Notice we have intentionally avoided completing the family at $0\in S$. That would involve the choice of a model of $\pi:X\to S$, namely a normal flat projective $S$-scheme $\mathcal{X}$ together with an isomorphism with $X$ over the punctured curve $S\setminus\{0\}$. It is called an snc model if $\mathcal{X}$ is smooth, and the central fibre over $0\in S$ is a simple normal crossing divisor in $\mathcal{X}$. If furthermore the central fibre is reduced, it is called a semistable snc model. Models can be analogously defined over the formal disc. The existence of snc models is a consequence of Hironaka’s resolution theorem. They are highly nonunique. By the semistable reduction theorem [27, chapter 2], after finite base change to another smooth algebraic curve $S^{\prime}$, we can always find some semistable snc model for the degeneration family $X\times_{S}(S^{\prime}\setminus\{0\})$, so the existence of a semistable snc model is not a substantial assumption. Everything here is quasi-projective. The choice of a model is very useful, but not intrinsic to the degenerating CY metrics.