Theorem 7.3.7 (Bedford–Taylor)D87C13B3250A

Proof. We argue by induction on $q$. Step 1 derives the unweighted assertion and the symmetry from the weighted assertion for fewer factors. The other steps prove the weighted assertion on a ball, after a modification near its boundary: every weak limit of the weighted products is at most the expected limit (Step 4), and repeated integration by parts gives the reverse inequality for the total masses (Step 5).

Step 1: the induction. We prove the second (weighted) assertion by induction on $q$, simultaneously for all open sets $\Omega \subset \mathbb C^ n$, all $r\ge q$, all closed positive currents $T$ of bidimension $(r,r)$ on $\Omega $ and all decreasing sequences as in the statement. For $q=0$ it states that $u_{0,j}T\to u_0T$; this follows from dominated convergence with respect to the total variations of the coefficient measures of $T$, since $u_0\le u_{0,j}\le u_{0,1}$. Let $q\ge 1$ and assume the weighted assertion for fewer than $q$ factors. Applying it to the $q-1$ factors $u_{2,j},\ldots ,u_{q,j}$ with the weight $u_{1,j}$, and then applying the weakly continuous operator $\mathrm{dd}^{\mathrm{c}}$, we obtain the unweighted assertion for $q$ factors.

Next we prove symmetry for $q$ factors. Let $u_1,\ldots ,u_ q$ be locally bounded and plurisubharmonic, let $\varepsilon >0$, and choose $\varepsilon _ k\downarrow 0$ with $\varepsilon _ k<\varepsilon $. The regularizations $u_ l*\rho _{\varepsilon _ k}$ are smooth plurisubharmonic functions on $\Omega _\varepsilon $ decreasing to $u_ l$ (proposition 5.9.2). By the unweighted assertion, the product $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge T$ on $\Omega _\varepsilon $ is therefore the weak limit of exterior products of smooth forms with $T$ (proposition 7.2.2(1)), and these are symmetric in the factors because $2$-forms commute. Since the sets $\Omega _\varepsilon $ exhaust $\Omega $, products with $q$ factors are symmetric. It remains to prove the weighted assertion for $q$ factors; in doing so we may use the unweighted assertion and the symmetry for $q$ factors.

Step 2: reduction to a common collar. By lemma 7.3.6, it suffices to prove the convergence near every point. Fix a ball $B=B(a,R)$, a larger ball $B_1$ with $\overline B\subset B_1\Subset \Omega $, and a compact set $K\subset B$; we prove the convergence on a neighborhood of $K$. Since $u_ l\le u_{l,j}\le u_{l,1}$, all the functions $u_{l,j}$ and $u_ l$ are bounded on $B_1$ uniformly in $j$. Subtracting a constant from $u_{l,j}$ and $u_ l$ changes neither the Hessians nor, since the unweighted assertion is already known, the validity of the weighted assertion. We may therefore assume that $-N\le u_{l,j},u_ l\le -1$ on $B_1$ for all $l$ and $j$, where $N\ge 1$. Put $\psi =|z-a|^2-R^2$, choose $\delta >0$ with $K\subset \{ \psi <-\delta \} $, and replace every function $v$ among the $u_{l,j}$ and $u_ l$ by the function $\max \{ v,A\psi \} $ of (7.3.2), with $U=B_1$ and $A=N/\delta $. The new functions are plurisubharmonic on $B_1$, and the new sequences still decrease to the new limits. The functions are unchanged on the neighborhood $\{ \psi <-\delta \} $ of $K$, and by locality so are their products. From now on, $u_{l,j}$ and $u_ l$ denote the new functions; they take values in $[-N,0)$ on $B$ and coincide with $A\psi $ on the collar $V=\{ z\in B_1:\psi (z)>-1/A\} $ of $\partial B$.

Step 3: smooth approximation with the same collar. Fix $\eta >0$ such that $\{ \psi \le \eta \} \subset B_1$, and let $V'=\{ z:-1/(2A)<\psi (z)<\eta \} $; it is a collar of $\partial B$ contained in $V$. Fix $l$ and $j$, and let $\tilde u$ be the function $u_{l,j}$ before the modification of Step 2, so that $\tilde u\le -1$ on $B_1$ and $u_{l,j}=\max \{ \tilde u,A\psi \} $. For $0<\varepsilon <1/4$ smaller than the distance from $\{ \psi \le \eta \} $ to $\partial B_1$, put

with $M_\varepsilon $ as in lemma 7.3.5. The functions $\tilde u*\rho _\varepsilon $ are smooth and plurisubharmonic on $\{ \psi <\eta \} $ and decrease to $\tilde u$ as $\varepsilon \downarrow 0$ (proposition 5.9.2), so the $u^\varepsilon _{l,j}$ are smooth and plurisubharmonic on $\{ \psi <\eta \} $ by lemma 7.3.5. Since $\tilde u*\rho _\varepsilon \le -1$, on $V'$ we have $A\psi -\tilde u*\rho _\varepsilon >1/2>2\varepsilon $, and therefore $u^\varepsilon _{l,j}=A\psi $. Since $M_\varepsilon $ decreases to $\max $, is increasing in each variable and satisfies $\max \le M_\varepsilon \le \max +\varepsilon $, the functions $u^\varepsilon _{l,j}$ decrease to $\max \{ \tilde u,A\psi \} =u_{l,j}$. In particular $u^\varepsilon _{l,j}\ge u_{l,j}\ge u_ l$, and all the functions $u_{l,j}$, $u_ l$ and $u^\varepsilon _{l,j}$ coincide with $A\psi $ on $V'$.

Step 4: an upper bound for weak limits. Put

By Step 1, $S_ j\to S$ weakly. On $B$ the weights $u_{0,j}$ take values in $[-N,0)$, so the currents $-u_{0,j}S_ j$ are positive there, and by theorem 7.2.5, since $|u_{l,j}|\le N$ on $B$, their masses on compact subsets of $B$ are bounded uniformly in $j$. By the compactness part of theorem 5.3.7, every subsequence of $(u_{0,j}S_ j)$ has a further subsequence converging weakly on $B$. Let $\Theta $ be the limit of such a subsequence. Let $\varphi $ be a strongly positive test form on $B$ of complementary bidegree. The measures $\mu _ j=S_ j\wedge \varphi $ are positive and converge weakly to $S\wedge \varphi $, and along the chosen subsequence $u_{0,j}\mu _ j\to \Theta \wedge \varphi $. Hence lemma 7.3.4, applied on $B$ with the real-valued upper semicontinuous functions $f_ j=u_{0,j}$, gives $\Theta \wedge \varphi \le u_0S\wedge \varphi $. Evaluating this inequality of measures on a nonnegative test function equal to $1$ on $\operatorname{Supp}\varphi $, we get $(u_0S-\Theta )(\varphi )\ge 0$; thus $u_0S-\Theta $ is a positive current on $B$. By locality (proposition 7.2.2), all the currents $u_{0,j}S_ j$, and also $u_0S$, coincide on $V'$ with the fixed current $A\psi \, (A\mathrm{dd}^{\mathrm{c}}\psi )^ q\wedge T$; hence so does $\Theta $, and $u_0S-\Theta $ has compact support in $B$.

Step 5: comparison of masses. Set $\gamma =\beta ^{r-q}$ and

We claim that $I\le I_ j$ for every $j$. Fix $j$ and parameters $\varepsilon _0,\ldots ,\varepsilon _ q$, and abbreviate $w_ l=u^{\varepsilon _ l}_{l,j}$. For $0\le k\le q$ put

so that $J_0=I$. The measure in $J_ k$ is positive and $u_ k\le w_ k$, so $J_ k$ does not decrease if $u_ k$ is replaced by $w_ k$. If $k<q$, symmetry of products with $q$ factors and proposition 7.2.2(4) let us write that measure as $\mathrm{dd}^{\mathrm{c}}u_{k+1}\wedge R_ k$, where

is a closed positive current of bidimension $(1,1)$ on $\{ \psi <\eta \} $; then lemma 7.3.3, with $a=w_ k$ and $b=u_{k+1}$, which both equal $A\psi $ on the collar $V'$, gives

Hence

Now let $\varepsilon _ q\downarrow 0$, then $\varepsilon _{q-1}\downarrow 0$, and so on down to $\varepsilon _1\downarrow 0$, keeping $\varepsilon _0$ fixed. We claim that the right-hand side of the last inequality then tends to

When $\varepsilon _ q\downarrow 0$, the measure is fixed and the bounded weights $w_ q$ decrease to $u_{q,j}$, so dominated convergence applies. When $\varepsilon _ k\downarrow 0$ for some $1\le k\le q-1$, the integrand is, after moving the smooth factors next to $T$ by symmetry,

where $T_ k=\mathrm{dd}^{\mathrm{c}}w_0\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}w_{k-1}\wedge T$ is a fixed closed positive current on $\{ \psi <\eta \} $ of bidimension $(r-k,r-k)$. There are $q-k\le q-1$ Hessian factors and the fixed weight $u_{q,j}$. Along any sequence $\varepsilon _ k\downarrow 0$, the functions $u^{\varepsilon _ k}_{k,j}$ decrease to $u_{k,j}$ (Step 3), so the induction hypothesis, applied on $\{ \psi <\eta \} $ with the base current $T_ k$, constant sequences for the other factors and the constant weight $u_{q,j}$, shows that these currents converge weakly; so do their exterior products with $\gamma $.

In each case the convergence holds for the integrals over $B$, and not only against compactly supported test forms, because all the currents involved coincide on the collar $V'$ with the same fixed current: if $\chi \in C^\infty _ c(B)$ equals $1$ on $B\setminus V'$, the integrals against $\chi $ converge, and the integrals against $1-\chi $ do not depend on the parameters.

By lemma 7.3.3, with $a=w_0$ and $b=u_{q,j}$, and by symmetry of products with $q$ factors, the limit integral equals

Hence $I\le \int _ Bu^{\varepsilon _0}_{0,j}S_ j\wedge \gamma $ for every $\varepsilon _0$, and dominated convergence with respect to the finite measure $S_ j\wedge \gamma $ on $B$ gives $I\le I_ j$ as $\varepsilon _0\downarrow 0$.

Step 6: conclusion. Along the subsequence defining $\Theta $, splitting the integrals with a cutoff as in Step 5 shows that $I_ j\to \int _ B\Theta \wedge \gamma $. Therefore

Since $u_0S-\Theta $ is positive with compact support in $B$, its trace measure $(u_0S-\Theta )\wedge \gamma $ is a positive measure of total mass at most zero, hence zero; by the coefficient bounds of theorem 5.3.7, $u_0S-\Theta =0$ on $B$. Thus every subsequence of $(u_{0,j}S_ j)$ has a further subsequence converging weakly to $u_0S$ on $B$. Hence the whole sequence converges to $u_0S$ on $B$: otherwise there would be a test form $\varphi $ on $B$ and a subsequence along which $|(u_{0,j}S_ j-u_0S)(\varphi )|$ stays bounded away from zero, and no further subsequence of it could converge to $u_0S$. The replacement by maxima in Step 2 changes neither the functions nor, by locality, the weighted products on the neighborhood $\{ \psi <-\delta \} $ of $K$, and the subtraction of constants does not affect the convergence; hence the original weighted products converge on this neighborhood. As $K$ and $B$ were arbitrary, this proves the weighted assertion for $q$ factors on $\Omega $ and completes the induction. $\square$