Let $(V,\Delta)$ be a klt compact Kähler pair.

If $(V,\Delta)$ is $\mathbb{Q}$-CY it carries a singular Ricci flat metric with adapted volume form in any Kähler class of $V$, this current being smooth outside $\Delta\cup V^{sing}$ if $V$ projective and the Kähler class is rational.

If it is canonically polarized it carries a unique singular KE metric in the cohomology class of $K_{V}+\Delta$, regular outside $\Delta\cup V^{sing}$.