Lemma 5.42.4.D0B76A103405 For $K$ sufficiently large, there is an increasing function $\epsilon (\delta )\geq 0$ tending to zero at zero such that, for every small $\delta >0$ and every $c>0$, the function
\[ \Phi _{c,\delta } =\inf _{0<t\leq \delta } \{ R_ tu+K(t^2-\delta ^2)+K(t-\delta )-c\log (t/\delta )\} \]
is bounded and quasi-plurisubharmonic and satisfies
\[ \theta +\mathrm{dd}^{\mathrm{c}}\Phi _{c,\delta } \geq -(Ac+\epsilon (\delta ))\omega . \]
The infimum is attained away from zero. Moreover, $R_ tu+Kt^2\downarrow u$ as $t\downarrow 0$.
Proof.
Take $K\geq K_0$ and put $H(z,t)=R_ tu(z)+Kt^2+Kt$. As a function of $\log t$, $H$ is increasing and strictly convex, because $R_ tu+K_0t^2$ has nonnegative increasing logarithmic derivative and the added terms have positive second logarithmic derivative. Its logarithmic derivative $\widetilde\lambda =t\partial _ tH$ is at least $\lambda $.
The complex Hessian of $K|w|$ has a positive vertical term proportional to $K/|w|$. For sufficiently small $|w|=t$, this term absorbs the vertical error and, by the inequality
\[ C|\zeta ||\eta | \leq C' t|\zeta |^2+\frac{K}{8t}|\eta |^2, \]
also the mixed error. After increasing the constants and replacing $\lambda $ by the larger $\widetilde\lambda $, we find that the complex Hessian of $(z,w)\mapsto H(z,|w|)$ is bounded below by the base form
\[ -\theta -(A\widetilde\lambda +\epsilon (t))\omega , \]
where $\epsilon $ is an increasing majorant of $\epsilon _0(t)+C't$.
Since $u$ is bounded, the term $-c\log (t/\delta )$ tends uniformly to $+\infty $ as $t\downarrow 0$, so the minimum occurs on a compact interval of radii away from zero. Strict convexity in $\log t$ makes the minimizing radius unique. At an interior minimum $\widetilde\lambda =c$, while at the upper endpoint $t=\delta $ one has $\widetilde\lambda \leq c$. No differentiability of the minimizing radius is needed.
In more detail, localize near a point of $X$ and fix $a>0$. By continuity of the minimizers and of $\widetilde\lambda $, there is an interval of radii containing the minimizers at all nearby points on which $\widetilde\lambda \leq c+a$. If the minimum is attained at the upper endpoint, keep the upper endpoint $\delta $; an open interval approaching it has the same infimum. Because of the uniform logarithmic term, the lower endpoint can be chosen below all minimizers, and since $\widetilde\lambda $ is increasing, the bound $\widetilde\lambda \leq c+a$ holds throughout the chosen interval. The local Kiselman minimum principle for functions depending only on $|w|$, applied after a local quadratic correction, now gives the lower curvature bound with error $A(c+a)+\epsilon (\delta )$, up to an arbitrarily small error from freezing the smooth base forms. Letting these errors and $a$ tend to zero proves the asserted current inequality and the quasi-plurisubharmonicity. Finally, $K\geq K_0$ preserves the monotone convergence of $R_ tu+Kt^2$ to $u$.
$\square$
Pome
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