Lemma 13.8.2984FFB015C24

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. $\square$