Proposition 20.3.17.F8A7A320265A Choose a smooth positive form $\eta $ of integral one on $Y$ and choose $A>D_ u(Y)$ so large that every restricted approximant $u_ m|_ Y$ is $A\eta $-plurisubharmonic. Define $w$ and $w_ m$ as above, with a fixed nonpositive Green kernel. Then
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20.3.18
\begin{equation} d_{S,A\eta }(w,u_ m|_ Y) =d_{S,A\eta }(w,w_ m) =\frac12\bigl (D_ u(Y)-D_ m(Y)\bigr ) \longrightarrow 0. \end{equation}
In particular, the atomic trace has $\mathcal I$-good singularities on the curve, in the sense recalled in section 18.2. If $X$ is Kähler, the atomic trace represents the $P$-equivalence class of the Kähler trace.
Proof.
The curvature bounds of corollary 5.36.2, restricted to $Y$, are dominated by a single form $A\eta $ once $A$ is large. Since $D_ m\leq D_ u$ and $G\leq 0$, we have $w\leq w_ m$. Also
\[ A\eta +\mathrm{dd}^{\mathrm{c}}w=D_ u+(A-D_ u(Y))\eta , \qquad A\eta +\mathrm{dd}^{\mathrm{c}}w_ m=D_ m+(A-D_ m(Y))\eta . \]
Hence their extended masses are $A-D_ u(Y)$ and $A-D_ m(Y)$, both positive. For $a,b\in \mathrm{PSH}(Y,A\eta )$ with $a\leq b$, the formula for the distance of ordered potentials [Lemma 6.2.2, SGPT] reads
\[ d_{S,A\eta }(a,b) =\frac12\left(\int _ Y\mathrm{MA}_{A\eta }(b) -\int _ Y\mathrm{MA}_{A\eta }(a)\right). \]
Substituting $a=w$ and $b=w_ m$ proves the second equality in (20.3.18). Since $u_ m|_ Y$ and $w_ m$ are ordinarily equivalent, their distance vanishes, which proves the first equality. The convergence of $D_ m$ in total variation proves the limit.
Each $w_ m$ has finitely many logarithmic poles with rational coefficients, hence analytic singularities. The characterization in Theorem 18.1.2, applied on the compact Kähler curve $Y$, shows that their $d_ S$-limit $w$ is $\mathcal I$-good.
Finally, suppose that $X$ is Kähler. The Kähler trace is defined in Definition 20.2.6, Proposition 20.2.4 through a quasi-equisingular approximation of $u$ with analytic types $(2^{-j},\mathcal I(2^ ju))$, whose restriction to $Y$ has the ordinary type of $w_{2^ j}$. The same distance calculation shows that these restrictions converge to $w$ for $d_{S,A\eta }$, so $w$ represents the Kähler trace up to $P$-equivalence.
$\square$
Pome
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