Lemma 10.9.340E5831EBB76

Proof. (1) We freeze the coefficients of $\omega $ on small balls. Fix $\varepsilon >0$, a coordinate chart and a compact subset $L$ of it. By uniform continuity of the coefficients of $\omega $ on a compact neighborhood of $L$ in the chart, there is $r>0$ such that, for every $a\in L$, the ball $B(a,r)$ lies in the chart and the form $\gamma _ a=\omega (a)+\tfrac \varepsilon 2\beta $ satisfies $\omega \leq \gamma _ a\leq \omega +\varepsilon \beta $ on $B(a,r)$. Let $g_ a$ be the real quadratic polynomial with $\mathrm{dd}^{\mathrm{c}}g_ a=\gamma _ a$ described at the beginning of this section. If $u\in \mathrm{PSH}(X,\omega )$, then $u+g_ a$ is quasi-plurisubharmonic with $\mathrm{dd}^{\mathrm{c}}(u+g_ a)=\omega _ u+(\gamma _ a-\omega )\geq 0$, so it is plurisubharmonic on $B(a,r)$; in the same way $v+g_ a$ is plurisubharmonic on $D\cap B(a,r)$.

Let $w$ be the function in question: a finite maximum $\max _ iu_ i$, a regularized supremum $(\sup _ iu_ i)^*$, or the pasted function. We show that $w+g_ a$ is plurisubharmonic on $B(a,r)$. In the first case $w+g_ a$ is the maximum of the plurisubharmonic functions $u_ i+g_ a$, which is plurisubharmonic by proposition 5.9.5. In the second case $w+g_ a=(\sup _ i(u_ i+g_ a))^*$, because $g_ a$ is continuous, and it is plurisubharmonic by theorem 5.10.2, since the family $(u_ i+g_ a)$ is uniformly bounded above on $B(a,r)$. In the third case $w+g_ a$ equals $\max \{ u+g_ a,v+g_ a\} $ on $D\cap B(a,r)$ and $u+g_ a$ elsewhere; every boundary point $x$ of $D\cap B(a,r)$ in $B(a,r)$ lies in $\partial D$, and $\varlimsup _{y\to x}(v+g_ a)(y)\leq u(x)+g_ a(x)$ because $g_ a$ is continuous, so proposition 5.9.5 applies on $B(a,r)$. Hence $w$ is quasi-plurisubharmonic on $B(a,r)$ and $\omega +\mathrm{dd}^{\mathrm{c}}w\geq \omega -\gamma _ a\geq -\varepsilon \beta $ there. The balls $B(a,r)$, $a\in L$, cover a neighborhood of $L$, and positivity of currents is a local property, so $\omega +\mathrm{dd}^{\mathrm{c}}w\geq -\varepsilon \beta $ on a neighborhood of $L$. Letting $\varepsilon \downarrow 0$ gives $\omega +\mathrm{dd}^{\mathrm{c}}w\geq 0$ on the interior of $L$, and since the chart and $L$ were arbitrary, $w\in \mathrm{PSH}(X,\omega )$.

(2) We first prove positivity on a coordinate ball $B$ whose closure lies in a chart, choosing $q=C|z|^2$ in (10.9.2). Let $B'\Subset B$ be a smaller concentric ball, let $\rho _ k$ be radial mollifiers whose radii decrease to zero and are smaller than the distance from $B'$ to $\partial B$, and put $u_{l,k}=(u_ l+q)*\rho _ k-q$ on $B'$. Since $\mathrm{dd}^{\mathrm{c}}q$ has constant coefficients and $\int \rho _ k=1$, we have $\mathrm{dd}^{\mathrm{c}}u_{l,k}=(\mathrm{dd}^{\mathrm{c}}u_ l)*\rho _ k$, where the convolution acts on the coefficients, and $(\omega +\mathrm{dd}^{\mathrm{c}}u_ l)*\rho _ k\geq 0$ because mollification preserves positivity (section 5.3). By uniform continuity of the coefficients of $\omega $, there are $\tau _ k\to 0$ with $\omega *\rho _ k\leq \omega +\tau _ k\beta $ on $B'$. Hence the smooth $(1,1)$-forms $\omega _{u_{l,k}}+\tau _ k\beta =(\omega +\mathrm{dd}^{\mathrm{c}}u_ l)*\rho _ k+(\omega +\tau _ k\beta -\omega *\rho _ k)$ are positive, and so is their exterior product, by proposition 5.3.2. Expanding this product of the forms $\mathrm{dd}^{\mathrm{c}}((u_ l+q)*\rho _ k)+(\sigma +\tau _ k\beta )$, we obtain

As $k\to \infty $, the smooth plurisubharmonic functions $(u_ l+q)*\rho _ k$ decrease to $u_ l+q$ by proposition 5.9.2(2), so the Bedford–Taylor products converge weakly by theorem 7.3.7; their masses are bounded on compact subsets of $B'$ uniformly in $k$ by theorem 7.2.5, and the coefficients of $(\sigma +\tau _ k\beta )^{p-|J|}$ converge uniformly to those of $\sigma ^{p-|J|}$. Hence each term converges weakly to the corresponding term of (10.9.2), the product is a weak limit of positive forms on $B'$, and it is positive. Since such balls $B'$ cover $X$, the product is positive on $X$.

Cover $X$ by finitely many such balls $B'$, with $q$ defined on the larger balls $B$. On compact subsets of $B$, theorem 7.2.5 bounds the trace masses of the Bedford–Taylor products in (10.9.2) by a constant times a product of the numbers $\sup _ B|u_ l+q|$, and theorem 5.3.7 bounds the total variations of their coefficient measures in the same way. Since $\sigma $ and $\omega $ are smooth, the measure $(\omega _{u_1}\wedge \cdots \wedge \omega _{u_ p})\wedge \omega ^{n-p}$ is a sum of the coefficient measures of these Bedford–Taylor products, and of a smooth density for $J=\varnothing $, multiplied by bounded smooth functions. Summing over the finite cover gives the asserted bound for the total mass of the trace measure. By corollary 5.47.11(3), every pluripolar set has outer measure zero for the total variations of the coefficient measures of Bedford–Taylor products of locally bounded plurisubharmonic functions; it is also Lebesgue null (remarks following definition 5.47.2). The trace measure is a positive measure bounded on each $B'$ by a finite sum of such total variations and of a multiple of Lebesgue measure, so every pluripolar set has outer measure zero for it.

Finally, let $\mathrm{dd}^{\mathrm{c}}g\geq \omega $ on $D$. We may take $q=g$ in (10.9.2) on $D$. Put $A=\mathrm{dd}^{\mathrm{c}}(u+g)$ and $\gamma =\mathrm{dd}^{\mathrm{c}}g-\omega \geq 0$, so that $\sigma =-\gamma $ and (10.9.2) reads $\omega _ u^ k=\sum _{i=0}^ k\binom ki A^ i\wedge (-\gamma )^{k-i}$, where $A^ i$ is a Bedford–Taylor power. Using $\binom nk\binom ki=\binom ni\binom {n-i}{k-i}$ and $\sum _{m=0}^{n-i}\binom {n-i}m(-1)^ m=0$ for $i<n$, we obtain

The currents $\omega _ u^ k$ are positive and the forms $\gamma ^{n-k}$ are strongly positive, so every term on the left is a positive measure, and $(\mathrm{dd}^{\mathrm{c}}(u+g))^ n=A^ n\geq \omega _ u^ n$.

(3) On a coordinate ball, add a function $q$ as in (10.9.2). In the decreasing case, $u_ l$ is bounded, the functions $u_{l,j}+q$ are plurisubharmonic and decrease to $u_ l+q$, which is bounded below, so $u_ l+q$ is plurisubharmonic and $u_{l,j}+q\to u_ l+q$ in $L^1_{\mathrm{loc}}$ by theorem 5.10.1. Hence $u_ l$ is quasi-plurisubharmonic, and $\omega +\mathrm{dd}^{\mathrm{c}}u_ l$ is the weak limit of the positive currents $\omega +\mathrm{dd}^{\mathrm{c}}u_{l,j}$, so $u_ l\in \mathrm{PSH}(X,\omega )$. The Bedford–Taylor products of the functions $u_{l,j}+q$ converge weakly by theorem 7.3.7. In the increasing case, $u_ l\in \mathrm{PSH}(X,\omega )$ by (1), and $u_ l$ is bounded. The plurisubharmonic functions $u_{l,j}+q$ increase, and $u_ l+q$ is the upper semicontinuous regularization of their limit, because $q$ is continuous; so $u_{l,j}+q\to u_ l+q$ almost everywhere by theorem 5.10.2, and theorem 7.6.1, applied on the ball, gives the weak convergence of the Bedford–Taylor products. In both cases we multiply by the fixed smooth forms $\sigma ^{p-|J|}$ and sum; a partition of unity subordinate to finitely many balls gives weak convergence on $X$.

(4) Choose finitely many coordinate balls $B_ k$, with closures in charts, that cover $X$, concentric balls $B'_ k\Subset B_ k$ that still cover $X$, and smooth functions $q_ k$ on neighborhoods of $\overline{B_ k}$ with $\mathrm{dd}^{\mathrm{c}}q_ k\geq \omega $. For every $u\in \mathrm{PSH}(X,\omega )$ the functions $u+q_ k$ are plurisubharmonic on $B_ k$. Two quasi-plurisubharmonic functions that agree almost everywhere on an open set agree everywhere on it, by lemma 6.14.4(1). We may therefore regard $\mathrm{PSH}(X,\omega )$ as a subset of $L^1(X)$; on compact subsets of a chart, $\omega ^ n$ is bounded above and below by positive multiples of Lebesgue measure.

Continuity of the supremum. Let $u_ j,u\in \mathrm{PSH}(X,\omega )$ with $u_ j\to u$ in $L^1(X)$; then $u_ j+q_ k\to u+q_ k$ in $L^1_{\mathrm{loc}}(B_ k)$ for every $k$. Choose points $x_ j$ with $u_ j(x_ j)=\sup _ Xu_ j$, which exist because $u_ j$ is upper semicontinuous on the compact manifold $X$, and a subsequence along which $\sup _ Xu_ j\to \varlimsup _ j\sup _ Xu_ j$ and $x_ j\to x$. Choose $k$ with $x\in B_ k$ and a real number $c>u(x)$, and a closed ball $K\subset B_ k$ centered at $x$ on which $u<c$. Since $c+q_ k$ is continuous and $u+q_ k<c+q_ k$ on $K$, the second part of theorem 5.10.4 gives $u_ j+q_ k<c+q_ k$ on $K$ for large $j$; hence $u_ j(x_ j)<c$ for large $j$. Letting $c\downarrow u(x)$ we obtain $\varlimsup _ j\sup _ Xu_ j\leq u(x)\leq \sup _ Xu$. Conversely, choose a subsequence along which $\sup _ Xu_ j$ tends to $s=\varliminf _ j\sup _ Xu_ j$ and $u_ j\to u$ almost everywhere. Then $u\leq s$ almost everywhere, so $s>-\infty $ because $u$ is integrable, and $u\leq s$ everywhere by lemma 6.14.4(1), applied to $u$ and the constant $s$. Thus $\sup _ Xu\leq \varliminf _ j\sup _ Xu_ j$, and the supremum is continuous.

Compactness. Let $(u_ j)$ be a sequence in $\mathrm{PSH}(X,\omega )$ with $\sup _ Xu_ j=0$. On each $B_ k$ the plurisubharmonic functions $u_ j+q_ k$ are bounded above by $\sup _{B_ k}q_ k$. Choose $x_ j$ with $u_ j(x_ j)=0$, pass to a subsequence with $x_ j\to x$, and choose $k_0$ with $x\in B'_{k_0}$. On $B_{k_0}$ the functions $u_ j+q_{k_0}$ do not converge locally uniformly to $-\infty $, since for large $j$ their values at the points $x_ j\in \overline{B'_{k_0}}$ are at least $\min _{\overline{B'_{k_0}}}q_{k_0}$. By theorem 5.10.4, a subsequence converges in $L^1_{\mathrm{loc}}(B_{k_0})$.

If $B_ l$ meets a ball $B_ k$ on which $(u_ j+q_ k)$ converges in $L^1_{\mathrm{loc}}(B_ k)$, then on the nonempty open set $B_ k\cap B_ l$ the functions $u_ j+q_ l=(u_ j+q_ k)+(q_ l-q_ k)$ converge in $L^1_{\mathrm{loc}}$ to a locally integrable function, so they do not tend to $-\infty $ locally uniformly on $B_ l$, and a further subsequence converges in $L^1_{\mathrm{loc}}(B_ l)$. Since $X$ is connected, every ball is joined to $B_{k_0}$ by a finite chain of balls, each meeting the next, so after finitely many extractions $(u_ j+q_ k)$ converges in $L^1_{\mathrm{loc}}(B_ k)$ to a plurisubharmonic function $v_ k$ for every $k$ (theorem 5.10.4). Then $(u_ j)$ converges in $L^1(B'_ k)$ for every $k$, hence in $L^1(X)$.

It remains to identify the limit. The quasi-plurisubharmonic functions $v_ k-q_ k$ and $v_ l-q_ l$ agree almost everywhere on $B_ k\cap B_ l$, hence everywhere, so they define a quasi-plurisubharmonic function $u$ on $X$, and $\omega +\mathrm{dd}^{\mathrm{c}}u\geq 0$ as a distributional limit of the positive currents $\omega +\mathrm{dd}^{\mathrm{c}}u_ j$. Thus $u\in \mathrm{PSH}(X,\omega )$, $u_ j\to u$ in $L^1(X)$, and $\sup _ Xu=\lim _ j\sup _ Xu_ j=0$ by the continuity just proved. The set of normalized functions is therefore sequentially compact, hence compact, in the metric space $L^1(X)$, and the continuous function $u\mapsto \| u\| _{L^1(X)}$ is bounded on it. $\square$