2.3. Local character of the measures [01JE]

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2.3. Local character of the measures

The definition of the measures associated to metrized line bundles is global in nature. Still, the main result of this section implies that they are local.

Definition 2.3.1.

Let $\mathrm{X}$ be an analytic space. A function on $\mathrm{X}$ is said to be strongly pluriharmonic if it is locally a uniform limit of functions of the form $a\log\left|{u}\right|$ , where $a\in{\mathbf{R}}$ and $u$ is holomorphic and nonvanishing.

There is a general theory of harmonic functions on curves due to Thuillier [51] (see also [31, 6] on the projective line ; note that the definition of a strongly harmonic function of the latter reference is different from the one adopted here). Strongly pluriharmonic functions are harmonic in their sense. Indeed, logarithms of absolute values of invertible holomorphic functions are harmonic, and harmonic functions are preserved by uniform limits (Prop. 2.3.20 and 3.1.2 of [51]). In fact, when the residue field of $K$ is algebraic over a finite field, any harmonic function is locally the logarithm of the absolute value of an invertible function (loc.cit., Theorem 2.3.21).

This is not necessarily the case for more general fields $K$ : there are harmonic functions over analytic curves which are not locally equal to the logarithm of the absolute value of an invertible function ; examples require to consider curves of genus $\geq 1$ . In a conversation with A. Ducros, we devised the following example of a one-dimensional affinoid space. Let $\mathfrak{E}$ be an elliptic scheme over $K^{\circ}$ , let $o$ be the origin in $\mathfrak{E}_{\tilde{K}}$ and let $p$ be a non-torsion rational point in $\mathfrak{E}_{\tilde{K}}$ ; let $\mathfrak{X}$ be the blow-up of $\mathfrak{E}$ at the point $p$ . Let then $\mathfrak{U}$ be its open subset obtained by removing the point $o$ as well as a smooth point in the exceptional divisor of the blow-up ; its generic fiber $\mathrm{U}$ is the desired affinoid space — it is the complementary subset in the elliptic curve $\mathrm{E}_{K}$ to two small disjoint disks. One can prove that the space of harmonic functions on $\mathrm{U}$ is 2-dimensional, and that all holomorphic invertible functions on $\mathrm{U}$ have constant absolute value.

I do not know whether any harmonic function on a curve is locally a uniform limit of logarithms.

Definition 2.3.2.

Let $\overline{L}$ be a metrized line bundle on an analytic space $\mathrm{X}$ and let $\mathrm{U}$ be an open subset of $\mathrm{X}$ . One says that $\overline{L}$ is strongly pluriharmonic on $\mathrm{U}$ if for any local frame $s$ of $L$ defined on an open subset $\mathrm{V}\subset\mathrm{U}$ , $\log\left\|{s}\right\|^{-1}$ is strongly pluriharmonic on $\mathrm{V}$ .

Equivalently, a metrized line bundle is pluriharmonic on $\mathrm{U}$ if it admits, in a neighbourhood of any point of $\mathrm{U}$ a local frame whose norm is identically equal to $1$ .

Proposition 2.3.3.

Let $\overline{X}$ a the analytic space associated to a proper $K$ -scheme. Let $\overline{L}_{1},\overline{L}_{2},\dots,\overline{L}_{k}$ be admissible metrized line bundles on $\mathrm{X}$ . Let $\mathrm{Z}$ be a $k$ -dimensional Zariski closed subset of $\mathrm{X}$ . Assume that $\overline{L}_{1}$ is strongly pluriharmonic on $\mathrm{U}$ . Then, the support of the measure $c_{1}(\overline{L}_{1})\dots c_{1}(\overline{L}_{k})\delta_{\mathrm{Z}}$ is disjoint from $\mathrm{U}$ .

Démonstration.

One has to show that for any continuous function $\varphi$ with compact support contained in $\mathrm{U}$

\int_{\mathrm{X}}\varphi\,c_{1}(\overline{L}_{1})\dots c_{1}(\overline{L}_{k})\delta_{\mathrm{Z}}=0.

By Gubler’s theorem, the space of smooth functions is dense in the space of smooth functions on $\mathrm{X}$ . Using the fact that the maximum and the minimum of smooth functions are still smooth, one proves that the space of smooth functions with compact support contained in $\mathrm{U}$ is dense in the space of continuous functions with compact support contained in $\mathrm{U}$ , for the topology of uniform convergence. We thus may assume that $\varphi$ is smooth, with compact support contained in $\mathrm{U}$ . Finally, we may also assume that the metric on the line bundles $\overline{L}_{2},\dots,\overline{L}_{k}$ are smooth.

We may argue locally and assume that $L_{1}$ has a meromorphic section $s$ whose divisor $\operatorname{div}(s)$ is disjoint from $\mathrm{U}$ . Up to shrinking $\mathrm{U}$ again, we may assume that there exists a sequence $(u_{n})$ of rational functions without zeroes nor poles on $\mathrm{U}$ such that $\log\left\|{s}\right\|=\lim\log\left|{u_{n}}\right|^{1/n}$ .

According to Prop. 1.3.2, one has

\int_{\mathrm{X}}\varphi\,c_{1}(\overline{L}_{1})\dots c_{1}(\overline{L}_{k})\delta_{\mathrm{Z}}\\ =\int_{\mathrm{X}}\varphi c_{1}(\overline{L}_{2})\dots c_{1}(\overline{L}_{k})\delta_{\operatorname{div}(s|_{\mathrm{Z}})}+\int_{\mathrm{X}}\log\left\|{s}\right\|^{-1}\mathop{\mathrm{d}\mathrm{d}^{c}}\varphi c_{1}(\overline{L}_{2})\dots c_{1}(\overline{L}_{k})\delta_{\mathrm{Z}},

The first term vanishes because $\operatorname{div}(s|_{\mathrm{Z}})$ and the support of $\varphi$ are disjoint. The second is the limit of

\int_{\mathrm{X}}\log\left|{u_{n}}\right|^{-1/n}\mathop{\mathrm{d}\mathrm{d}^{c}}\varphi c_{1}(\overline{L})^{k-1}\delta_{\mathrm{Z}}.

Using the fact that $\operatorname{div}(u_{n})\cap\mathrm{U}$ is empty and applying the same computation, the term of index $n$ equals

\frac{1}{n}\int_{\mathrm{X}}\varphi\,c_{1}(\overline{M}_{n})c_{1}(\overline{L}_{2})\dots c_{1}(\overline{L}_{k})\delta_{\mathrm{Z}}.

where $\overline{M}_{n}$ is the trivial metrized line bundle $\mathscr{O}_{X}$ , and its meromorphic section $u_{n}$ replacing $s$ . But this integral is zero, by definition of the measures associated to smooth metrized line bundles. ∎