3.2. The skeleton of a good minimal d l t -model

(3.2.1) In the following subsections, we will make use of the weight function

\mathrm{wt}_{\omega}:X_{K}^{\mathrm{an}}\to\mathbb{R}\cup\{+\infty\}

associated to a non-zero $m$ -pluricanonical form on $X_{K}$ , for any $m>0$ . Its construction and main properties are described in [MN13, 4.4.5]. For us, its most important features are the following: if $\mathscr{X}$ is an $snc$ -model of $X$ over $\mathscr{C}$ and $x$ is a point of $\mathrm{Sk}(\mathscr{X})$ , then

\mathrm{wt}_{\omega}(x)=v_{x}(\mathrm{div}_{\mathscr{X}}(\omega)+m(\mathscr{X}_{s})_{\mathrm{red}})

(here we use the notation recalled in (2.1)). Moreover, for every point $y$ of $\widehat{\mathscr{X}}_{\eta}$ , we have

\mathrm{wt}_{\omega}(y)\geq\mathrm{wt}_{\omega}(\rho_{\mathscr{X}}(y))

with equality if and only if $y$ lies on $\mathrm{Sk}(\mathscr{X})$ . In [MN13, 4.4.5] there is no properness assumption on $X_{K}$ ; this allows us to deal with rational pluricanonical forms by removing the locus of poles from $X_{K}$ .

(3.2.2) It will often be useful to interpret the weight function in terms of logarithmic differential forms. Let $\mathscr{Y}$ be a regular separated $R$ -scheme of finite type such that $\mathscr{Y}_{k}$ is a divisor with strict normal crossings. We write $S^{+}$ for the log scheme associated to $R\setminus\{0\}\to R$ and $\mathscr{Y}^{+}$ for the log scheme obtained by endowing $\mathscr{Y}$ with the divisorial log structure associated to $\mathscr{Y}_{k}$ . Then $\mathscr{Y}^{+}$ is log smooth over $S^{+}$ . If we denote by $j:\mathscr{Y}_{K}\to\mathscr{Y}$ the natural open immersion, then a simple computation shows that the sub- $\mathcal{O}_{\mathscr{Y}}$ -module $\omega_{\mathscr{Y}^{+}/S^{+}}$ of $j_{*}\omega_{\mathscr{Y}_{K}/K}$ is equal to $\omega_{\mathscr{Y}/R}((\mathscr{Y}_{k})_{\mathrm{red}}-\mathscr{Y}_{k})$ (it suffices to check that these line bundles coincide at the generic points of the special fiber $\mathscr{Y}_{k}$ ). Thus if $\omega$ is an $m$ -pluricanonical form on $X_{K}$ and $\mathscr{X}$ is an $snc$ -model of $X$ over $\mathscr{C}$ , then

\mathrm{wt}_{\omega}(x)=v_{x}(\mathrm{div}_{\mathscr{X}^{+}}(\omega))+m

for every point $x$ of $\mathrm{Sk}(\mathscr{X})$ , where we denote by $\mathrm{div}_{\mathscr{X}^{+}}(\omega)$ the divisor on $\mathscr{X}_{R}$ associated to $\omega$ viewed as a rational section of the line bundle $\omega^{\otimes m}_{\mathscr{X}^{+}_{R}/S^{+}}$ .

Lemma 3.2.3.

Let $\mathscr{X}$ be a $dlt$ -model of $X$ and let $h:\mathscr{Y}\to\mathscr{X}$ be a log resolution of $(\mathscr{X},\mathscr{X}_{s})$ . Denote by $\Delta$ the log pullback of $(\mathscr{X}_{s})_{\mathrm{red}}$ to $\mathscr{Y}$ . Let $x$ be a point of $\mathrm{Sk}(\mathscr{Y})$ such that $\mathrm{red}_{\mathscr{X}}(x)$ does not lie in $\mathscr{X}^{\mathrm{snc}}$ . Then $\Delta<(\mathscr{Y}_{s})_{\mathrm{red}}$ locally at $\mathrm{red}_{\mathscr{Y}}(x)$ .

Proof.

By the definition of a $dlt$ -model, we know that $\Delta\leq(\mathscr{Y}_{s})_{\mathrm{red}}$ . Thus it suffices to show that these divisors are different locally at $\mathrm{red}_{\mathscr{Y}}(x)$ . Since $x$ lies on $\mathrm{Sk}(\mathscr{Y})$ , its reduction $\mathrm{red}_{\mathscr{Y}}(x)$ is a generic point of the intersection of the irreducible components of $\mathscr{Y}_{s}$ that contain $\mathrm{red}_{\mathscr{Y}}(x)$ . Thus if we denote by $h^{\prime}:\mathscr{Y}^{\prime}\to\mathscr{Y}$ the blow-up of $\mathscr{Y}$ at the closure of $\mathrm{red}_{\mathscr{Y}}(x)$ , then $\mathscr{Y}^{\prime}$ is again an $snc$ -model of $X$ .

We denote by $\Delta^{\prime}$ the log pullback of $\Delta$ to $\mathscr{Y}^{\prime}$ . The image of the exceptional divisor $E$ of $h^{\prime}$ in $\mathscr{X}$ is the closure of $\mathrm{red}_{\mathscr{X}}(x)=h(\mathrm{red}_{\mathscr{Y}}(x))$ and thus disjoint from $\mathscr{X}^{\mathrm{snc}}$ . By the definition of a $dlt$ -model, we know that the multiplicity of $E$ in $\Delta^{\prime}$ is strictly smaller than $1$ . Since the log pullback of $(\mathscr{Y}_{s})_{\mathrm{red}}$ to $\mathscr{Y}^{\prime}$ is equal to $(\mathscr{Y}^{\prime}_{s})_{\mathrm{red}}$ , we see that $\Delta<(\mathscr{Y}_{s})_{\mathrm{red}}$ locally at $\mathrm{red}_{\mathscr{Y}}(x)$ . ∎

Proposition 3.2.4.

Let $\mathscr{X}$ be a $dlt$ -model of $X$ over $\mathscr{C}$ , let $\mathscr{Y}$ be a proper $snc$ -model of $X$ over $\mathscr{C}$ and let $h:\mathscr{Y}\to\mathscr{X}$ be a morphism of $\mathscr{C}$ -models. Denote by $\Delta$ the log pullback of $(\mathscr{X}_{s})_{\mathrm{red}}$ to $\mathscr{Y}$ . If we set

S=\{x\in\mathrm{Sk}(\mathscr{Y})\,|\,v_{x}(\Delta)=v_{x}((\mathscr{Y}_{s})_{\mathrm{red}})\}

then $\mathrm{Sk}(\mathscr{X})=S$ .

Proof.

Applying [MN13, 3.1.7] to the proper morphism $h^{-1}(\mathscr{X}^{\mathrm{snc}})\to\mathscr{X}^{\mathrm{snc}}$ , we see that $\mathrm{Sk}(\mathscr{X})$ is contained in $\mathrm{Sk}(\mathscr{Y})$ . Moreover, it follows from Lemma 3.2.3 that for every point $x$ of $S$ , the reduction $\mathrm{red}_{\mathscr{X}}(x)$ must be contained in $\mathscr{X}^{\mathrm{snc}}$ . Now let $x$ be any point in $\mathrm{Sk}(\mathscr{Y})$ such that $\mathrm{red}_{\mathscr{X}}(x)$ lies in $\mathscr{X}^{\mathrm{snc}}$ . We must show that $v_{x}(\Delta)=v_{x}((\mathscr{Y}_{s})_{\mathrm{red}})$ if and only if $x$ lies in $\mathrm{Sk}(\mathscr{X})$ , or, equivalently, $x$ is equal to its projection

x^{\prime}=\rho_{\mathscr{X}}(x)

to the skeleton of $\mathscr{X}$ . Let $\omega$ be a local generator of $\omega_{\mathscr{X}^{\mathrm{snc}}/\mathscr{C}}$ at $\mathrm{red}_{\mathscr{X}}(x)$ . It induces a rational section of the canonical bundle $\omega_{X_{K}/K}$ by base change. By [MN13, 4.4.5], we know that $x=x^{\prime}$ if and only if

\mathrm{wt}_{\omega}(x)=\mathrm{wt}_{\omega}(x^{\prime}).

Since the divisor of $\omega$ is zero in a neighbourhood of $\mathrm{red}_{\mathscr{X}}(x^{\prime})$ , we have

\mathrm{wt}_{\omega}(x^{\prime})=v_{x^{\prime}}((\mathscr{X}_{s})_{\mathrm{red}})=v_{x}((\mathscr{X}_{s})_{\mathrm{red}}).

On the other hand, computing $\mathrm{wt}_{\omega}(x)$ on the model $\mathscr{Y}$ we get

\mathrm{wt}_{\omega}(x)=v_{x}(\mathrm{div}_{\mathscr{Y}}(\omega)+(\mathscr{Y}_{s})_{\mathrm{red}})=v_{x}((\mathscr{X}_{s})_{\mathrm{red}})+v_{x}((\mathscr{Y}_{s})_{\mathrm{red}}-\Delta).

Thus we see that $\mathrm{Sk}(\mathscr{X})=S$ . ∎

Corollary 3.2.5.

Let $\mathscr{X}_{1}$ and $\mathscr{X}_{2}$ be two $dlt$ -models of $X$ over $\mathscr{C}$ . If $\mathscr{X}_{1}$ and $\mathscr{X}_{2}$ are crepant birational, then $\mathrm{Sk}(\mathscr{X}_{1})=\mathrm{Sk}(\mathscr{X}_{2})$ .

Proof.

This follows immediately from Proposition 3.2.4. ∎

(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 $X_{K}^{\mathrm{an}}$ . ∎

ScalingStacks

3.2. The skeleton of a good minimal d l t -model [04V7]