Semistable arithmetic Yau–Tian–Donaldson conjecture

Let $(X,L)$ be a smooth projective variety over a number field $K$. Semistable arithmetic Yau–Tian–Donaldson conjecture. The following conditions are equivalent: $(X(\mathbb{C}),L(\mathbb{C}))$ is K-semistable; $(X,L)$ is Arakelov K-semistable, meaning that $h_K(\mathcal{X},\mathcal{L},h)$ is uniformly lower bounded over all models and positively curved metrics; for an integral model over $\mathcal{O}_K$, every maximal-ideal reduction is K-semistable; and for every integral model, all but finitely many maximal-ideal reductions are K-semistable, a condition called arithmetic K-semistability. This conjecture links differential-geometric K-semistability, lower bounds for Arakelov modular heights, and K-semistability of reductions; the source provides no resolution evidence.