Thresholds, valuations, and K-stability

By H. Blum, Mattias Jonsson. Advances in Mathematics, 2017

182 citations recorded by the source corpus.

Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as another invariant, the stability threshold. The latter generalizes a notion by Fujita and Odaka, and can be used to characterize when a Q-Fano variety is K-semistable or uniformly K-stable. It can also be used to generalize volume bounds due to Fujita and Liu. The two thresholds can be written as infima of certain functionals on the space of valuations on X. When L is ample, we prove that these infima are attained. In the toric case, toric valuations acheive these infima, and we obtain simple expressions for the two thresholds in terms of the moment polytope of L.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/54ff386b30f0fccd71227f66b0ecebd64f1277e2

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In Geometry and Topology

1. Mattias Jonsson
2. S. Boucksom
3. Chi Li
4. Robert J. Berman
5. C. Favre
6. M. Mustaţă
7. H. Blum
8. Yuchen Liu

In Mathematical Physics

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

Citations in the corpus

• K-stability of Fano varieties via admissible flags (2020, unknown): “Equivalently [2], it can also be defined as the limit δ(L) = limm→∞ δm (L), where δm(L) = sup{λ ≥ 0 | (X,Δ + λD) is lc for all m-basis type Q-divisors D ∼Q L}.”
• A non-Archimedean approach to K-stability (2018, methodology): “Further, the Alexander-Taylor capacity T ( x ) coincides with the invariant T in [BlJ17] .”