PomeLemma 17.8.1.950F051CF55A Let $h$ be a maximal plurisubharmonic function on a domain $\Omega \subset \mathbb C^ d$, $d\geq 1$, in the sense of definition 7.7.6. Then the top-degree non-pluripolar product $\langle (\mathrm{dd}^{\mathrm{c}}h)^ d\rangle $ is locally finite and vanishes. Here $h$ need not be locally bounded.
Lemma 17.8.1950F051CF55A
Proof. Fix a ball $B\Subset \Omega $ and a concentric ball $B_1$ with $B\Subset B_1\Subset \Omega $. Put $h_ j=\max \{ h,-j\} $ and let $\widehat h_ j$ be the function supplied by corollary 7.7.5 for $h_ j$ and $B$. These are locally bounded plurisubharmonic functions with $\widehat h_ j\geq h_ j\geq h$, equal to $h_ j$ outside $B$, with $(\mathrm{dd}^{\mathrm{c}}\widehat h_ j)^ d=0$ on $B$; they decrease with $j$, because $\widehat h_ j$ is increasing in $h_ j$. Their decreasing limit $v$ is plurisubharmonic by theorem 5.10.1, since it dominates $h$, and it equals $h$ outside $B$. Maximality gives $v\leq h$, hence $v=h$.
Fix $C>0$ and $\varepsilon >0$, and put
\[ z_ j=\max \{ \widehat h_ j,-C\} ,\quad a_ j=\frac{z_ j+C}{z_ j+C+\varepsilon },\quad z=\max \{ h,-C\} ,\quad a=\frac{z+C}{z+C+\varepsilon }. \]
The functions $z_ j$ are plurisubharmonic, locally uniformly bounded (between $-C$ and $\widehat h_1$), and decrease to $z$, so they converge to $z$ locally in capacity on the bounded open set $B_1$ (proposition 7.5.6). The weights take values in $[0,1]$, and since $s\mapsto (s+C)/(s+C+\varepsilon )$ is Lipschitz with constant $\varepsilon ^{-1}$ on $[-C,\infty )$, we have $|a_ j-a|\leq \varepsilon ^{-1}|z_ j-z|$, so $a_ j\to a$ locally in capacity on $B_1$. The limit weight $a$ is quasi-continuous on $B_1$, because $z$ is (theorem 7.5.2) and $a$ is a continuous function of $z$. On $B$, the weight $a_ j$ vanishes where $\widehat h_ j\leq -C$, and on the plurifine open set $\{ \widehat h_ j>-C\} $ we have $z_ j=\widehat h_ j$; hence plurifine locality of Bedford–Taylor products (theorem 5.47.22) gives
\[ a_ j(\mathrm{dd}^{\mathrm{c}}z_ j)^ d=a_ j(\mathrm{dd}^{\mathrm{c}}\widehat h_ j)^ d=0\quad \text{on }B. \]
The last assertion of theorem 7.5.8, applied on $B_1$ with $d$ factors $z_ j$ and the weights $a_ j$, gives
\[ a_ j(\mathrm{dd}^{\mathrm{c}}z_ j)^ d\longrightarrow a(\mathrm{dd}^{\mathrm{c}}z)^ d\quad \text{weakly on }B_1, \]
so $a(\mathrm{dd}^{\mathrm{c}}z)^ d=0$ on $B$. Since $a>0$ exactly on $\{ h>-C\} $ and $z=\max \{ h,-C\} $, the measure $\mathbf1_{\{ h>-C\} }(\mathrm{dd}^{\mathrm{c}}z)^ d$ vanishes on $B$. Taking $C=k$ for all integers $k$, the truncated products in the definition of $\langle (\mathrm{dd}^{\mathrm{c}}h)^ d\rangle $ all vanish on $B$; hence the non-pluripolar product is locally finite and zero on $B$. A cover of $\Omega $ by such balls proves the lemma. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.