(2.1) Let $X$ be a Calabi-Yau variety over $K$. The essential skeleton $\mathrm{Sk}(X)$ of $X$ was first defined by Kontsevich and Soibelman in [KS06]. The construction was then refined and generalized in [MN15]. Let $\omega$ be a volume form on $X$. Then one can attach to the pair $(X,\omega)$ a weight function

$$\mathrm{wt}_{\omega}\colon X^{\mathrm{an}}\to\mathbb{R}\cup\{+\infty\}$$

that measures the degeneration of $(X,\omega)$ at $t=0$ along points of $X^{\mathrm{an}}$; see [MN15, §4.5]. The essential skeleton $\mathrm{Sk}(X)$ is the locus of points in $X^{\mathrm{an}}$ where $\mathrm{wt}_{\omega}$ reaches its minimal value. This definition only depends on $X$, and not on $\omega$, because multiplying $\omega$ with a scalar $\lambda\in K^{\ast}$ shifts the weight function by the constant $\mathrm{ord}_{t}\lambda$. The essential skeleton is a non-empty compact subspace of $X^{\mathrm{an}}$, which can be explicitly computed in the following way. Let $\mathscr{X}$ be an snc-model of $X$, with special fiber $\mathscr{X}_{k}=\sum_{i\in I}N_{i}E_{i}$. If we view $\omega$ as a rational section of the line bundle $\omega_{\mathscr{X}/R}(\mathscr{X}_{k,\mathrm{red}})$, then it defines a Cartier divisor on $\mathscr{X}$ that we denote by $\mathrm{div}_{\mathscr{X}}(\omega)$. It is supported on $\mathscr{X}_{k}$ because $\omega$ is nowhere vanishing on $X$; thus we can write $\mathrm{div}_{\mathscr{X}}(\omega)=\sum_{i\in I}\nu_{i}E_{i}$. If we denote by $\Delta(\mathscr{X})$ the dual intersection complex of $\mathscr{X}_{k}$, then $\mathrm{Sk}(X)$ is canonically homeomorphic to the sub-$\Delta$-complex of $\Delta(\mathscr{X})$ spanned by the vertices corresponding to the components $E_{i}$ for which $\nu_{i}/N_{i}$ is minimal (see Theorem 3 in [KS06, §6.6] and Theorem 4.7.5 in [MN15]). In particular, $\mathrm{Sk}(X)$ is homeomorphic to a finite $\Delta$-complex of dimension $\leq\dim(X)$.