Stability of valuations and Kollár components

By Chi Li. Journal of the European Mathematical Society (Print), 2016

85 citations recorded by the source corpus.

We prove that among all Kollar components obtained by plt blow ups of a klt singularity $o \in (X, D)$, there is at most one that is (log-)K-semistable. We achieve this by showing that if such a Kollar component exists, it uniquely minimizes the normalized volume function introduced in [Li15a] among all divisorial valuations. Conversely, we show any divisorial minimizer of the normalized volume function yields a K-semistable Kollar component. We also prove that for any klt singularity, the infimum of the normalized function is always approximated by the normalized volumes of Kollar components.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4721a8b8c630d6ec366140318db34060ca11e52b

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In Applied Mathematics

1. Chi Li
2. H. Blum
3. Yuchen Liu
4. Feng Wang
5. G. Tian
6. Ziquan Zhuang
7. Mattias Jonsson
8. Chenyang Xu

In Geometry and Topology

1. Chi Li
2. H. Blum
3. Yuchen Liu
4. Chenyang Xu
5. Feng Wang
6. G. Tian
7. Ziquan Zhuang
8. Mattias Jonsson

Citations in the corpus

• Stability of Valuations: Higher Rational Rank (2017, methodology): “onjecture B for klt pair (X,D) in [JM12] implies that any minimizer vof volc (X,D) is quasi-monomial. Another characterization of the normalized volume is by using the volumes of models. Theorem 2.5 ([LX16]). For any model Y → Xwhich is isomorphic over X\ {x} with a nontrivial exceptional divisor, w”
• Existence of valuations with smallest normalized volume (2016, result): “rime divisor on a normal variety which is proper and birational over X, then (1)the graded O X-algebra O X a 1(v) a 2(v) is nitely generated, (2)the valuation v corresponds to a Kollar component (See [LX16]), (3)the number vol(cv) is rational. The previous proposition was independently observed in [”
• On the Yau‐Tian‐Donaldson Conjecture for Singular Fano Varieties (2017, methodology): “,B)(F), ˜δ(M) = δ˜(M,∅), where Franges over all divisorial valuations over M. Theorem 2.10. Assume (M,B) is a log Fano pair with log terminal singularities. Then we have the following criterion: (1) ([34, 37], [24]) (M,B) is log-K-semistable if and only if δ˜(M,B) ≥ 1. (2) ([25], see also [10]) (M,B”
• Kähler–Einstein metrics and volume minimization (2016, result): “[Li15a]. Further consequences of our main results, in particular, their relation to the (sharp) de-Fernex-Ein-Musta¸ta type inequalities for Q-Gorenstein klt singularities, will appear in [Liu16] and [LX16]. 1.2 Statement of main results Let (Vn−1,En−2) be a log-Fano pair. By this we mean that −(K V”