22.1.66 Valuations618C84432CDA

Let $X$ be an irreducible reduced variety over $\mathbb {C}$ of dimension $n$. We recall the Berkovich analytification $X^{\mathrm{An}}$ of $X$ with respect to the trivial valuation on $\mathbb {C}$.

In the remainder of this section, we assume that $X$ is projective.

As a set, $X^{\mathrm{An}}$ is the set of semi-valuations on $X$, in other words, real-valued valuations $v$ on irreducible reduced subvarieties $Y$ of $X$ that are trivial on $\mathbb {C}$. We call $Y$ the support of the semi-valuation $v$. In other words,

We will write $v_{\mathrm{triv}}\in X^{\mathrm{An}}$ for the trivial valuation on $X$: $v_{\mathrm{triv}}(f)=0$ for any $f\in \mathbb {C}(X)^{\times }$.

We endow $X^{\mathrm{An}}$ with the coarsest topology such that

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   for any Zariski open subset $U\subseteq X$, the subset $U^{\mathrm{An}}$ of $X^{\mathrm{An}}$ consisting of semi-valuations whose supports meet $U$ is open;

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   for each Zariski open subset $U\subseteq X$ and each $f\in \mathrm{H}^0(U,\mathcal{O}_ X)$ (here $\mathcal{O}_ X$ is the sheaf of regular functions), the map $|f|\colon U^{\mathrm{An}}\rightarrow \mathbb {R}$ sending $v$ to $\exp (-v(f))$ is continuous.

We will be most interested in divisorial valuations. Recall that a divisorial valuation on $X$ is a valuation of the form $t\operatorname{ord}_ E$, where $t\in \mathbb {Q}_{>0}$ and $E$ is a prime divisor over $X$. The set of divisorial valuations on $X$ is denoted by $X^{\operatorname{div}}$. When $\mathbb {Q}_{>0}$ is replaced by $\mathbb {R}_{>0}$, we can similarly define a space $X^{\operatorname{div}}_{\mathbb {R}}$.

Given any coherent ideal $\mathfrak {a}$ on $X$ and any $v\in X^{\mathrm{An}}$, we define

where $c(v)$ is the center of the valuation $v$ on $X$.

Given any valuation $v$ on $X$, the Gauss extension of $v$ is a valuation $\sigma (v)$ on $X\times \mathbb {A}^1$:

Here $t$ is the standard coordinate on $\mathbb {A}^1=\operatorname{Spec}\mathbb {C}[t]$. The key property is that when $v$ is a divisorial valuation, then so is $\sigma (v)$. See [Lemma 4.2, BHJ17].