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20.3.23 The trace envelope53884234AA76

We return to the general setting, in which $Y\subset X$ has arbitrary positive dimension. The curvature errors of the restricted approximants tend to zero, so normalized compactness gives a limit potential in the original background $\theta |_ Y$, whose divisorial Lelong numbers dominate the trace data. These inequalities define an envelope, and comparison with the resolved restricted ideals identifies it as the limit of the restricted analytic envelopes.

Proposition 20.3.24.787C2A9E0C8D There is a potential $v\in \mathrm{PSH}(Y,\theta |_ Y)$ such that

\[ \nu (v\circ \pi ,E)\geq \tau _{Y,u}(E) \quad \hbox{for every divisor }E\hbox{ over }Y. \]

In particular, $v\in \mathrm{PSH}(Y,\Omega _0)$ for every Hermitian form $\Omega _0\geq \theta |_ Y$.

Proof. The restrictions $u_ m|_ Y$ have a common smooth lower bound for their curvature. After subtracting their suprema, normalized compactness gives a subsequence converging in $L^1$ to some $v$. The inequalities $(\theta +\epsilon _ m\omega )|_ Y+\mathrm{dd}^{\mathrm{c}}(u_ m|_ Y)\geq 0$ pass to the limit as $\epsilon _ m\to 0$, giving $\theta |_ Y+\mathrm{dd}^{\mathrm{c}}v\geq 0$.

For each fixed compact modification $\pi :Z\to Y$, lemma 5.43.19 gives $L^1$ convergence of the pullbacks along the same subsequence, and lemma 5.38.7 gives, at every point $z$ of a prime divisor $E\subset Z$,

\[ \nu (v\circ \pi ,z) \geq \limsup _ m\nu ((u_ m|_ Y)\circ \pi ,z) \geq \lim _ m\frac{\operatorname {ord}_ E\mathfrak a_ m(u;Y)}m =\tau _{Y,u}(E). \]

Taking the generic value along $E$ proves the assertion. No diagonal extraction over divisors is needed, since the pullback lemma applies to each modification once the subsequence on $Y$ has been fixed. $\square$

For a smooth real $(1,1)$-form $\Omega $ on $Y$ for which the following family is nonempty, define

AA3FC60BB1F2 20.3.25
\begin{equation} \begin{split} \mathcal T_\Omega (u;Y):=\Bigl (\sup \{ a\in \mathrm{PSH}(Y,\Omega ):\ & a\leq 0,\\ & \nu (a\circ \pi ,E)\geq \tau _{Y,u}(E) \text{ for every }E\} \Bigr )^*. \end{split} \end{equation}

Here $E$ ranges over all divisors over $Y$, and we call the members of this family trace competitors for $\Omega $. By the preceding proposition, $v-\sup _ Yv$ is a trace competitor for $\Omega =\theta |_ Y$ and for every larger background. The envelope $\mathcal T_\Omega (u;Y)$ depends only on $\Omega $ and on the multiplier-ideal class of $u$ on $X$.

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