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.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.