Corollary 10.3.5B2C28BB645D3

Proof. Suppose that no such constant exists. Since $\sup _ X\varphi -\sup _ K\varphi $ does not change when a constant is added to $\varphi $, there are $\varphi _ j\in \mathrm{PSH}(X,\theta )$ with $\sup _ X\varphi _ j=0$ and $\sup _ K\varphi _ j\leq -2^ j$. Choose $C\geq 0$ with $\theta \leq C\omega $ (lemma 6.14.1) and put $\psi _ N=\sum _{j=1}^ N2^{-j}\varphi _ j$. Since $\sum _{j\leq N}2^{-j}\leq 1$, we have

The sequence $(\psi _ N)$ is decreasing because $\varphi _ j\leq 0$, and $\| \psi _ N\| _{L^1}\leq \sup _ j\| \varphi _ j\| _{L^1}<\infty $ by proposition 10.3.1. By monotone convergence the limit $\psi $ is integrable, so it is not identically $-\infty $, and theorem 5.10.1, applied on coordinate balls after adding local potentials of $C\omega $, shows that $\psi \in \mathrm{PSH}(X,C\omega )$. On $K$ every term $2^{-j}\varphi _ j$ is at most $-1$, so $\psi =-\infty $ on $K$. On every coordinate ball $U$ with a smooth function $h$ such that $\mathrm{dd}^{\mathrm{c}}h=C\omega $, the function $\psi +h$ is plurisubharmonic and equals $-\infty $ on $K\cap U$. Hence $K$ is pluripolar, a contradiction. The last assertion follows: bounds for $\sup _ K\varphi $ give bounds for $\sup _ X\varphi $, and proposition 10.3.1 applies. $\square$