We return to the compact setting, and until subsection 20.3.23 we assume that $Y$ is a curve. Since every modification of a compact smooth curve is an isomorphism, the divisors over $Y$ are its points, and the trace data prescribes an atomic curvature measure. Stokes' theorem bounds its total mass by the degree of the background on $Y$, and filling the remaining mass with a smooth measure gives a positive current whose potential realizes the data.
Set
\[ D_ u=\sum _{p\in Y}\tau _{Y,u}(p)\, \delta _ p, \]
where the sum is understood as the supremum of its finite subsums.
Theorem 20.3.10.879739B42EBB Put $M=\int _ Y\theta |_ Y$. Then
29F47A0F8A5D
20.3.11
\begin{equation} 0\leq D_ u(Y)\leq M. \end{equation}
The finite atomic measure $D_ u$ determines a canonical ordinary singularity type of quasi-plurisubharmonic functions on $Y$: for every smooth positive form $\eta $ on $Y$ with $\int _ Y\eta =1$, this type has a representative $v\in \mathrm{PSH}(Y,\theta |_ Y)$ with
57C6BC849EC1
20.3.12
\begin{equation} \theta |_ Y+\mathrm{dd}^{\mathrm{c}}v=D_ u+(M-D_ u(Y))\eta . \end{equation}
The extended Monge–Ampère measure of $v$ is $(M-D_ u(Y))\eta $. The ordinary type does not depend on $\eta $, on the choice of the smooth form $\theta $ with $u\in \mathrm{PSH}(X,\theta )$, or on the choice of $u$ within its multiplier-ideal class.
Proof.
For integer $m$ let
\[ D_ m=\frac1m\sum _{p\in Y} \operatorname {ord}_ p\mathfrak a_ m(u;Y)\, \delta _ p. \]
A nonzero coherent ideal on a compact smooth curve has finitely many zeros, so this sum is finite. Since $u_ m|_ Y$ has analytic type $(1/m,\mathfrak a_ m(u;Y))$, the measure $D_ m$ is the atomic part of the positive current $(\theta +\epsilon _ m\omega )|_ Y+\mathrm{dd}^{\mathrm{c}}(u_ m|_ Y)$, and Stokes' theorem on the curve gives
\[ D_ m(Y)\leq M+\epsilon _ m\int _ Y\omega |_ Y. \]
For any finite subset of $Y$, passing to the limit in the sum of the corresponding coefficients by theorem 20.3.2, and then taking the supremum over finite subsets, gives (20.3.11). In particular, the positive coefficients have at most countable support and define a finite measure. Write $d=D_ u(Y)$.
Choose a Green kernel $G$ for $\mathrm{dd}^{\mathrm{c}}$ on the curve, normalized so that
\[ \mathrm{dd}^{\mathrm{c}}_ xG(x,p)=\delta _ p-\eta ,\qquad G(x,p)\leq 0. \]
Such a kernel exists: the Green kernel of the scalar Laplacian on the compact Riemann surface has, in a local coordinate $z$, the singularity $\log |z(x)-z(p)|^2$, it is uniformly bounded above, and $\sup _ p\int _ Y|G(x,p)|\eta (x)<\infty $; subtracting a constant gives the sign condition. By Fubini's theorem,
\[ w(x)=\int _ YG(x,p)\, \mathrm{d}D_ u(p) \]
is integrable, and it is quasi-subharmonic with $\mathrm{dd}^{\mathrm{c}}w=D_ u-d\, \eta $. Its Lelong number at every point $p$ is the mass of $D_ u$ at $p$, namely $\tau _{Y,u}(p)$.
Since $M\eta -\theta |_ Y$ is smooth with integral zero, the Poisson equation gives a smooth $s$ with $\mathrm{dd}^{\mathrm{c}}s=M\eta -\theta |_ Y$. Then $v=w+s$ satisfies (20.3.12), and the right-hand side is positive because $M\geq d$. At every atom of $D_ u$, the positive Lelong number forces $v=-\infty $. The extended measure of $v$ is the restriction of the current in (20.3.12) to the set where $v>-\infty $; this set contains no atom of $D_ u$, and its complement is polar and therefore not charged by $\eta $. Hence the extended measure is $(M-d)\eta $.
If another smooth form $\eta $ or another smooth background $\theta $ is used, the two constructed representatives have the same atomic curvature, and all other terms in their equations are smooth. Their difference therefore has smooth $\mathrm{dd}^{\mathrm{c}}$, so it is smooth by elliptic regularity, and in particular bounded. This proves independence of these choices at the level of ordinary types. Independence of the choice of $u$ within its multiplier-ideal class holds because the numbers defining $D_ u$ depend only on that class.
$\square$
We denote this ordinary type by $\operatorname {Tr}^{\mathrm{at}}_ Y(u)$ and call it the atomic trace of $u$ on the curve $Y$; the same symbol also denotes any of its representatives. All potentials on a curve with these Lelong numbers have the same multiplier ideals, but they need not have the same ordinary type; the construction selects the type whose curvature, apart from the atoms, is a smooth measure. Additivity of the trace data (theorem 20.3.4) and linearity of the Green operator give
\[ \operatorname {Tr}^{\mathrm{at}}_ Y(u+v) \sim \operatorname {Tr}^{\mathrm{at}}_ Y(u)+\operatorname {Tr}^{\mathrm{at}}_ Y(v), \qquad \operatorname {Tr}^{\mathrm{at}}_ Y(au)\sim a\operatorname {Tr}^{\mathrm{at}}_ Y(u) \quad (a>0). \]
Here $v\in \mathrm{QPSH}(X)$ is any function that also has zero generic Lelong number along $Y$.
Keep $D_ m$, $G$ and $w$ from the proof, and put $w_ m=\int _ YG(\cdot ,p)\, \mathrm{d}D_ m(p)$.
Proposition 20.3.13.B3BE3C08418A The measures $D_ m$ satisfy $D_ m\leq D_ u$ and converge to $D_ u$ in total variation, and they increase along dyadic indices. For a fixed nonpositive Green kernel, the potentials $w_{2^ j}$ decrease to $w$, and $w_ m\to w$ in $L^1(Y)$. Each $w_ m$ has the ordinary singularity type of $u_ m|_ Y$.
If the restriction $u|_ Y$ is not identically $-\infty $, it is at least as singular as the atomic trace:
\[ u|_ Y\preceq \operatorname {Tr}^{\mathrm{at}}_ Y(u). \]
If $u$ has analytic singularities, the two are ordinarily equivalent.
Proof.
The supremum formula in (20.3.3) gives $D_ m\leq D_ u$ coefficient by coefficient. Since the coefficients converge and are dominated by those of the finite measure $D_ u$, dominated convergence gives $D_ m(Y)\to D_ u(Y)$, and hence convergence in total variation. Superadditivity with $s=t=m$ gives $D_{2m}\geq D_ m$, and since $G\leq 0$ the potentials $w_{2^ j}$ decrease. Monotone convergence for $-G$ identifies their pointwise limit with $w$, including at poles. The uniform $L^1$ bound for the kernel also gives
\[ \| w_ m-w\| _{L^1(\eta )} \leq C\| D_ m-D_ u\| _{\mathrm{TV}}\longrightarrow 0. \]
Each $D_ m$ is finitely supported, and locally both $w_ m$ and $u_ m|_ Y$ are the sum of the logarithmic singularities prescribed by $D_ m$ and a bounded function. Thus their ordinary types agree.
If $a=u|_ Y$ is not identically $-\infty $, proposition 20.3.6 gives $\nu (a,p)\geq \tau _{Y,u}(p)$ at every point. Since $\mathrm{dd}^{\mathrm{c}}w=D_ u-D_ u(Y)\eta $, lemma 5.38.5 implies $a\preceq w$. If $u$ has analytic singularities, proposition 20.3.6 identifies the finitely many logarithmic coefficients of $u|_ Y$ with those of $D_ u$, and the bounded local remainders give ordinary equivalence.
$\square$
Pome
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