Let $f:\mathcal{X}\rightarrow D$ be a proper flat family of relative dimension $n$, where $D$ is a disk and $\mathcal{X}$ is a complex analytic space (not necessarily non-singular). We say $f$ is a toric degeneration of Calabi-Yau varieties if

1. (1)

   $\mathcal{X}_{t}$ is an irreducible normal Calabi-Yau variety with only canonical singularities for $t\not=0$. (The reader may like to assume $\mathcal{X}_{t}$ is smooth for $t\not=0$).

2. (2)

   If $\nu:\widetilde{\mathcal{X}}_{0}\to\mathcal{X}_{0}$ is the normalization, then $\widetilde{\mathcal{X}}_{0}$ is a disjoint union of toric varieties, the conductor locus $C\subseteq\widetilde{\mathcal{X}}_{0}$ is reduced, and the map $C\to\nu(C)$ is unramified and generically two-to-one. The square

   $$\begin{CD}C@>{}>{}>\widetilde{\mathcal{X}}_{0}\\ @V{}V{}V@V{}V{\nu}V\\ \nu(C)@>{}>{}>\mathcal{X}_{0}\end{CD}$$

   is cartesian and cocartesian.

3. (3)

   $\mathcal{X}_{0}$ is a reduced Gorenstein space and the conductor locus $C$ restricted to each irreducible component of $\widetilde{\mathcal{X}}_{0}$ is the union of all toric Weil divisors of that component.

4. (4)

   There exists a closed subset $Z\subseteq\mathcal{X}$ of relative codimension $\geq 2$ such that $Z$ satisfies the following properties: $Z$ does not contain the image under $\nu$ of any toric stratum of $\widetilde{\mathcal{X}}_{0}$, and for any point $x\in\mathcal{X}\setminus Z$, there is a neighbourhood $\widetilde{U}_{x}$ (in the analytic topology) of $x$, an $n+1$-dimensional affine toric variety $Y_{x}$, a regular function $f_{x}$ on $Y_{x}$ given by a monomial, and a commutative diagram

   $$\begin{matrix}\widetilde{U}_{x}&\smash{\mathop{\longrightarrow}\limits^{\psi_{x}}}&Y_{x}\cr\Big\downarrow\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle f|_{\widetilde{U}_{x}}$}}$\hss}&&\Big\downarrow\hbox to0.0pt{$\vbox{\hbox{$\scriptstyle f_{x}$}}$\hss}\cr D^{\prime}&\smash{\mathop{\longrightarrow}\limits^{\varphi_{x}}}&\mathbb{C}\cr\end{matrix}$$

   where $\psi_{x}$ and $\varphi_{x}$ are open embeddings and $D^{\prime}\subseteq D$. Furthermore, $f_{x}$ vanishes precisely once on each toric divisor of $Y_{x}$.