Metrized line bundles and the reduction graph [01JB]

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Metrized line bundles and the reduction graph

A construction of S. Zhang [57], building on prior results of Chinburg–Rumely [20], furnishes continuous metrics on divisors from continuous functions on the reduction graph $R(\mathfrak{X})$ . It works as follows. First of all, if $P\in\mathrm{X}(K)$ is a rational point, there is a unique morphism $\varepsilon_{P}\colon\operatorname{Spec}K^{\circ}\rightarrow\mathfrak{X}$ which extends the point $P$ viewed as a morphism from $\operatorname{Spec}K$ to $X$ . The image of this section is a divisor $D_{P}$ on $\mathfrak{X}$ and the line bundle $\mathscr{O}(D_{P})$ on $\mathfrak{X}$ defines a smooth metric on $\mathscr{O}(P)$ ; we write $\overline{\mathscr{O}(P)}_{\mathfrak{X}}$ for the corresponding metrized line bundle. We also define $\mu_{P}$ as the Dirac measure at the vertex of the graph corresponding to the (unique) irreducible component of the special fiber by which $D_{P}$ passes through. The construction and the notation is extended by additivity for divisors which are sums of rational points. More generally, if $P$ is only a closed point of $X$ , we do this construction after the finite extension $K(P)/K$ , so that $P$ becomes a sum of rational points, using for model the minimal resolution of $\mathfrak{X}\otimes K(P)^{\circ}$ described earlier.

If $f$ is any continuous function on $R(\mathfrak{X})$ and $D$ a divisor on $\mathrm{X}$ , the metrized line bundle $\mathscr{O}(D+f)_{\mathfrak{X}}$ is deduced from $\overline{\mathscr{O}(D)}_{\mathfrak{X}}$ by multiplying the metric by $e^{-f}$ . When $f$ is piecewise linear, this metrized line bundle is smooth. To prove that, we may extend the scalars and assume that $D$ is a sum of rational points $\sum n_{j}P_{j}$ and that $f$ is linear on each edge corresponding to an intersection point of components of the special fiber. Letting $(V_{i})$ being the family of these components, and writing $v_{i}$ for the vertex of $R(\mathfrak{X})$ corresponding to $V_{i}$ , the divisor

\sum_{j}n_{j}D_{P_{j}}+\sum_{i}f(v_{i})V_{i}

(2.2.1)

defines the metrized line bundle $\mathscr{O}(D+f)_{\mathfrak{X}}$ .

In this context, Zhang has defined a curvature operator, which associates to a metrized line bundle a distribution on the graph $R(\mathfrak{X})$ , defined in such a way that

—

for any divisor $D$ on $\mathrm{X}$ , ${\operatorname{curv}}(\overline{\mathscr{O}(D)}_{\mathfrak{X}})=\mu_{D}$ ;
—

for any continuous function $f$ , ${\operatorname{curv}}(\overline{O}(f)_{\mathfrak{X}})=-\Delta f$ , where $\Delta$ is the Laplacian operator of the graph $R(\mathfrak{X})$ ,

and depending linearly on the metrized line bundle. The following lemma compares this construction with the general one on Berkovich spaces.

Lemma 2.2.2.

Let $\overline{L}=\overline{\mathscr{O}(D+f)}_{\mathfrak{X}}$ be a metrized line bundle on $\mathrm{X}$ associated to a divisor $D$ on $\mathrm{X}$ and a continuous function $f$ on the graph $R(\mathfrak{X})$ . If it is semi-positive, resp. admissible in the sense of [57] then it is semi-positive, resp. admissible in the sense of this article, and one has

c_{1}(\overline{L})=\iota_{*}{\operatorname{curv}}(\overline{\mathscr{O}(D+f)})\log\left|{\pi}\right|^{-1}.

In other words, the measure $c_{1}(\overline{L})$ is supported by the graph $R(\mathfrak{X})$ where it coincides essentially with Zhang’s curvature.

Démonstration.

We first assume that $f$ is linear on each edge of $R(\mathscr{X})$ and that $D$ is a sum of rational points of $X$ . Then, $\overline{L}$ corresponds to the line bundle $\mathfrak{L}$ on the model $\mathfrak{X}$ given by Equation 2.2.1. By definition, the measure $c_{1}(\mathfrak{L})$ is computed as follows. It is a sum, for all components $V_{i}$ of the special fiber, of $\deg(\mathfrak{L}|V_{i})\log\left|{\pi}\right|^{-1}$ times the Dirac measure at the corresponding point $v_{i}$ of $R(\mathfrak{X})$ . In particular, it is supported by $R(\mathfrak{X})$ . Then,

\deg(\mathfrak{L}|V_{i})=\sum_{j}n_{j}\left\{\begin{array}[]{cc}1&\text{if $D_{P_{j}}$ passes through $V_{i}$ ;}\\ 0&\text{otherwise}\end{array}\right\}+\sum_{j}f(V_{j})(V_{i},V_{j}),

where $(V_{i},V_{j})$ is the intersection number of the divisors $V_{i}$ and $V_{j}$ . That $D_{P_{j}}$ passes through $V_{i}$ means exactly that $\rho(P_{j})=v_{i}$ . Moreover, if $j\neq i$ , then $(V_{i},V_{j})=m_{i,j}$ is just the number of intersection points of $V_{i}$ and $V_{j}$ , while

(V_{i},V_{i})=(V_{i},\sum_{j}V_{j})-\sum_{j\neq i}(V_{i},V_{j})=-\sum_{j\neq i}(V_{i},V_{j}),

since the whole special fiber is numerically equivalent to zero. Consequently,

\sum_{j}f(v_{j})(V_{i},V_{j})=\sum_{j\neq i}m_{i,j}\big(f(V_{j})-f(V_{i})\big).

Observe that this is the sum, over all edges from $V_{i}$ , of the derivative of $f$ along this edge. Comparing with the definitions given by Zhang in [57], one finds, for any function $g$ on $R(\mathfrak{X})$

	$\displaystyle\sum_{i}\deg(\mathfrak{L}\|V_{i})g(v_{i})$	$\displaystyle=\sum_{j}n_{j}g(\rho(P_{j}))+\sum_{i}\langle\delta f(v_{i}),g\rangle$
		$\displaystyle=\int_{R(\mathfrak{X})}g\,(\mu_{D}+\delta f)$
		$\displaystyle=\int_{R(\mathfrak{X})}g\,{\operatorname{curv}}(\overline{\mathscr{O}(D+f)}).$

This proves the claimed formula when $f$ is linear on each edge of $\mathfrak{X}$ and $D$ is a sum of rational points.

By working over an appropriate finite extension of $K$ , it extends to the case where $f$ is only piecewise linear, $D$ being any divisor on $X$ .

Zhang defines $\overline{\mathscr{O}(D+f)}_{\mathfrak{X}}$ to be semi-positive if $f$ is uniform limit of piecewise linear functions $f_{n}$ such that ${\operatorname{curv}}(\overline{\mathscr{O}(D+f_{n})}_{\mathfrak{X}})\geq 0$ . The metrized line bundle $\overline{L}$ is then the limit of the metrized line bundles $\overline{L}_{n}$ corresponding models $\mathfrak{L}_{n}$ (on appropriate models $\mathfrak{X}_{n}$ of $\mathrm{X}$ after some extension of scalars) of $\mathscr{O}(D)$ . By the previous computation, these metrics are smooth and $c_{1}(\overline{L}_{n})\geq 0$ . Reversing the computation, this means that $\mathfrak{L}_{n}$ is numerically effective on $\mathfrak{X}_{n}$ , hence $\overline{L}$ is semi-positive. By definition of the measure $c_{1}(\overline{L})$ , one has

	$\displaystyle c_{1}(\overline{L})$	$\displaystyle=\lim_{n}c_{1}(\overline{L}_{n})=\lim_{n}\iota_{*}{\operatorname{curv}}(\overline{\mathscr{O}(D+f_{n})})$
		$\displaystyle=\iota_{}\lim_{n}{\operatorname{curv}}(\overline{\mathscr{O}(D+f_{n})})=\iota_{}{\operatorname{curv}}(\overline{\mathscr{O}(D+f)}).$

The case of an admissible metrized line bundle follows by linearity. ∎