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arXiv · 2610.34145

K-energy for the coupled constant scalar curvature Kähler equation, I: Slope formula and semistability

Akito Futaki·Jianwei Shi
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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 $\mathcal{M}_{\operatorname{ccscK}}$) integrates the moment map considered by Datar-Pingali and the other (denoted $\mathcal{M}'$) 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 $\mathcal{M}_{\operatorname{ccscK}}$ to $\mathcal{H}^{1,\bar{1}}$ and the finite energy class $\mathcal{E}^1$, 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.

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Akito Futaki, Jianwei Shi. 2026-10-25. K-energy for the coupled constant scalar curvature Kähler equation, I: Slope formula and semistability. https://arxiv.org/abs/2610.34145

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