Corollary 10.3.3C7609FD8FC8C

Proof. (If $\mathrm{PSH}(X,\theta )$ is empty, any $\alpha ,C>0$ will do.) Choose finitely many coordinate unit balls $B_ j$ such that the balls $\frac14B_ j$ with the same centers and a quarter of the radii cover $X$, and smooth functions $\rho _ j$ near $\overline{B_ j}$ with $\mathrm{dd}^{\mathrm{c}}\rho _ j=\theta $. For $u$ as in the statement put $v_ j=u+\rho _ j-\sup _{B_ j}\rho _ j$ on $B_ j$; then $v_ j$ is plurisubharmonic and $v_ j\leq u\leq 0$. By proposition 10.3.1 and comparison of the coordinate volume densities, $\| v_ j\| _{L^1(B_ j)}\leq M_ j$ for a constant $M_ j\geq 1$ independent of $u$. Applying proposition 5.10.12 to $v_ j/M_ j$, we see that $\int _{\frac14B_ j}\mathrm e^{-\alpha _ jv_ j}$ is bounded independently of $u$, where $\alpha _ j=\alpha _ n/M_ j$. Put $\alpha =\min _ j\alpha _ j>0$. Since $v_ j\leq u\leq 0$ and $\alpha \leq \alpha _ j$, we have $\mathrm e^{-\alpha u}\leq \mathrm e^{-\alpha _ ju}\leq \mathrm e^{-\alpha _ jv_ j}$ on $B_ j$. Summing over $j$ and comparing the coordinate volume densities with $\omega ^ n$ gives (10.3.4). $\square$