Arithmetic Yau–Tian–Donaldson conjecture

Let $F$ be a number field, $(X_{\eta},L_{\eta})$ a normal polarized projective variety over $F$, and let $(X,L,h_L)$ range over all metrized polarized normal models over $\mathcal{O}_{F'}$, where $F'$ ranges over all finite extensions of $F$. For each reduction $(X_{\mathfrak p},L_{\mathfrak p})$, let K-semistability have its usual meaning, and let $\omega_{h_L}$ denote the positive first Chern form of the hermitian metric $h_L$. Arithmetic Yau–Tian–Donaldson conjecture. The Arakelov K-energy $h_K(X,L,h_L)$ attains its minimum if and only if all reductions $(X_{\mathfrak p},L_{\mathfrak p})$ are K-semistable and $\omega_{h_L}$ is a Kähler form with constant scalar curvature. This conjecture combines the expected non-archimedean K-stability criterion with the constant-scalar-curvature condition at the complex place; the source states that the paper partially proves it, but gives no resolution of the full equivalence.