Lemma 15.16.5DA20F7BB0188

Proof. The normalized potential satisfies

Fix a Gauduchon metric $\gamma $ and write $\gamma ^ n=G\, \mathrm{d}V$. Since $v_\lambda \leq 0$ and $0<\lambda \leq 1$, (15.14.3) gives

The integrals on the left have a positive lower bound independent of $\lambda $. Otherwise, the compactness of normalized quasi-plurisubharmonic functions would give an almost everywhere convergent sequence of the normalized $v_\lambda $ whose integrals tend to zero, with a $\theta $-plurisubharmonic limit; since $f$ is nonzero, dominated convergence would give a positive limit integral. Thus $c_\lambda \leq C$.

To control the singularity type, we use uniform exponential integrability (lemma 5.26.2) and Hölder's inequality to choose $\varepsilon >0$ and $1<q<p$ such that $\| \mathrm{e}^{-\varepsilon v_\lambda }f\| _ q$ is uniformly bounded. Lemma 15.14.1, applied with the reference $V_\theta -1$ and the barrier $\rho -1$, gives $\eta _\lambda \in \mathrm{PSH}(X,\theta )$ with

for a fixed $m>0$. The scalar order in corollary 15.13.3, together with $\mathrm{MA}_\theta (v_\lambda )\leq c_\lambda f\, \mathrm{d}V$, gives $c_\lambda \geq m$. Set $D=\varepsilon ^{-1}\max \{ 0,\log (C/m)\} $ and $w_\lambda =\eta _\lambda -D$. Then

since $\lambda v_\lambda \leq 0$ and $\eta _\lambda \leq 0$. The exponential order in corollary 15.13.3 gives $v_\lambda \geq w_\lambda $, which proves the uniform bounds with $B=1+D$. $\square$