Proposition 13.8.5A00B4940B406

Proof. The potential $\Phi $. Each function in the supremum is quasi-plurisubharmonic and $\widetilde\theta $-plurisubharmonic: its coefficients of the $\theta $-plurisubharmonic functions $\varphi _0$ and $\varphi $ are nonnegative and add up to one, and so do its coefficients of the functions $s_ i$, since $(1-t(x))+\sum _ ix_ i=1$. The $s_ i$ are $\omega _{\mathrm{FS}}$-plurisubharmonic: in the standard chart $\{ Z_ l\neq 0\} $, where $\omega _{\mathrm{FS}}=\mathrm{dd}^{\mathrm{c}}\log \sum _ m|Z_ m/Z_ l|^2$ as in example 10.6.7, the function $s_ i+\log \sum _ m|Z_ m/Z_ l|^2=\log |Z_ i/Z_ l|^2$ is plurisubharmonic. The family is uniformly bounded above, because $\varphi _0$ and $\varphi $ are bounded above on the compact manifold $X$ and $s_ i\leq 0$. Hence, by theorem 5.10.2 applied after adding smooth local potentials of $\widetilde\theta $, its regularized supremum $\Phi $ is $\widetilde\theta $-plurisubharmonic. The branch $x=0$ is $\tau =\varphi _0+s_0$, which is quasi-plurisubharmonic, equal to $-\infty $ exactly on $A$, and locally bounded outside $A$, because $\varphi _0$ is smooth on $X\setminus P$ and $s_0$ is smooth on $\{ Z_0\neq 0\} $. Since $\Phi \geq \Phi _0\geq \tau $ and $\Phi $ is bounded above, $\Phi $ is locally bounded outside $A$; the set $A$ is closed because $P$ is.

Local form. Let $W\subset X\setminus P$ be a coordinate chart, choose a smooth function $h$ on $W$ with $\mathrm{dd}^{\mathrm{c}}h=\theta $, and use the affine coordinates $z_ i=Z_ i/Z_0$ on $\{ Z_0\neq 0\} =\mathbb C^ N$. There $s_0=-\log (1+|z|^2)$ and $s_ i=\log |z_ i|^2-\log (1+|z|^2)$, and $\log (1+|z|^2)$ is a potential of $\omega _{\mathrm{FS}}$. Since the coefficients of the $s_ i$ add up to one, adding the smooth function $h+\log (1+|z|^2)$ to each function in the supremum turns $\Phi _0$ into the function $F$ of lemma 13.8.2, formed with $u=\varphi _0+h$, which is smooth and plurisubharmonic on $W$, and $v=\varphi +h$, which is plurisubharmonic on $W$. By that lemma, $F$ is plurisubharmonic, hence upper semicontinuous, so $\Phi _0$ is upper semicontinuous on the open set $(X\setminus P)\times \{ Z_0\neq 0\} =(X\times \mathbb P^ N)\setminus A$, and $\Phi =\Phi _0$ there. Thus $F$ is a locally bounded plurisubharmonic potential of $\widetilde\theta +\mathrm{dd}^{\mathrm{c}}\Phi $ on $W\times \mathbb C^ N$, and the non-pluripolar product $\langle (\widetilde\theta +\mathrm{dd}^{\mathrm{c}}\Phi )^{n+N}\rangle $ is the Bedford–Taylor measure $(\mathrm{dd}^{\mathrm{c}}F)^{n+N}$ there.

The mass formula. For $C>0$ put $v_ C=\max \{ v,u-C\} $, a locally bounded plurisubharmonic function on $W$, and let $F_ C$ be the function of lemma 13.8.2 formed with $u$ and $v_ C$. At base points $w$ with $v(w)>u(w)-C$ we have $v_ C(w)=v(w)$, so $F_ C=F$ on the set $Y_ C=\{ (w,z):v(w)>u(w)-C\} $. This set is plurifine open in $W\times \mathbb C^ N$, because $(w,z)\mapsto v(w)$ is plurisubharmonic and $u$ is continuous. By plurifine locality (theorem 5.47.22), $(\mathrm{dd}^{\mathrm{c}}F)^{n+N}$ and $(\mathrm{dd}^{\mathrm{c}}F_ C)^{n+N}$ agree on $Y_ C$.

Similarly, for $0\leq t\leq 1$, the function $(1-t)u+tv$ is a local potential of the closed positive current $\theta +\mathrm{dd}^{\mathrm{c}}((1-t)\varphi _0+t\varphi )$, whose non-pluripolar products are finite by theorem 13.5.4, and on the plurifine open set $\{ v>u-C\} $ it coincides with the bounded function $(1-t)u+tv_ C$; so by proposition 13.2.6,

Let $g\geq 0$ be a bounded Borel function with compact support in $W$. Applying lemma 13.8.2 to $F_ C$ with the Borel weight $g\mathbf1_{\{ v>u-C\} }$, we obtain

Let $C\to \infty $. The sets $\{ v>u-C\} $ increase to $\{ v>-\infty \} $, whose complement is the pole set of $v$; this set and its product with $\mathbb C^ N$ are pluripolar Borel sets, which carry no mass for the non-pluripolar products on $W$ and for the Bedford–Taylor measure $(\mathrm{dd}^{\mathrm{c}}F)^{n+N}$ (corollary 5.47.11). By monotone convergence, first in the inner integrals and then in $t$,

The non-pluripolar product on $X\times \mathbb P^ N$ gives zero mass to the pluripolar Borel set $A$, and the products on $X$ give zero mass to $P$. Hence the left side is the integral of $g\circ p_ X$ over $X\times \mathbb P^ N$. Now cover $X\setminus P$ by countably many compact sets $K_ k$, each contained in a chart $W_ k\subset X\setminus P$, and put $B_ k=K_ k\setminus \bigcup _{j<k}K_ j$. For a nonnegative Borel function $g$ on $X$, applying the identity to the functions $\min \{ g,M\} \mathbf1_{B_ k}$ and using monotone convergence in $M$ and in the sum over $k$, we obtain the identity for $g\mathbf1_{X\setminus P}$, and hence for $g$, since both sides give $P$ zero mass. Taking $g=1$ gives (13.8.6).

Existence of $\varphi _0$. If $\{ \theta \} $ is big, it contains a Kähler current $\theta +\mathrm{dd}^{\mathrm{c}}\psi \geq \delta \omega $ with $\delta >0$, that is, $\psi \in \mathrm{PSH}(X,\theta -\delta \omega )$. By lemma 6.17.1(1), applied to $\psi $ with the background $\theta -\delta \omega $, for every integer $m\geq 1$ there is a quasi-plurisubharmonic function $G_ m=m^{-1}\log R_ m$ with $R_ m$ smooth and $\theta -\delta \omega +\mathrm{dd}^{\mathrm{c}}G_ m\geq -(c'/m)\omega $, where the constant $c'>0$ does not depend on $m$. For $m\geq c'/\delta $, the function $\varphi _0=G_ m$ is $\theta $-plurisubharmonic; its pole set $\{ R_ m=0\} $ is closed, and $\varphi _0$ is smooth on its complement, where $R_ m>0$. $\square$