K-semistability is equivariant volume minimization

By Chi Li. 2015

203 citations recorded by the source corpus.

This is a continuation to the paper \cite{Li15a} in which a problem of minimizing normalized volumes over $\mathbb{Q}$-Gorenstein klt singularities was proposed. Here we consider the case when there is a $\mathbb{C}^*$-action on $(X,o)$. In particular, we prove that for a $\mathbb{Q}$-Fano variety $V$, the K-semistability of $(V, -K_V)$ is equivalent to the condition that the normalized volume is minimized among $\mathbb{C}^*$-invariant valuations at the canonical valuation ${\rm ord}_V$ on the cone associated to any positive Cartier multiple of $-K_V$. In this case, it's shown that ${\rm ord}_V$ is the unique minimizer among all $\mathbb{C}^*$-invariant quasi-monomial valuations. These results partially confirm parts of the conjecture in \cite{Li15a}. We also get a characterization of K-semistability using divisorial valuations of $\mathbb{C}(V)$.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7061df6c1646cc609203b5be76ff47bceeb9e900

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

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

In Geometry and Topology

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

Citations in the corpus

• Minimizing normalized volumes of valuations (2015, background): “ ambient Cn+1. As special examples, we conjecture that the weights found in Example3.8are indeed global minimizers of vol(cv). Postscript Note: The 4th part of the above conjecture has been proved in [Li15b] and [LL16]. Results on the existence and uniqueness of minimizers will appear in [Li15b] and”
• A non-Archimedean approach to K-stability (2018, methodology): “Theorem E and Theorem E’ generalize Li and Fujita’s valuative criterion [Li17, Fuj16] for K-stability to the adjoint setting.”
• Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (2015, background): “— The filtration defined by a valuation and its relation to K-stability has been recently studied by Fujita [38, 39, 40], Li [57, 56] and Liu [60].”
• Kähler–Einstein metrics and volume minimization (2016, methodology): “The main idea, which is similar to that in [42,41], is that the formula appearing in log version of Fujita’s result should be viewed as a derivative of normalized volumes.”
• On the K-stability of Fano varieties and anticanonical divisors (2016, result): “ simply refer to the original [Tia97], [Don02], and for uniform version, to [Der14], [BHJ15]. The reason is that, as in the next section (Theorem 1.1) we start from review of the results of [Fjt16b], [Li16] which can be seen as giving an alternative definition of the (uniform) K-(semi)stability of Fa”
• A Guided Tour to Normalized Volume (2018, methodology): “Such a question was first extensively studied in [Li17].”
• A minimizing valuation is quasi-monomial (2019, background): “We note that the Stable Degeneration Conjecture has been intensively studied (see e.g. [Li17,Blu18,LL19,LX16,LX18]).”
• Openness of K-semistability for Fano varieties (2019, methodology): “[Li17] C. Li, K-semistability is equivariant volume minimization, Duke Math.”
• On the Yau‐Tian‐Donaldson Conjecture for Singular Fano Varieties (2017, methodology): “Then we have the following criterion: (1) ([36, 41], [25]) (M,B) is log-K-semistable if and only if δ̃(M,B) ≥ 1.”
• Uniqueness of K-polystable degenerations of Fano varieties (2018, methodology): “More precisely, we will provide information on the language of valuations and filtrations following [BHJ17,Fuj17a,Li17] and the invariants β and δ associated to them following [Fuj17a, FO16, BJ17].”
• On positivity of the CM line bundle on K-moduli spaces (2019, methodology): “Our invariant βX,∆,δ is a natural extension of the β-invariant, defined in [Fuj19, Li17] for valuations, to more general filtrations.”
• Uniqueness of the minimizer of the normalized volume function (2020, methodology): “There has been a lot of progress on the solution of different parts of the Stable Degeneration Conjecture in [Blu18,Li17,LX18,LX16,Xu20].”