Theorem 13.7.567C4F7EC316E

Proof. Lower inequality. Work on a coordinate ball $B$ such that $\overline B$ lies in a larger coordinate ball, add fixed smooth local potentials of the $\theta _ j$, defined near $\overline B$, and denote the resulting plurisubharmonic functions by $v_{j,k}$ and $v_ j$. By lemma 13.7.3 (including its independence assertion, which allows us to use the chart $B$), $v_{j,k}\to v_ j$ in extended capacity on $B$, and the $v_{j,k}$ are uniformly bounded above on $B$, because the suprema of the $u_{j,k}$ are bounded above. The local non-pluripolar products of the $v_{j,k}$ and of the $v_ j$ are the restrictions of $R_ k$ and $R$ to $B$, and they are finite by theorem 13.5.4. For $C,\varepsilon >0$ let $v_{j,k}^ C$, $v_ j^ C$, $a_ k$ and $a$ be as in lemma 13.7.4. By that lemma, the weights take values in $[0,1]$, $a$ is quasi-continuous, $a_ k\to a$ locally in capacity, and

on $B$. Moreover, the plurisubharmonic functions $v_{j,k}^ C\geq -C$ are uniformly bounded on $B$, and they converge to $v_ j^ C$ locally in capacity, by lemma 13.7.3.

Let $\chi \geq 0$ be a smooth function with compact support in $B$. By the weighted capacity continuity of theorem 7.5.8,

where the inequality holds because $0\leq a_ k\leq 1$ and $R_ k\wedge \chi \eta $ is a positive measure. Now let $\varepsilon \downarrow 0$ and then $C\to \infty $. By lemma 13.7.4(3), the weights $a$ increase to the indicator function of the complement of the union of the pole sets of the $v_ j$, which $R$ does not charge; so monotone convergence gives $\int \chi R\wedge \eta \leq \varliminf _ k\int \chi R_ k\wedge \eta $. This is the local lower inequality; it used only the finiteness of the local products, the extended capacity convergence and the local upper bounds. A finite partition of unity by such functions $\chi $ (proposition 5.2.6) gives the global inequality.

Convergence. By (13.5.5), the masses $\int _ XR_ k\wedge \omega ^{n-p}$ are bounded, since the shifted backgrounds depend only on the fixed forms $\theta _ j$. Near a compact subset $L$ of a chart with Euclidean form $\beta $, we have $\beta \leq c\, \omega $ for some $c>0$ (lemma 6.14.1), and hence $R_ k\wedge \beta ^{n-p}\leq c^{n-p}R_ k\wedge \omega ^{n-p}$ by (13.2.8). Therefore the trace measures of the $R_ k$ are locally uniformly bounded, and by theorem 5.3.7 every subsequence of $(R_ k)$ has a weakly convergent subsequence. Its limit $R'$ is positive, and $R'-R$ is positive, by the lower inequality applied to all smooth strongly positive forms $\eta $ on $X$. The additional hypothesis gives $\int _ X(R'-R)\wedge \omega ^{n-p}\leq 0$, because $\omega ^{n-p}$ is a test form on the compact manifold $X$. Hence the positive measure $(R'-R)\wedge \omega ^{n-p}$ vanishes, so the trace measures of $R'-R$, which are bounded on compact subsets of charts by multiples of it, vanish as well, and $R'-R=0$ by theorem 5.3.7. Thus every subsequence of $(R_ k)$ has a further subsequence converging weakly to $R$, which proves $R_ k\to R$. $\square$