, so that (L) 1. These volume bounds were explored in [SS17] and [LX19]. Next we investigate whether the inma in Theorem C are attained. We say that a valuation v 2Val”
• K-stability of Fano varieties via admissible flags (2020, unknown): “However, both criteria apply exclusively to certain Fano varieties of index one, and except in a few sporadic cases, it is unclear how to attack the problem when the required conditions in neither criterion are satisfied; see for example [1, 16, 38, 43].”
• A Guided Tour to Normalized Volume (2018, background): “lystable) cubic threefolds are also K-semi-stable (resp. K-polystable). Our main estimate is in Theorem 5.8, which heavily depends on classifications of canonical threefold singularities. Theorem 5.8 ([LX17b]). Let x ∈X be a 3-dimensional non-smooth klt singularity. Then vol(cx,X) ≤16 and the equalit”
• K-stability of Fano varieties: an algebro-geometric approach (2020, background): “ey ingredient to prove the following theorem, which gives the first example of using the explicit Kmoduli space to find out new smooth K-stable Fano manifolds in dimension at least three. Theorem 10.2 ([LX19]). The GIT moduli of cubic threefolds is the same as the K-moduli. Sketch of the proof. By the”