Journal
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
Volume 21, Issue 9, Pages 2905-2944Publisher
EUROPEAN MATHEMATICAL SOC
DOI: 10.4171/JEMS/894
Keywords
K-stability; Kahler geometry; canonical metrics; non-Archimedean geometry
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Funding
- ANR
- JSPS KAKENHI [25-6660, 15H06262]
- NSF [DMS-1266207]
- Knut and Alice Wallenberg foundation
- United States-Israel Binational Science Foundation
- Grants-in-Aid for Scientific Research [15H06262] Funding Source: KAKEN
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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 many common functionals in Kahler 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 [BHJ17]) 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 defined in [Der 15, BHJ17]. As a partial converse, we show that uniform K-stability implies coercivity of the Mabuchi functional when restricted to Bergman metrics.
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