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

The pair $(V,\Delta)$ is said to be $\mathbb{Q}$-CY, iff there is some multiple $N^{\prime}$ of $index(X,\Delta)$ such that $H^{0}(V,\mathcal{O}_{V}(N^{\prime}(K_{V}+\Delta)))=\mathbb{C}\alpha$ where $\alpha$ is a global generator of $\mathcal{O}_{V}(N^{\prime}(K_{V}+\Delta))$.

The pair $(V,\Delta)$ is canonically polarized iff $K_{V}+\Delta$ is ample.