On the K-stability of Fano varieties and anticanonical divisors

By Kento Fujita, Y. Odaka. Tohoku mathematical journal, 2016

172 citations recorded by the source corpus.

We apply a recent theorem of Li and the first author to give some criteria for the K-stability of Fano varieties in terms of anticanonical Q-divisors. First, we propose a condition in terms of certain anticanonical Q-divisors of given Fano variety, which we conjecture to be equivalent to the K-stability. We prove that it is at least sufficient condition and also relate to the Berman-Gibbs stability. We also give another algebraic proof of the K-stability of Fano varieties which satisfy Tian's alpha invariants condition.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dc2fdd042e30c31e68256ec311778080f86d7e04

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In Applied Mathematics

1. Chi Li
2. Chenyang Xu
3. Mattias Jonsson
4. Yuchen Liu
5. H. Blum
6. S. Boucksom
7. Cristiano Spotti
8. Robert J. Berman

In Geometry and Topology

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

Citations in the corpus

• A non-Archimedean approach to K-stability (2018, methodology): “Now Theorem 5.16 below shows that δ ( − K X ) agrees with the invariant considered in [FO16, BlJ17].”
• Openness of K-semistability for Fano varieties (2019, methodology): “n{δ(Xb,∆b),1} are constructible and lower semicontinuous. Recall that the α-invariant of a log Fano pair (X,∆) was introduced in [Tia87] and the stability threshold (also known as the δ-invariant) in [FO18]. It was shown in [FO18,BJ17] that δ(X,∆) ≥ 1 if and only if (X,∆) is K-semistable, based on t”
• Explicit Gromov-Hausdorff compactifications of moduli spaces of K\"ahler-Einstein Fano manifolds (2017, result): “s of Ka¨hler-Einstein Fano manifolds. By the result in [13] this amounts to verify the K-stability for a given Fano manifold. There have been important recent progresses towards this, see for example [28, 26, 27, 51] as well as older results based on α-invariant/log canonical threshold and multiplie”
• K-stability of Fano varieties: an algebro-geometric approach (2020, methodology): “first approach dated back to the invention of α-invariant in [Tia87] (and also [OS12]), which gave a sufficient condition. Then after the δ-invariant (see Section 3.2) was defined in a similar fashion in [FO18], people have developed a number of ways to estimate δ(X) for a Fano variety X, see e.g. [Fuj1”
• Thresholds, valuations, and K-stability (2017, background): “o and L= K X. The second invariant measures the \average" singularities and was introduced by Fujita and Odaka in the Fano case, where it is relevant for K-stability, see [FO18, PW16]. Following [FO18] we say that an eective Q-divisor D˘ Q Lon Xis of m-basis type, where m1, Date: February 11, 2”
• K-stability of Fano varieties via admissible flags (2020, unknown): “The proof is based on the following lemma, which is known to imply Tian’s criterion (this is the strategy used in [24]).”