Let $\mathcal X\to \mathrm{Spec}(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$. Let $\omega$ be a Kähler current of $\mathcal X(\mathbb C)$. Assume that the arithmetic first Chern class $c_1(\mathcal X)$ vanishes.

Then what can we say about $\mathrm{Ric}(\omega)$? Is there any Calabi-Yau like theorem?

Have we such formula $$c_1(\mathcal X(\mathbb C))=[\mathrm{Ric}(\omega)]$$

In fact arithmetic first Chern class is in Chow group and '$\mathrm{Ric}$' is in $H^2$ and they have different cohomology.

More generally, If the anti-canonical arithmetic line bundle is negative, then can we say $\mathrm{Ric}(\omega)=-\omega$ ?