K-energy for the coupled constant scalar curvature Kähler equation, I: Slope formula and semistability
Abstract
In this paper we study coupled constant scalar curvature Kähler metrics defined by Datar-Pingali extending coupled Kähler-Einstein metrics defined by Hultgren-Witt Nyström. We consider two K-energy functionals of which one (denoted ℳccscK) integrates the moment map considered by Datar-Pingali and the other (denoted ℳ') is designed so that its slope at infinity for subgeodesic rays for ample test configurations with reduced central fiber coincides with the Donaldson-Futaki invariant considered by Hultgren-Witt Nyström and Datar-Pingali. We also extend ℳccscK to ℋ1,1̅ and the finite energy class ℰ1, and prove its convexity along tuples of weak geodesics and finite energy geodesics, respectively. As a corollary, we obtain a semistability condition necessary for the existence of ccscK metrics.
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BibTeX
@misc{futaki2026kenergy,
title = {{K-energy for the coupled constant scalar curvature Kähler equation, I: Slope formula and semistability}},
author = {Akito Futaki and Jianwei Shi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34145},
eprint = {2610.34145},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34145},
}
RIS
TY - UNPB
TI - K-energy for the coupled constant scalar curvature Kähler equation, I: Slope formula and semistability
AU - Akito Futaki
AU - Jianwei Shi
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34145
UR - https://arxiv.org/abs/2610.34145
AB - In this paper we study coupled constant scalar curvature Kähler metrics defined by Datar-Pingali extending coupled Kähler-Einstein metrics defined by Hultgren-Witt Nyström. We consider two K-energy functionals of which one (denoted ℳ_ccscK) integrates the moment map considered by Datar-Pingali and the other (denoted ℳ') is designed so that its slope at infinity for subgeodesic rays for ample test configurations with reduced central fiber coincides with the Donaldson-Futaki invariant considered by Hultgren-Witt Nyström and Datar-Pingali. We also extend ℳ_ccscK to ℋ^1,1̅ and the finite energy class ℰ^1, and prove its convexity along tuples of weak geodesics and finite energy geodesics, respectively. As a corollary, we obtain a semistability condition necessary for the existence of ccscK metrics.
ER -