Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$. Let $\omega$ be a Kaehler current of $\mathcal X(\mathbb C)$. Assume that the arithmetic first Chern class $c_1(\mathcal X)$ vanishes.
Then what can we say about $Ric(\omega)$? Is there any Calabi-Yau like theorem
Have we such formula $$c_1(\mathcal X(\mathbb C))=[Ric(\omega)]$$
In fact arithmetic first Chern class is in Chow group and $Ric$ is in $H^2$ and they have different chhomology
More generally, If the anti-canonical arithmetic line bundle is negative, then can we say $Ric(\omega)=-\omega$ ?