Subsubsection [04VG]

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(3.2.6) Corollary 3.2.5 implies, in particular, that the skeleta $\mathrm{Sk}(\mathscr{X}_{1})=\mathrm{Sk}(\mathscr{X}_{2})$ are isomorphic as topological spaces with piecewise affine structure, by [MN13, §3.2]. Since $\mathrm{Sk}(\mathscr{X}_{i})$ is canonically homeomorphic to the dual complex associated to the reduced special fiber of $\mathscr{X}_{i}$ , for $i=1,2$ , this also follows from Proposition 11 in [dFKX12], whose proof relies on Weak Factorization. The proofs of Corollary 3.2.5 and [MN13, §3.2] do not use Weak Factorization.

Corollary 3.2.7.

If $K_{X}$ is semi-ample, then the skeleton of a good minimal $dlt$ -model of $X$ does not depend on the choice of the good minimal $dlt$ -model.

Proof.

This follows from Theorem 2.2.6(2) and Corollary 3.2.5. ∎

Theorem 3.2.8.

Assume that $K_{X}$ is semi-ample over $C$ . If $\mathscr{X}$ is a good minimal $dlt$ -model of $X$ and $\mathscr{Y}$ is any $dlt$ -model of $X$ , then $\mathrm{Sk}(\mathscr{X})$ is contained in $\mathrm{Sk}(\mathscr{Y})$ . Moreover, $\mathrm{Sk}(\mathscr{X})$ can be obtained from $\mathrm{Sk}(\mathscr{Y})$ (as a topological subspace of $\mathrm{Sk}(\mathscr{Y})$ with piecewise affine structure) by a finite number of elementary collapses.

Proof.

For the definition of an elementary collapse in a simplicial topological space, we refer to Definition 18 in [dFKX12]. By Corollary 3.2.7, we can assume that the good minimal $dlt$ -model $\mathscr{X}$ is the result of running MMP for $(\mathscr{Y},(\mathscr{Y}_{s})_{\mathrm{red}})$ . Now the statement follows from Corollary 22 in [dFKX12]. When $k$ is not algebraically closed, see also §31 in [dFKX12]. ∎

Corollary 3.2.9.

If $\mathscr{X}$ is a good minimal $dlt$ -model of $X$ , then $\mathrm{Sk}(\mathscr{X})$ is a strong deformation retract of $X_{K}^{\mathrm{an}}$ .

Proof.

Let $\mathscr{Y}\to\mathscr{X}$ be a log resolution of $(\mathscr{X},\mathscr{X}_{s})$ . By Theorem 3.2.8, the skeleton $\mathrm{Sk}(\mathscr{X})$ is a strong deformation retract of $\mathrm{Sk}(\mathscr{Y})$ . By Theorem 3.1.3, $\mathrm{Sk}(\mathscr{Y})$ is a strong deformation retract of $X_{K}^{\mathrm{an}}$ . ∎