3.4. Functions on dual complexes [01EU]

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3.4. Functions on dual complexes

Proposition 3.9.

Let $\mathcal{X}$ be an SNC model of $X$ and let $\mathfrak{a}$ be a vertical fractional ideal sheaf on $\mathcal{X}$ . Then $\varphi:=\log|\mathfrak{a}|\in\mathcal{D}(X)$ satisfies:

(i)

$\varphi\circ\emb_{\mathcal{X}}$ is piecewise affine and convex on each face of $\Delta_{\mathcal{X}}$ ;
(ii)

$\varphi\leq\varphi\circ p_{\mathcal{X}}$ .

Combining this result with Proposition 2.2 we see that for any model function $\psi\in\mathcal{D}(X)$ , the composition $\psi\circ\emb_{\mathcal{X}}$ is piecewise affine on each face of $\Delta_{\mathcal{X}}$ . In fact, it is affine on each face iff $\psi$ is determined on $\mathcal{X}$ .

Proof.

Upon multiplying by $\varpi^{m}$ with $m\gg 1$ , we may assume that $\mathfrak{a}\subset\mathcal{O}_{\mathcal{X}}$ is a vertical ideal sheaf. Pick $J\subset I$ such that $E_{J}$ is non-empty, choose a point $\xi\in E_{J}$ and let $f_{1},\dots,f_{m}$ be generators of $\mathfrak{a}$ at $\xi$ . With the notation introduced in the proof of Theorem 3.1 we then have

(3.5)

\log|\mathfrak{a}|(\emb_{\mathcal{X}}(s))=\max\left\{-\val_{\mathcal{X},s}(f_{i}),\,i=1,\dots,m\right\}.

By (3.4) each function $s\mapsto-\val_{\mathcal{X},s}(f_{i})$ is piecewise affine and convex on $\sigma_{J}$ , proving (i). To prove (ii), pick any $x\in X$ , set $\xi:=c_{\mathcal{X}}(x)$ and let $J\subset I$ be the set of indices $j\in I$ such that $\xi\in E_{j}$ . Arguing similarly with generators of $\mathfrak{a}$ , it is enough to show that $|f(x)|\leq|f(p_{\mathcal{X}}(x))|$ for each $f\in\mathcal{O}_{\mathcal{X},\xi}$ . Note that the seminorm $f\mapsto|f(x)|$ extends by continuity to $\widehat{\mathcal{O}}_{\mathcal{X},\xi}$ since $\xi=c_{\mathcal{X}}(x)$ . Writing in the notation of Remark 3.8 $f=\sum_{\alpha\in\mathbf{N}^{L}}f_{\alpha}z^{\alpha}\in\widehat{\mathcal{O}}_{\mathcal{X},\xi}$ we then have

|f(x)|\leq\sup_{f_{\alpha}\neq 0}\prod_{j\in J}|z_{j}(x)|^{\alpha_{j}}

by the ultrametric property, using that $|f_{\alpha}(x)|=1$ since each non-zero $f_{\alpha}\in\mathcal{O}_{\mathcal{X},\xi}$ is a unit. On the other hand, if we set $s_{j}:=-\log|z_{j}(x)|$ for $j\in J$ then we have by definition $p_{\mathcal{X}}(x)=\emb_{\mathcal{X}}(s)$ , hence

\sup_{f_{\alpha}\neq 0}\prod_{j\in J}|z_{j}(x)|^{\alpha_{j}}=|f(p_{\mathcal{X}}(x))|

and the result follows. ∎

Let $\PA(\Delta_{\mathcal{X}})_{\mathbf{Z}}$ be the set of all continuous functions $h:\Delta_{\mathcal{X}}\to\mathbf{R}$ whose restriction to each face of $\Delta_{\mathcal{X}}$ is piecewise affine, with gradients given by $\mathbf{Z}$ -divisors $D\in\Div_{0}(\mathcal{X})$ .

Definition 3.10.

Let $h\in\PA(\Delta_{\mathcal{X}})_{\mathbf{Z}}$ . For each $J\subset I$ such that $E_{J}\neq\emptyset$ we set $\mathcal{X}_{J}:=\mathcal{X}\setminus\bigcup_{i\in I\setminus J}E_{i}$ and define a vertical fractional ideal sheaf $\mathfrak{a}_{h}$ on $\mathcal{X}$ by letting for each $J$

(3.6)

\mathfrak{a}_{h}|_{\mathcal{X}_{J}}:=\sum\left\{\mathcal{O}_{\mathcal{X}_{J}}(D),\,D\in\Div_{0}(\mathcal{X})\text{ such that }\langle D,\cdot\rangle\leq h\text{ on }\sigma_{J}\right\}.

Note that these locally defined sheaves glue well together, and that $\log|\mathfrak{a}_{h}|\circ\emb_{\mathcal{X}}$ is equal to the convex envelope of $h$ on each face of $\Delta_{\mathcal{X}}$ .