PomeLemma 20.3.65.72230836D2EE There are $U_ L\sim U$ with $\sup U_ L=0$, $\mathrm{dd}^{\mathrm{c}}U_ L\geq -\epsilon _ Lw$ and $\epsilon _ L\downarrow 0$, such that $U_ L\to 0$ in $L^1(\mathbb P^1)$. Each has zero Lelong number at $p$ and is locally bounded off $p$.
20.3.63 Ordinary trace volumes on a nonclosed ruled surface070063ADB7C2
The preceding product formula computes the mass of a separated representative. When the background itself varies along the curve, even bounded changes of the representative can change its mass. On a product of projective lines we can optimize over the entire ordinary singularity type, including potentials depending on both factors. The residual pole will be unbounded with zero Lelong number. Its multiplier ideals see the imposed divisor, whereas its ordinary type also retains this residual pole.
In this section let $Y=\mathbb P^1\times \mathbb P^1$, let $w$ be the probability Fubini–Study form on $\mathbb P^1$, and denote its pullbacks by $w_1,w_2$. Fix $p\in \mathbb P^1$, and let $G_ p\leq 0$ be the Green potential satisfying $\mathrm{dd}^{\mathrm{c}}G_ p=[p]-w$ and $\sup G_ p=0$. Functions on the first factor are also viewed as functions on $Y$. Put
\[ D=\{ p\} \times \mathbb P^1,\qquad U=1-(1-G_ p)^\alpha ,\qquad r=bG_ p+U, \qquad 0<\alpha <1, \]
where $b\geq 0$. The term $bG_ p$ is identically zero when $b=0$. Choose $c>0$ and a smooth positive function $a$ on the first curve, and set
Thus $\theta $ is Hermitian and is nonclosed precisely when $a$ is nonconstant. The function $r$ specifies an ordinary type; whether that type has $\theta $-plurisubharmonic representatives will follow from the next theorem.
Proof. For $t\leq 0$ put $\chi (t)=1-(1-t)^\alpha $. Then
\[ \chi '(t)=\alpha (1-t)^{\alpha -1}>0,\qquad \chi ''(t)=\alpha (1-\alpha )(1-t)^{\alpha -2}>0. \]
For $L>0$ set $\epsilon _ L=\chi '(-L)$, $C_ L=-\epsilon _ LL-\chi (-L)$, and define
\[ \chi _ L(t)= \begin{cases} \chi (t)+C_ L,& t<-L,\\ \epsilon _ Lt,& -L\leq t\leq 0. \end{cases} \]
The values and first derivatives match, and $\chi _ L$ is convex and increasing with derivative at most $\epsilon _ L$. Off $p$, the chain rule gives $\mathrm{dd}^{\mathrm{c}}\chi _ L(G_ p)\geq -\epsilon _ Lw$. Canonical extension across $p$ retains this bound by the removable-singularity theorem, so $U_ L=\chi _ L(G_ p)$ has the required curvature. Near $p$ it differs from $U$ by $C_ L$, and elsewhere both are bounded. Its growth is sublogarithmic, hence its Lelong number is zero.
Writing $x=1+L$, we have $C_ L=(1-\alpha )x^\alpha +\alpha x^{\alpha -1}-1\geq 0$. Also $\chi (t)\leq \epsilon _ Lt$ for $-L\leq t\leq 0$, by integrating $\chi '\geq \epsilon _ L$ on this interval. It follows that $U\leq U_ L\leq 0$. Since $\epsilon _ L\to 0$, the functions tend to zero off $p$, and dominated convergence gives the $L^1$ limit. $\square$
Theorem 20.3.66.9EDE5892D63B The ordinary type of $r$ has $\theta $-plurisubharmonic representatives if and only if $b<c$. In this range it contains a strict current: some $v\sim r$ satisfies $\theta +\mathrm{dd}^{\mathrm{c}}v\geq \delta (w_1+w_2)$ with $\delta >0$. Every $v\in \mathrm{PSH}(Y,\theta )$ with $v\sim r$ has positive finite extended mass, given by
EEBF58093B80
20.3.67
\begin{equation} \int _ Y\mathrm{MA}^{\mathrm{ext}}_\theta (v) =\int _ Y\theta ^2-2b a(p)+2\int _ Yv\, \mathrm{dd}^{\mathrm{c}}\theta . \end{equation}
The volumes of definition 16.28.58 are
9099C8F4472B
20.3.68
\begin{equation} \underline{\operatorname {vol}}_\theta (v)=2(c-b)\min a, \qquad \overline{\operatorname {vol}}_\theta (v)=2(c-b)\max a. \end{equation}
Both values are limits of masses of representatives of this ordinary type.
Proof. We first remove the logarithmic divisor. If $v\sim r$, then $\nu (v,D)=b$, so lemma 5.38.1 gives $\theta +\mathrm{dd}^{\mathrm{c}}v-b[D]\geq 0$. Taking $q=v-bG_ p$ in $L^1$ and then its quasi-plurisubharmonic representative gives
\[ q\in \mathrm{PSH}(Y,\beta ),\qquad q\sim U,\qquad \beta =kw_1+aw_2,\qquad k=c-b. \]
Indeed the ordinary inequalities hold off $D$ and extend canonically across it. Conversely such a $q$ gives $v=bG_ p+q$ with
\[ \theta +\mathrm{dd}^{\mathrm{c}}v=b[D]+\beta +\mathrm{dd}^{\mathrm{c}}q. \]
Their locally bounded products agree off $D$, and both extended measures give $D$ zero mass. Thus their extended measures agree everywhere.
If $k>0$, choose $L$ with $\epsilon _ L<k/2$. The potential $bG_ p+U_ L$ belongs to the specified type and satisfies
\[ \theta +\mathrm{dd}^{\mathrm{c}}(bG_ p+U_ L) \geq b[D]+(k-\epsilon _ L)w_1+aw_2 \geq \min (k/2,\min a)(w_1+w_2). \]
For necessity, average $q$ over the unitary group acting transitively on the second projective line, with its Haar probability measure. The background is invariant. The average has a genuine first-factor representative $q_0\sim U$ and $kw+\mathrm{dd}^{\mathrm{c}}q_0\geq 0$. One can first average in $L^1$ and then take the canonical representative; the curvature inequality and the ordinary bounds pass to it. The positive curve measure has total mass $k$, excluding $k<0$. When $k=0$, $q_0$ is harmonic on the compact curve and therefore constant, contradicting $q_0\sim U$.
It remains to compute the mass for every competitor, before using the average. Choose a closed Kähler form $K=C(w_1+w_2)\geq \beta $ and a fixed $u_0=U_ L$ as above. The closed positive current $S_0=K+\mathrm{dd}^{\mathrm{c}}u_0$ has no coefficient mass on $D$, since its first-factor curvature has no atom at $p$. The product formula gives
\[ \langle S_0\rangle =S_0,\qquad \langle S_0^2\rangle =2C(Cw_1+\mathrm{dd}^{\mathrm{c}}u_0)\wedge w_2, \qquad \int _ Y\langle S_0^2\rangle =\int _ YK^2. \]
For $S=K+\mathrm{dd}^{\mathrm{c}}q$, both $S$ and $S_0$ are closed positive currents in the same Kähler class, with bounded difference of potentials and small unbounded locus $D$. The classical closed-class comparison of non-pluripolar products, applied in both singularity orders in degrees one and two, gives
\[ \{ \langle S\rangle \} =\{ K\} ,\qquad \int _ Y\langle S^2\rangle =\int _ YK^2. \]
Here the degree-one non-pluripolar current is the restriction of $S$ off $D$ and is closed. Hence the positive current $S-\langle S\rangle $ has zero trace against $K$, and vanishes. This proves $\langle S\rangle =S$. Equivalently, absence of polar coefficient mass follows from proposition 5.46.3. The closed comparison is used only for $K$.
Put $R=\beta -K$. Bedford–Taylor expansion off $D$, followed by extension by zero, now yields
\[ \mathrm{MA}^{\mathrm{ext}}_\beta (q) =\langle S^2\rangle +2R\wedge S+R^2. \]
The first term is finite by closed comparison, the second has finite variation because $S$ has finite trace, and the last is smooth. Since none has coefficient mass on $D$, this is also an identity on $Y$. Distributional Stokes therefore gives
F060F71204CB
20.3.69
\begin{equation} \int _ Y\mathrm{MA}^{\mathrm{ext}}_\beta (q) =\int _ Y\beta ^2+2\int _ Yq\, \mathrm{dd}^{\mathrm{c}}\beta =2\int _{\mathbb P^1}a(kw+\mathrm{dd}^{\mathrm{c}}q_0). \end{equation}
The pairing uses only $q\in L^1$. Averaging preserves it, since $\mathrm{dd}^{\mathrm{c}}\beta =\mathrm{dd}^{\mathrm{c}}a\wedge w_2$. The last measure is positive of total mass $k$, proving the bounds in (20.3.68) for all original competitors. Substituting $q=v-bG_ p$ and $\int G_ p\mathrm{dd}^{\mathrm{c}}a=a(p)-\int aw$ proves (20.3.67).
To approach either bound, choose a point $x$ where $a$ takes the corresponding extreme value, smooth strictly positive measures $\zeta _ j$ of mass $k$ tending weakly to $k\delta _ x$, and smooth $g_ j$ with $\mathrm{dd}^{\mathrm{c}}g_ j=\zeta _ j-kw$. Choose $L_ j$ so that $\epsilon _{L_ j}$ is less than half the minimum density of $\zeta _ j$ relative to $w$ and $|\int U_{L_ j}\mathrm{dd}^{\mathrm{c}}a|<1/j$. Then
\[ v_ j=bG_ p+g_ j+U_{L_ j}\sim r,\qquad \beta +\mathrm{dd}^{\mathrm{c}}(g_ j+U_{L_ j})\geq \tfrac 12\zeta _ j+aw_2>0. \]
Their masses tend to $2ka(x)$ by (20.3.69). $\square$
No finite first moment is used here. For the reference $u_0=U_ L$, the closed first moment $\int _ Y(-u_0)\langle (K+\mathrm{dd}^{\mathrm{c}}u_0)^2\rangle $ is finite exactly when $\alpha <1/2$. Near $p$ put $t=-G_ p$. Then $-u_0\asymp t^\alpha $ and $\chi ''(G_ p)\asymp t^{\alpha -2}$, while angular integration of $\mathrm dG_ p\wedge \mathrm d^{\mathrm c}G_ p$ is comparable to $\mathrm dt$. The singular contribution is thus comparable to $\int ^\infty t^{2\alpha -2}\, \mathrm dt$. The smooth contributions are integrable, and the mass formula includes the divergent range.
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