Lemma 5.43.11CE5106030996

Proof. Composing a local analytic presentation with $p$ proves the forward implication. The pulled-back generators are not all identically zero, since a nonconstant map to a curve is open.

Conversely fix $s\in C$ and $x\in p^{-1}(s)$. In coordinates, choose a complex line through $x$ on which $p-s$ has its finite order $e$ and is nonconstant, as in the Lelong lemma. On that line write

where $h$ is a local biholomorphism fixing zero; take an analytic $e$th root of the nonvanishing factor of the germ. The restriction of $v\circ p$ to the line is not identically $-\infty $, since $v$ is finite almost everywhere and this map is open. By proposition 5.34.1, it has analytic singularities. In one variable its presentation is $r\log |\zeta |^2+O(1)$ with $r\in \mathbb Q_{\geq 0}$, obtained by taking the least order of its finitely many generators. Changing to $w=h(\zeta )$ gives

Every sufficiently small $t$ has an $e$th root $w$, so $v(s+t)=(r/e)\log |t|^2+O(1)$. This is an analytic presentation on the base, including a bounded one when $r=0$. $\square$