Uniform K-stability and asymptotics of energy functionals in Kähler geometry

By S. Boucksom, Tomoyuki Hisamoto, Mattias Jonsson. Journal of the European Mathematical Society (Print), 2016

95 citations recorded by the source corpus.

Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in K\"ahler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper) at the non-Archimedean metric on L defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability. As a partial converse, we show that uniform K-stability implies coercivity of the Mabuchi functional when restricted to Bergman metrics.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/355c2b70141f34ff4e3c5cce472a146e27a3aa2c

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In Applied Mathematics

1. Chi Li
2. Mattias Jonsson
3. S. Boucksom
4. Robert J. Berman
5. Feng Wang
6. G. Tian
7. Yuchen Liu
8. H. Blum

In Geometry and Topology

1. Chi Li
2. Mattias Jonsson
3. S. Boucksom
4. Robert J. Berman
5. Feng Wang
6. G. Tian
7. Yuchen Liu
8. H. Blum

In Mathematical Physics

1. Chi Li
2. Mattias Jonsson
3. S. Boucksom
4. G. Tian
5. Feng Wang
6. H. Blum
7. Robert J. Berman
8. Yuchen Liu

Citations in the corpus

• A non-Archimedean approach to K-stability (2018, background): “en the classical entropy of with respect to is dened as R log d d d, when ˝, and +1otherwise. Our notion of entropy can be seen as a non-Archimedean degeneration of the usual notion, see [BHJ16]. The entropy behaves well with respect to regularizations of measures: Lemma 2.3. For any snc model Xo”
• On the Yau‐Tian‐Donaldson Conjecture for Singular Fano Varieties (2017, background): “he implication 1 ⇒ 2, and 3 ⇒ 4 ⇒ 5. The equivalence of 3 and 5 was proved in the case when B= 0 in [4] and the more general log version was proved in [24]. The implication 1 ⇒ 3, 2 ⇒ 4 was proved in [9]. So one just needs to show 4 ⇒ 2, which will be proved by contradiction. Assume 2 is not true, t”