Coupled free energy and a diagonal approach to coupled Kähler-Einstein metrics
Abstract
In this paper, we introduce the coupled free energy F−γcp which is a natural generalization of the free energy functional defined by Berman. In the Fano case, our coupled free energy provides a diagonal approach to the coercivity of the coupled Ding functional. We also define the non-Archimedean coupled free energy F−γcp,NA on the space of finite energy measures on the Berkovich analytification. As a consequence of the regularization process of the non-Archimedean entropy established by Boucksom-Jonsson, Chi Li and Trusiani, we prove that the existence of a unique coupled Kähler-Einstein metric is equivalent to the stability condition that there exists ε>0 such that F−1−εcp,NA ≥ 0 on the space of measures associated with a test configuration for the anticanonical line bundle. As a coupled version of the work by Berman, we show that the law of the empirical measure satisfies a large deviation principle whose rate function is the coupled free energy under the assumption that a conjecture by Hultgren is true. We also introduce a coupled version of the microscopic stability thresholds and prove that a uniformly coupled Gibbs stable Fano manifold has a unique coupled Kähler-Einstein metric. This is a generalization of the results by Fujita-Odaka and Berman.
Related
- Instability of conformally Kähler, Einstein metrics
- Psh variation of singular Kähler-Einstein metrics and Invariance of plurigenera for smooth Kähler families
- Limiting properties of Hermitian-Einstein metrics under variation of the Kähler class
- Kähler-Einstein cone metrics and parabolic bundles
- The curvature estimation of the complete Kähler-Einstein metrics on the disk bundles
BibTeX
@misc{aoi2026coupled,
title = {{Coupled free energy and a diagonal approach to coupled Kähler-Einstein metrics}},
author = {Takahiro Aoi and Satoshi Nakamura},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06968},
eprint = {2611.06968},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06968},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Coupled free energy and a diagonal approach to coupled Kähler-Einstein metrics
AU - Takahiro Aoi
AU - Satoshi Nakamura
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06968
UR - https://arxiv.org/abs/2611.06968
AB - In this paper, we introduce the coupled free energy F_−γ^cp which is a natural generalization of the free energy functional defined by Berman. In the Fano case, our coupled free energy provides a diagonal approach to the coercivity of the coupled Ding functional. We also define the non-Archimedean coupled free energy F_−γ^cp,NA on the space of finite energy measures on the Berkovich analytification. As a consequence of the regularization process of the non-Archimedean entropy established by Boucksom-Jonsson, Chi Li and Trusiani, we prove that the existence of a unique coupled Kähler-Einstein metric is equivalent to the stability condition that there exists ε>0 such that F_−1−ε^cp,NA ≥ 0 on the space of measures associated with a test configuration for the anticanonical line bundle. As a coupled version of the work by Berman, we show that the law of the empirical measure satisfies a large deviation principle whose rate function is the coupled free energy under the assumption that a conjecture by Hultgren is true. We also introduce a coupled version of the microscopic stability thresholds and prove that a uniformly coupled Gibbs stable Fano manifold has a unique coupled Kähler-Einstein metric. This is a generalization of the results by Fujita-Odaka and Berman.
ER -