1.4 Skeletons [04MJ]

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1.4 Skeletons

Let $X$ be a smooth proper variety over $K$ . To every dlt model $\mathscr{X}$ of $X$ , with special fiber $\mathscr{X}_{k}=\sum_{i\in I}a_{i}D_{i}$ , we can associate a cell complex encoding the combinatorics of the intersections of the components $D_{i}$ , whose faces are in one-to-one correspondence with strata of $\mathscr{X}_{k}$ .

Definition 1.4.1.

We call simplex a topological space, endowed with a $\mathbb{Z}$ -affine structure, which is $\mathbb{Z}$ -affine isomorphic to a space of the form:

\tau=\{\sum_{i=0}^{m}a_{i}w_{i}=1\}\subset\mathbb{R}^{m+1},\quad\textrm{ for some $a_{i}\in\mathbb{N}_{\geqslant 0}$}.

Definition 1.4.2.

Let $\mathscr{X}$ be a dlt model of $X$ . To each stratum $Y$ of $\mathscr{X}_{k}$ which is a connected component of $D_{J}$ , we associate a simplex:

\tau_{Y}=\{w\in\mathbb{R}_{\geqslant 0}^{|J|}\,|\sum_{j\in J}a_{j}w_{j}=1\}.

We define the cell complex $\mathcal{D}(\mathscr{X}_{k})$ by the following incidence relations: $\tau_{Y}$ is a face of $\tau_{Y^{\prime}}$ if and only if $Y^{\prime}\subset Y$ .

Given any dlt model $\mathscr{X}$ of $X$ over $R$ , there is a natural embedding $i_{\mathscr{X}}$ of the dual complex $\mathcal{D}(\mathscr{X}_{k})$ into $X^{\text{an}}$ , given as follows. The vertices $v_{i}$ of $\mathcal{D}(\mathscr{X}_{k})$ are in one-to-one correspondence with irreducible components $D_{i}$ of the special fiber $\mathscr{X}_{k}=\lx@nobreakspace\sum_{i\in I}a_{i}D_{i}$ , so that we set

i_{\mathscr{X}}(v_{i})=v_{D_{i}}\coloneqq a^{-1}_{i}\ord_{D_{i}},

where the valuation $\ord_{D_{i}}$ associates to a meromorphic function $f\in K(X)\simeq K(\mathscr{X})$ its vanishing order along $D_{i}$ - the normalisation by $a^{-1}_{i}$ ensuring that $v_{D_{i}}(t)=1$ . A valuation given in this way, for some dlt model $\mathscr{X}$ of $X$ , is called divisorial. One can now somehow interpolate between those divisorial valuations using quasi-monomial valuations, in order to embed $\mathcal{D}(\mathscr{X}_{k})$ into $X^{\text{an}}$ :

Proposition 1.4.3 ([MN15, Proposition 2.4.4]).

Let $\mathscr{X}$ be a dlt model of $\mathscr{X}$ , with special fiber $\mathscr{X}_{k}=\sum_{i\in I}a_{i}D_{i}$ . Let $J\subset I$ such that $D_{J}=\cap_{j\in J}D_{j}$ is non-empty, and $Y$ a connected component of $D_{J}$ , with generic point $\eta$ . We furthermore fix a local equation $z_{j}\in\mathcal{O}_{\mathscr{X},\eta}$ for $D_{j}$ , for any $j\in J$ .
Then, for any $w\in\tau_{Y}=\{w\in\mathbb{R}^{|J|}_{\geqslant 0}\,|\sum_{j\in J}a_{j}w_{j}=1\}$ , there exists a unique valuation

v_{w}:\mathcal{O}_{\mathscr{X},\eta}\longrightarrow\mathbb{R}_{\geqslant 0}\cup\{+\infty\}

such that for every $f\in\mathcal{O}_{\mathscr{X},\eta}$ , with expansion $f=\sum_{\beta\in\mathbb{N}^{|J|}}c_{\beta}z^{\beta}$ (with $c_{\beta}$ either zero or unit), we have:

v_{w}(f)=\min\{(w\cdot\beta)\,|\beta\in\mathbb{N}^{|J|},c_{\beta}\neq 0\},

where $(\;\cdot\;)$ is the usual scalar product on $\mathbb{R}^{|J|}$ .

The above valuation is called the quasi-monomial valuation associated with the data $(Y,w)$ . Then

	$\displaystyle i_{\mathscr{X}}:$	$\displaystyle\quad\mathcal{D}(\mathscr{X}_{k})\rightarrow X^{\text{an}}$
		$\displaystyle\quad\tau_{Y}\ni w\mapsto v_{w}$

gives a well-defined continuous injective map from $\mathcal{D}(\mathscr{X}_{k})$ to $X^{\text{an}}$ .

Definition 1.4.4.

We call the image of $\mathcal{D}(\mathscr{X}_{k})$ by $i_{\mathscr{X}}$ the skeleton of $\mathscr{X}$ , written as $\Sk(\mathscr{X})\subset X^{\emph{an}}$ . It is a cell complex of dimension at most $\dim X$ .

By compactness of $\mathcal{D}(\mathscr{X}_{k})$ , $i_{\mathscr{X}}$ induces a homeomorphism between $\mathcal{D}(\mathscr{X}_{k})$ and $\Sk(\mathscr{X})$ , so that we will sometimes abusively identify $\mathcal{D}(\mathscr{X}_{k})$ with $\Sk(\mathscr{X})$ .

Definition 1.4.5.

Let $Y$ be a stratum of $\mathscr{X}_{k}$ . We define $\Star(\tau_{Y})$ as the union of open faces in $\Sk(\mathscr{X})$ whose closure contains $\tau_{Y}$ .