Theorem 20.3.39A30F4066C584

Proof. Lemma 5.37.7 and restriction to $Y$ give

The nonzero coherent ideal $\mathfrak a_ t(U;C)$ on the smooth curve is generated near $b$ by $t_ b^{k_ t(b)}$, where $k_ t(b)=\operatorname {ord}_ b\mathfrak a_ t(U;C)$. For a divisor $E$ with $h(E)=\{ b\} $, the order along $E$ of the extended ideal is therefore $k_ t(b)\operatorname {ord}_ E(t_ b\circ h)$. For a divisor dominating $C$, the generic order is zero, since the ideal on the curve has only finitely many zeros. Dividing by $t$ and letting $t\to \infty $ gives exactly the stated formula for $\tau _{Y,u}(E)$.

For the Lelong numbers of $v\circ \pi $, suppose first that $h(E)=\{ b\} $. At a general point of $E$, a local equation $z_1=0$ of $E$ satisfies $t_ b\circ h=z_1^ k g$ with $g$ holomorphic and nonvanishing, and $k=\operatorname {ord}_ E(t_ b\circ h)$. Replacing $z_1$ by $z_1g^{1/k}$ for a local holomorphic $k$th root, we may assume $t_ b\circ h=z_1^ k$. The Lelong number of the one-variable function $w(z_1^ k)$ at $0$ is $k\nu (w,b)=k\tau _{C,U}(b)$, as follows from the definition through circular means, and this is also the generic Lelong number of $v\circ \pi $ along $E$ in these product coordinates. If $h(E)=C$, the regular locus of $E$ contains a point mapped to a point where $w$ is finite, because the nonconstant holomorphic map from that locus to $C$ has open image; the pullback is finite there, so its generic Lelong number along $E$ is zero. This proves exact realization on every compact model.

By lemma 5.38.5 and (20.3.38), a quasi-plurisubharmonic function $a$ on $Y$ satisfies $a\preceq v$ if and only if $\nu (a,Y_ b)\geq \tau _{C,U}(b)$ for every $b$ with $\tau _{C,U}(b)>0$. These fiber inequalities are among the trace inequalities, and conversely domination by $v$ implies all trace inequalities on modifications, by the exact formula just proved. Thus the trace competitors for $\Omega $ are exactly the competitors for the ordinary envelope of $v$, and $\mathcal T_\Omega (u;Y)=P_\Omega [v]$.

For $\psi =P_\Omega [v]_{\mathcal I}$, the Lelong numbers along the fibers are those of $v$, since $\psi $ and $v$ have the same multiplier ideals, and the same domination lemma gives $\psi \preceq v$. Hence

Since $\psi \leq 0$, it is a competitor for the ordinary envelope of $v$, so $\psi \leq P_\Omega [v]$. This proves (20.3.40).

The potential $v-\sup _ Yv$ is a competitor for $P_{\Omega /2}(P_\Omega [v])$. The contact estimate for envelope fixed points, proposition 16.28.7, gives finite mass, and lemma 16.28.69 gives strict positivity. The two envelope properties give the model condition (16.28.67), so theorem 15.12.1 gives the stated equations.

Two Green constructions on $C$ differ by a smooth function, by the Poisson argument in the proof of theorem 20.3.10, so their pullbacks have the same ordinary type. Every background in the statement gives an envelope ordinarily equivalent to this pullback. This proves the asserted independence, at the level of ordinary types rather than of normalized potentials. $\square$