13.8 Mass comparison and projective fibersC76D4F3419F8

Proof. We replace the factors $T_ j$ by $T'_ j$ one at a time; the intermediate tuples satisfy the same hypotheses, so it suffices to treat one step, in which a single factor changes. Write it as $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ before and $\theta +\mathrm{dd}^{\mathrm{c}}\psi $ after the change, and let $\Theta $ be the non-pluripolar product of the remaining factors (some primed, some not) wedged with $\omega ^{n-p}$. By theorem 13.5.4, theorem 13.3.1, $\Theta $ is a closed positive current of bidimension $(1,1)$, and it gives zero mass to the set $A$, which is pluripolar and Borel, being closed and locally contained in the pole sets of plurisubharmonic functions.

On the open set $X\setminus A$ all potentials are locally bounded, so there the non-pluripolar products of the remaining factors are Bedford–Taylor products, and by proposition 13.4.3 the products of the two tuples, wedged with $\omega ^{n-p}$, are the Bedford–Taylor products $(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )\wedge \Theta $ and $(\theta +\mathrm{dd}^{\mathrm{c}}\psi )\wedge \Theta $ with base $\Theta $; here $(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )\wedge \Theta =\mathrm{dd}^{\mathrm{c}}(h+\varphi )\wedge \Theta $ for smooth local potentials $h$ of $\theta $. More generally, if $\Omega '$ is a smooth closed real $(1,1)$-form and $\varrho \in \mathrm{PSH}(X,\Omega ')$ is locally bounded on $X\setminus A$, then the measure $(\Omega '+\mathrm{dd}^{\mathrm{c}}\varrho )\wedge \Theta $ on $X\setminus A$ is the restriction of a global non-pluripolar product wedged with $\omega ^{n-p}$, so it has finite mass by theorem 13.5.4. Since none of these products charges $A$, it suffices to prove

We prove it by applying Stokes' formula on $X\setminus A$ to functions that equal $\varphi -k$ near $A$. After adding constants we may assume $\varphi \geq \psi $ and $\tau \leq 0$. Since $\tau $ is quasi-plurisubharmonic and $X$ is compact, there is $C_\tau \geq 0$ with $C_\tau \omega +\mathrm{dd}^{\mathrm{c}}\tau \geq 0$ (lemma 6.14.4(4)). Put $\Omega =(C_\tau +1)\omega $. Fix $0<\varepsilon \leq 1$ and put $f_ k=\max \{ \psi +\varepsilon \tau ,\varphi -k\} $. Since $\theta +\mathrm{dd}^{\mathrm{c}}\psi \geq 0$ and $\varepsilon (C_\tau \omega +\mathrm{dd}^{\mathrm{c}}\tau )\geq 0$, both $\psi +\varepsilon \tau $ and $\varphi -k$ are $(\theta +\Omega )$-plurisubharmonic, hence so is $f_ k$, and $f_ k$ is locally bounded on $X\setminus A$. Since $\psi +\varepsilon \tau -\varphi \leq \varepsilon \tau $, we have $f_ k=\varphi -k$ on the open neighborhood $\{ \tau <-k/\varepsilon \} $ of $A$. Hence on $X\setminus A$ the function $g_ k=f_ k-\varphi +k=\max \{ \psi +\varepsilon \tau -\varphi +k,0\} $ takes values in $[0,k]$, and it vanishes outside the compact set $L_ k=\{ \tau \geq -k/\varepsilon \} \subset X\setminus A$.

Stokes' formula. With a smooth local potential $h$ of $\theta +\Omega $, the Bedford–Taylor recursion gives, on $X\setminus A$,

because $\mathrm{dd}^{\mathrm{c}}(k\Theta )=0$. The signed measure $\lambda _ k=\mathrm{dd}^{\mathrm{c}}(g_ k\Theta )$ vanishes outside $L_ k$. Let $\kappa $ be a smooth function with compact support in $X\setminus A$ that equals one on a neighborhood of $L_ k$. Then $\lambda _ k(X\setminus A)=\int \kappa \, \mathrm d\lambda _ k=(g_ k\Theta )(\mathrm{dd}^{\mathrm{c}}\kappa )=0$, because $\mathrm{dd}^{\mathrm{c}}\kappa $ vanishes near the support of $g_ k\Theta $. Since the measures on the left have finite mass, we obtain

Passage to the limit. On $X\setminus A$ the functions $f_ k$ decrease to the locally bounded function $\psi +\varepsilon \tau $ as $k\to \infty $. By decreasing continuity with the fixed base $\Theta $ (theorem 7.3.7), applied locally, the positive measures $(\theta +\Omega +\mathrm{dd}^{\mathrm{c}}f_ k)\wedge \Theta $ converge weakly on $X\setminus A$ to $(\theta +\Omega +\mathrm{dd}^{\mathrm{c}}(\psi +\varepsilon \tau ))\wedge \Theta $. The mass of the limit on the open set $X\setminus A$ is at most the lower limit of the masses, because it is the supremum of the integrals of test functions $0\leq \kappa \leq 1$ supported in $X\setminus A$, and each such integral is the limit of the corresponding integrals for the approximating measures. Therefore

By additivity of Bedford–Taylor products, the measure on the left is the sum of the positive measures $(\theta +\mathrm{dd}^{\mathrm{c}}\psi )\wedge \Theta $, $\varepsilon (C_\tau \omega +\mathrm{dd}^{\mathrm{c}}\tau )\wedge \Theta $ and $(\Omega -\varepsilon C_\tau \omega )\wedge \Theta $, and the measure on the right is the sum of $(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )\wedge \Theta $ and $\Omega \wedge \Theta $; all these measures have finite mass on $X\setminus A$. Dropping the second term on the left, we get

and letting $\varepsilon \downarrow 0$ proves the desired inequality. If the differences are bounded, the same argument with the roles of $\varphi $ and $\psi $ exchanged gives equality. $\square$

Proof. After showing that $F$ is plurisubharmonic, we compute $(\mathrm{dd}^{\mathrm{c}}F)^{n+N}$ explicitly for smooth $v$, and we treat locally bounded $v$ by decreasing approximation. Points of $U\times \mathbb C^ N$ are also denoted by $q$.

Step 1: $F$ is plurisubharmonic. For fixed $x\in \Sigma _ N$, the function

is plurisubharmonic on $U\times \mathbb C^ N$, as a sum of nonnegative multiples of plurisubharmonic functions and a constant (proposition 5.9.5). Since $\phi _0=u$ and $\phi _ x\leq \max \{ u,v\} +\sum _ i\max \{ \log |z_ i|^2,0\} $ for all $x$, the family $(\phi _ x)$ is locally uniformly bounded above, and $F$ lies between $u$ and the right side of this inequality; hence $F$ is locally bounded. The function $(x,q)\mapsto \phi _ x(q)$ on $\Sigma _ N\times (U\times \mathbb C^ N)$ is upper semicontinuous. Indeed, if $c\geq 0$ is a continuous function of $x$ and $f$ is an upper semicontinuous function of $q$ with values in $[-\infty ,M]$ for some $M\in \mathbb R$, then $cf$ is upper semicontinuous, with the convention $0\cdot (-\infty )=0$: at a point where $c=0$, we have $cf\leq cM^+\to 0$ nearby, and at a point where $c>0$ this follows from the upper semicontinuity of $f$. This applies to the terms $(1-t(x))u$, $t(x)v$ and $x_ i\log |z_ i|^2$, and a finite sum of upper semicontinuous functions with values in $[-\infty ,\infty )$ is upper semicontinuous. Since $\Sigma _ N$ is compact, $F$ is upper semicontinuous: if $q_ k\to q$ and $x_ k\in \Sigma _ N$ satisfy $\phi _{x_ k}(q_ k)\geq F(q_ k)-1/k$, then along a subsequence with $\varlimsup _ kF(q_ k)$ as limit, $x_ k\to x$, and $\varlimsup _ kF(q_ k)\leq \phi _ x(q)\leq F(q)$. Hence $F$ coincides with its upper semicontinuous regularization, which is plurisubharmonic by theorem 5.10.2.

Step 2: regularity of $F$ for smooth $v$. Let $v$ be smooth, and work on $G=U\times (\mathbb C^*)^ N$. Put $a=v-u$, $y_ i=\log |z_ i|^2$ and $b_ i=(y_ i+a)/2$. Since

we have $F=u+\gamma (b)$ with $\gamma (b)=\max _{x\in \Sigma _ N}(2x\cdot b-|x|^2)=|b|^2-|b-P(b)|^2$, where $P(b)$ is the closest point of $\Sigma _ N$ to $b$; it exists by compactness and is unique because $x\mapsto |x-b|^2$ is strictly convex on the convex set $\Sigma _ N$. Differentiating $\sigma \mapsto |b-P(b)-\sigma (x'-P(b))|^2$ at $\sigma =0^+$ for $x'\in \Sigma _ N$ gives the variational inequality $(b-P(b))\cdot (x'-P(b))\leq 0$. Applying it to $b$ with $x'=P(b')$ and to $b'$ with $x'=P(b)$ and adding, we get $|P(b)-P(b')|^2\leq (b-b')\cdot (P(b)-P(b'))$, so $P$ is $1$-Lipschitz. Moreover $\gamma (b')\geq 2P(b)\cdot b'-|P(b)|^2=\gamma (b)+2P(b)\cdot (b'-b)$, and symmetrically, so that

Hence $\gamma $ is differentiable, with the Lipschitz gradient $2P$. Consequently $F$ is of class $C^1$ on $G$ with locally Lipschitz first derivatives, and

The projection is explicit. Let $\lambda (b)=0$ if $\sum _ i\max \{ b_ i,0\} \leq 1$; otherwise let $\lambda (b)>0$ be the unique solution of $\sum _ i\max \{ b_ i-\lambda ,0\} =1$, which exists because the left side is continuous, nonincreasing, and strictly decreasing where it is positive. The point $x$ with coordinates $\max \{ b_ i-\lambda (b),0\} $ lies in $\Sigma _ N$ and satisfies the variational inequality: if $I$ is the set of indices with $b_ i>\lambda (b)$, then for $x'\in \Sigma _ N$ the left side of the inequality is $\sum _{i\in I}\lambda (b)(x'_ i-x_ i)+\sum _{i\notin I}b_ ix'_ i\leq \lambda (b)(t(x')-t(x))\leq 0$, since $t(x)=1$ when $\lambda (b)>0$. A point $x\in \Sigma _ N$ satisfying the variational inequality for all $x'\in \Sigma _ N$ is the closest point, because $|b-x'|^2=|b-x|^2+|x'-x|^2-2(b-x)\cdot (x'-x)\geq |b-x|^2$; hence $P_ i(b)=\max \{ b_ i-\lambda (b),0\} $.

For a nonempty set $I\subset \{ 1,\ldots ,N\} $ put $\lambda _ I(b)=(\sum _{i\in I}b_ i-1)/|I|$, and consider the open sets

and $\Omega _{\varnothing ,0}=\{ b:b_ i<0\text{ for all }i\} $. On $\Omega _{I,0}$ we have $\lambda (b)=0$ and $P(b)=(b_ i\mathbf1_ I(i))_ i$, and on $\Omega _{I,+}$ we have $\lambda (b)=\lambda _ I(b)$, by the uniqueness of $\lambda $, and $P_ i(b)=(b_ i-\lambda _ I(b))\mathbf1_ I(i)$; in either case $P$ is affine there. Let $\mathcal H$ be the finite family of affine hyperplanes $\{ b_ i=0\} $, $\{ \sum _{i\in I}b_ i=1\} $ and $\{ b_ i=\lambda _ I(b)\} $, for $1\leq i\leq N$ and nonempty $I$, omitting the empty sets $\{ b_ i=\lambda _{\{ i\} }(b)\} $. Every $b$ outside $\bigcup \mathcal H$ lies in one of the sets $\Omega _{I,0}$, $\Omega _{I,+}$: if $\lambda (b)=0$, it lies in $\Omega _{I,0}$ with $I=\{ i:b_ i>0\} $, and if $\lambda (b)>0$, then $\lambda (b)=\lambda _ I(b)$ for $I=\{ i:b_ i>\lambda (b)\} $, and $b\in \Omega _{I,+}$ because no $b_ i$ equals $\lambda _ I(b)$.

Let $Z\subset G$ be the closed set of points with $b\in \bigcup \mathcal H$. For a fixed base point, each hyperplane of $\mathcal H$ is given by a nonconstant affine equation in $y=(\log |z_ i|^2)_ i$; in polar coordinates $z_ i=r_ i\mathrm e^{\mathrm i\alpha _ i}$, the corresponding set is the product of the image of a hyperplane of $\mathbb R^ N$ under the diffeomorphism $y\mapsto (\mathrm e^{y_ i/2})_ i$ with the torus of angles, so it is a Lebesgue null set. By Fubini's theorem, $Z$ is a Lebesgue null set. On $G\setminus Z$ the function $F=u+|b|^2-|b-P(b)|^2$ is smooth, because $P$ is locally affine there.

Step 3: for smooth $v$, $(\mathrm{dd}^{\mathrm{c}}F)^{n+N}$ is carried by the interior region. On the interior region $\Omega _{\{ 1,\ldots ,N\} ,0}=\{ b:b_ i>0,\ \sum _ ib_ i<1\} $, the interior of $\Sigma _ N$, we have $x=P(b)=b$ and $F=u+\sum _ ix_ i^2$. On every other region $\Omega $, the complex Hessian of $F$, which is positive semidefinite by proposition 5.9.2(1), has a vertical null vector, and hence zero determinant. Indeed, if $j\notin I$, then $x_ j=P_ j(b)=0$ on $\Omega $, so $\partial F/\partial z_ j=x_ j/z_ j=0$ there by (13.8.4), and all second derivatives $\partial ^2F/\partial z_ j\partial \bar z_ k$ vanish. If instead $I=\{ 1,\ldots ,N\} $ and $\lambda >0$, then $\lambda =\bar b-1/N$ with $\bar b=N^{-1}\sum _ ib_ i$, and since $|b|^2=\sum _ i(b_ i-\bar b)^2+N\bar b^2$,

For $\zeta \in \mathbb C$ near $0$, the point $(w,\mathrm e^\zeta z_1,\ldots ,\mathrm e^\zeta z_ N)$ stays in the region, every $y_ i$ increases by $2\operatorname{Re}\zeta $, and the differences $b_ i-\bar b=(y_ i-\bar y)/2$ do not change, so $F(w,\mathrm e^\zeta z)=F(w,z)+2\operatorname{Re}\zeta $ is harmonic in $\zeta $. Its Laplacian at $\zeta =0$ is a positive multiple of the value of the complex Hessian of $F$ on the vector $\xi =\sum _ iz_ i\, \partial /\partial z_ i$, which therefore vanishes; since the Hessian is positive semidefinite, $\xi $ is a null vector by the Cauchy–Schwarz inequality.

We now identify $(\mathrm{dd}^{\mathrm{c}}F)^{n+N}$ on $G$. Let $D$ be a first derivative of $F$ with respect to a real coordinate; by Step 2 it has a Lipschitz constant $L$ on a given ball $B\Subset G$. For a test function $\chi \geq 0$ supported in a smaller ball and a real coordinate $q_ l$,

so by corollary 2.8.2 the distributional derivative $\partial _{q_ l}D$ is a locally bounded function. Hence the distributional second derivatives of $F$ are locally bounded functions on $G$, and on the open set $G\setminus Z$ they coincide almost everywhere with the classical ones, since $F$ is smooth there. As $Z$ is a null set, the Hessian determinant $\det (F_{j\bar k})$, formed with these functions, is a locally bounded function that equals the classical determinant on $G\setminus Z$.

Let $F_\varepsilon =F*\rho _\varepsilon $ be the mollifications, defined, smooth and plurisubharmonic on a relatively compact open set $G'\Subset G$ for small $\varepsilon $ (proposition 5.9.2(2)). Their second derivatives are the mollifications of those of $F$ (proposition 2.5.3); by lemma 2.5.1 they converge in $L^1_{\mathrm{loc}}(G')$ and are locally uniformly bounded. Since the determinant is a polynomial in the entries, $\det (F_{\varepsilon ,j\bar k})\to \det (F_{j\bar k})$ in $L^1_{\mathrm{loc}}(G')$. On the other hand $F_\varepsilon \to F$ locally uniformly, because $F$ is continuous, so $(\mathrm{dd}^{\mathrm{c}}F_\varepsilon )^{n+N}\to (\mathrm{dd}^{\mathrm{c}}F)^{n+N}$ weakly by proposition 7.2.6. With (7.2.4) for the smooth functions $F_\varepsilon $, we conclude that $(\mathrm{dd}^{\mathrm{c}}F)^{n+N}=\frac{(n+N)!}{\pi ^{n+N}}\det (F_{j\bar k})\, \mathrm dV$ on $G$. In particular this measure does not charge the null set $Z$, and it vanishes on all regions except the interior one. It does not charge the pluripolar complement of $G$ either (corollary 5.47.11), so $(\mathrm{dd}^{\mathrm{c}}F)^{n+N}$ is carried by the set $G_0=\{ (w,z)\in G:b(w,z)\in \operatorname{Int}\Sigma _ N\} $ over the interior region.

Step 4: the formula for smooth $v$. On $G_0$ we have $F=u+\sum _ ix_ i^2$ with $x_ i=(y_ i+a)/2$. Since $\mathrm{dd}^{\mathrm{c}}(x_ i^2)=2x_ i\mathrm{dd}^{\mathrm{c}}x_ i+2\, \mathrm dx_ i\wedge \mathrm d^{\mathrm c}x_ i$ and $\mathrm{dd}^{\mathrm{c}}x_ i=\mathrm{dd}^{\mathrm{c}}a/2$, the functions $y_ i$ being pluriharmonic on $G$,

with $t=t(x)$. The form $\Lambda =(1-t)\mathrm{dd}^{\mathrm{c}}u+t\mathrm{dd}^{\mathrm{c}}v$ contains only differentials of the base variables, so $\Lambda ^{n+1}=0$, and $\Gamma _ i\wedge \Gamma _ i=0$. Since forms of even degree commute, the binomial expansion leaves only the term $\binom {n+N}{n}\Lambda ^ n\wedge (\sum _ i\Gamma _ i)^ N$, and $(\sum _ i\Gamma _ i)^ N=N!\, \Gamma _1\wedge \cdots \wedge \Gamma _ N$. The map $(w,z)\mapsto (w,x,\alpha )$, with $\alpha _ i=\arg z_ i$, is an orientation-preserving diffeomorphism of $G_0$ onto $U\times \operatorname{Int}\Sigma _ N\times (\mathbb R/2\pi \mathbb Z)^ N$, with inverse $z_ i=\mathrm e^{x_ i-a(w)/2}\mathrm e^{\mathrm i\alpha _ i}$. With our normalization, $\mathrm d^{\mathrm c}y_ i=\mathrm d\alpha _ i/(2\pi )$, so $\Gamma _ i=\mathrm dx_ i\wedge \mathrm d\alpha _ i/(2\pi )+\mathrm dx_ i\wedge \mathrm d^{\mathrm c}a$, and the second term contains a differential of the base variables and disappears in the product with $\Lambda ^ n$. Hence, in these coordinates,

Let $g\geq 0$ be a bounded Borel function with compact support in $U$. Since $(\mathrm{dd}^{\mathrm{c}}F)^{n+N}$ is carried by $G_0$, integrating $g(w)$ against this measure in the coordinates $(w,x,\alpha )$, first over the angles by Fubini's theorem, we obtain

where $\Phi (t)=\int _ Ug\, (\mathrm{dd}^{\mathrm{c}}((1-t)u+tv))^ n$. By multilinearity of Bedford–Taylor products, $\Phi $ is a polynomial in $t$. The image of Lebesgue measure on $\Sigma _ N$ under $t$ is $t^{N-1}\, \mathrm dt/(N-1)!$ on $[0,1]$, because $\{ x\in \Sigma _ N:t(x)\leq t_0\} =t_0\Sigma _ N$ has volume $t_0^ N|\Sigma _ N|$ and $|\Sigma _ N|=1/N!$ by iterated integration, and finite measures on $[0,1]$ that agree on the intervals $[0,t_0]$ agree (lemma 2.2.6). This proves (13.8.3) for smooth $v$; for signed bounded Borel $g$ split $g$ into its positive and negative parts.

Step 5: support for smooth $v$. If $a\geq -C$ at a base point, then at the points of $G_0$ over it we have $y_ i=2x_ i-a\leq 2+C$, because $x_ i<1$; hence $|z_ i|\leq \mathrm e^{1+C/2}$. Since $(\mathrm{dd}^{\mathrm{c}}F)^{n+N}$ is carried by $G_0$, this proves the support assertion for smooth $v$.

Step 6: general $v$. Let $v$ be locally bounded, let $U'\Subset U$ be open, and let $v_ l$ be smooth plurisubharmonic functions on a neighborhood of $\overline{U'}$ decreasing to $v$, for instance mollifications (proposition 5.9.2(2)). Let $F_ l$ be the corresponding functions. For each $x$, the branch of $F_ l$ exceeds that of $F$ by $t(x)(v_ l-v)\in [0,v_ l-v]$, so $F\leq F_ l\leq F+(v_ l-v)$, and the $F_ l$ decrease pointwise to $F$; by Step 1 they are locally uniformly bounded. By theorem 7.3.7, $(\mathrm{dd}^{\mathrm{c}}F_ l)^{n+N}\to (\mathrm{dd}^{\mathrm{c}}F)^{n+N}$ weakly on $U'\times \mathbb C^ N$. If $v-u\geq -C$ on an open set $E$, then also $v_ l-u\geq -C$ there, so by Step 5 the measures $(\mathrm{dd}^{\mathrm{c}}F_ l)^{n+N}$ give no mass to the open set of points over $E\cap U'$ at which some $|z_ i|>\mathrm e^{1+C/2}$. The weak limit gives no mass to this open set either, since the mass of an open set is the supremum of the integrals of test functions $0\leq \chi \leq 1$ supported in it. This proves the support assertion.

Now let $g$ be continuous with compact support in $U'$. Since $v$ is bounded below and $u$ is bounded above on a relatively compact neighborhood $E$ of $\operatorname{Supp}g$, there is $C$ with $v-u\geq -C$ on $E$, so all the measures $(\mathrm{dd}^{\mathrm{c}}F_ l)^{n+N}$ and $(\mathrm{dd}^{\mathrm{c}}F)^{n+N}$ over $E$ are carried by $E\times \{ |z_ i|\leq \mathrm e^{1+C/2}\} $. Multiplying $g\circ p$ by a test function of $z$ equal to one on this polydisc, we see that $\int (g\circ p)(\mathrm{dd}^{\mathrm{c}}F_ l)^{n+N}\to \int (g\circ p)(\mathrm{dd}^{\mathrm{c}}F)^{n+N}$; here we use that weakly convergent positive measures with locally bounded masses converge against continuous functions with compact support.

On the right side of (13.8.3), the functions $(1-t)u+tv_ l$ decrease to $(1-t)u+tv$ for each $t$, so $\int g\, (\mathrm{dd}^{\mathrm{c}}((1-t)u+tv_ l))^ n\to \int g\, (\mathrm{dd}^{\mathrm{c}}((1-t)u+tv))^ n$ by theorem 7.3.7; these integrals are bounded uniformly in $t\in [0,1]$ and $l$ by theorem 7.2.5, and dominated convergence in $t$ applies. Hence (13.8.3) holds on $U'$ when tested against continuous functions with compact support. Both sides are positive measures that are finite on compact subsets of $U'$, so they agree as measures by the uniqueness in theorem 2.5.6, and therefore also against bounded Borel functions with compact support. Exhausting $U$ by such sets $U'$ proves the lemma. $\square$

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 $R_ m>0$. $\square$

Proof. Suppose first that $\{ \theta \} $ is big. Choose a reference potential $\varphi _0$ as in proposition 13.8.5, which exists because the class is big, and construct the corresponding potentials $\Phi _ N$ from $\varphi $ and $\Psi _ N$ from $\psi $. If $\varphi \geq \psi -C$, then for every $x\in \Sigma _ N$ the branch defining $\Phi _ N$ is at least the branch defining $\Psi _ N$ minus $t(x)C\leq C$; hence the suprema and their regularizations satisfy $\Phi _ N\geq \Psi _ N-C$, with a constant independent of $N$. By proposition 13.8.5, both are locally bounded outside the same closed set $A$, which is the pole set of the quasi-plurisubharmonic function $\varphi _0+s_0$, and this function is locally bounded outside $A$. The product $X\times \mathbb P^ N$ is a connected compact Kähler manifold of dimension $n+N$, with the Kähler form $p_ X^*\omega +p_{\mathbb P^ N}^*\omega _{\mathrm{FS}}$, so the results of this section apply to it.

For $0\leq t\leq 1$ put

and define $f_\psi $ in the same way. Applying lemma 13.8.1 on $X\times \mathbb P^ N$, with all $n+N$ factors equal to $\widetilde\theta +\mathrm{dd}^{\mathrm{c}}\Phi _ N$, respectively $\widetilde\theta +\mathrm{dd}^{\mathrm{c}}\Psi _ N$, and cancelling the common binomial coefficient in (13.8.6), we obtain

Writing $\theta +\mathrm{dd}^{\mathrm{c}}((1-t)\varphi _0+t\varphi )=(1-t)T_0+tT_\varphi $ with $T_0=\theta +\mathrm{dd}^{\mathrm{c}}\varphi _0$ and $T_\varphi =\theta +\mathrm{dd}^{\mathrm{c}}\varphi $, multilinearity shows that $f_\varphi $ is a polynomial in $t$, in particular continuous on $[0,1]$, and $f_\varphi (1)$ is the mass of $\varphi $. The probability measures $Nt^{N-1}\, \mathrm dt$ concentrate at $1$ as $N\to \infty $, since their mass on $[0,1-\delta ]$ is $(1-\delta )^ N$. By uniform continuity the two integrals tend to $f_\varphi (1)$ and $f_\psi (1)$, which proves the theorem for big classes.

In general, $\{ \theta \} $ is pseudo-effective, because it contains $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $. For $\varepsilon >0$ the class of $\theta +\varepsilon \omega $ is big, since it contains the Kähler current $\theta +\varepsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi \geq \varepsilon \omega $, and $\varphi ,\psi $ are $(\theta +\varepsilon \omega )$-plurisubharmonic. By the big case, the mass of $\varphi $ with respect to $\theta +\varepsilon \omega $ is at least that of $\psi $. By multilinearity and theorem 13.5.4, both masses are polynomials in $\varepsilon $ with finite coefficients, and letting $\varepsilon \downarrow 0$ proves the theorem. $\square$

Proof. Bounded differences. For real numbers $a_0,a_1,\ldots ,a_ p>0$, apply theorem 13.8.7 in both directions to the potentials $\sum _ ja_ ju_ j$ and $\sum _ ja_ jv_ j$, whose difference is bounded, with respect to the background $a_0\omega +\sum _ ja_ j\theta _ j$. The two resulting masses are equal. By multilinearity, both are homogeneous polynomials of degree $n$ in $(a_0,\ldots ,a_ p)$, and they agree on an open set; fixing all but one variable and using the uniqueness of polynomials in one variable successively, we see that their coefficients agree. By proposition 13.4.3, the coefficient of $a_0^{n-p}a_1\cdots a_ p$ is $n!/(n-p)!$ times the mixed mass in the statement, which proves equality.

The ordered case. Put $v_{j,k}=\max \{ u_ j-k,v_ j\} $. Since $v_ j\leq u_ j+C_ j$ for constants $C_ j$, we have $u_ j-k\leq v_{j,k}\leq u_ j+C_ j$, so each tuple $(v_{j,k})_ j$ has the same mixed mass as $(u_ j)_ j$, by the first part. As $k\to \infty $, $v_{j,k}$ decreases to $v_ j$ and therefore converges to it in extended capacity (lemma 13.7.3). The lower inequality of theorem 13.7.5, with test form $\omega ^{n-p}$, bounds the mixed mass of the $v_ j$ by these equal masses. $\square$

Proof. The inequality $u_{j,k}\leq u_ j$ holds almost everywhere, hence everywhere, since $\max \{ u_{j,k},u_ j\} $ and $u_ j$ are quasi-plurisubharmonic functions that agree almost everywhere. On coordinate charts add smooth local potentials and truncate below at a fixed level $-C$. The resulting locally uniformly bounded plurisubharmonic sequences increase almost everywhere to the truncations of the limits, so they converge in capacity by theorem 7.6.1; by lemma 13.7.3, the original sequences converge in extended capacity. Since $u_{j,k}\leq u_ j$, corollary 13.8.8 bounds every mixed mass of the sequence by that of the limit, and theorem 13.7.5 gives weak convergence and convergence of the masses. This applies in particular to the regularized supremum of an increasing sequence in $\mathrm{PSH}(X,\theta _ j)$ that is bounded above: it belongs to $\mathrm{PSH}(X,\theta _ j)$ by theorem 5.10.2, and it differs from the pointwise supremum only on a pluripolar set, by corollary 5.47.11, hence only on a Lebesgue null set, because plurisubharmonic functions are locally integrable. $\square$