3BCA55154463
We return to the compact manifold $X$ with its Hermitian form $\omega $. Demailly's global regularization theorem glues the local Bergman constructions while controlling the curvature error. We use it as a classical input in the following form.
Theorem 5.36.1 (Demailly regularization).3CA98F11CC5B Let $u$ be quasi-plurisubharmonic on $X$ and let $\gamma $ be a continuous real $(1,1)$-form with $\mathrm{dd}^{\mathrm{c}}u\geq \gamma $. There are quasi-plurisubharmonic functions $u_ m$, defined for all sufficiently large positive integers $m$, and numbers $\epsilon _ m\downarrow 0$ such that
$u_ m$ has analytic type $(1/m,\mathcal I(mu))$;
$u_ m\geq u$ and $u_ m\to u$ pointwise and in $L^1(X)$;
$\mathrm{dd}^{\mathrm{c}}u_ m\geq \gamma -\epsilon _ m\omega $;
$\max \{ \nu (u,x)-n/m,0\} \leq \nu (u_ m,x)\leq \nu (u,x)$ for all $x$.
The sequence $(u_ m)$ need not be monotone.
This is the form of [Theorem 14.12, DemAM11] used here. The local generators in the construction are the weighted Bergman generators of $\mathcal I(mu)$; smooth local corrections do not change this ideal. On overlaps the local Bergman sums are comparable, so the glued logarithm keeps the analytic type $(1/m,\mathcal I(mu))$. The construction freezes the continuous lower bound $\gamma $ on sufficiently small coordinate balls, and its curvature error combines the modulus of continuity of $\gamma $ with an overlap term that tends to zero as $m$ increases. It is this global curvature estimate, and not merely the local convergence of proposition 5.35.1, that we take from the theorem.
Corollary 5.36.2.8F2184BBF4C4 Let $\theta $ be a smooth real $(1,1)$-form on $X$ and $u\in \mathrm{PSH}(X,\theta )$. The approximants $u_ m$ of theorem 5.36.1 for $\gamma =-\theta $ satisfy
\[ \theta +\mathrm{dd}^{\mathrm{c}}u_ m\geq -\epsilon _ m\omega . \]
In particular, they have analytic type $(1/m,\mathcal I(mu))$.
Proof.
Apply theorem 5.36.1 to $u$ with the lower bound $\gamma =-\theta $; the theorem does not require $\gamma $ to be closed. Adding $\theta $ to the curvature inequality gives the assertion. In the normalization of [DemAM11], the theorem is applied to the potential $u/2$, since $\frac{\mathrm{i}}{\pi }\partial \bar\partial (u/2)=\mathrm{dd}^{\mathrm{c}}u$.
$\square$
Thus the approximants keep the curvature lower bound up to an error that tends to zero. Their multiplier ideals determine the limit of their ordinary envelopes; see theorem 18.7.6.
We also use the regularization of closed currents in the form recalled in 5.18 Analytic singularities. Let $T=\eta +\mathrm{dd}^{\mathrm{c}}v$ be a closed real $(1,1)$-current, where $\eta $ is a smooth closed form and $v$ is quasi-plurisubharmonic, and set $\nu (T,x)=\nu (v,x)$. Suppose that $T\geq \gamma $ for a smooth real $(1,1)$-form $\gamma $. Smooth regularization gives smooth representatives $T_ j=\eta +\mathrm{dd}^{\mathrm{c}}v_ j$ with
\[ T_ j\geq \gamma -C\lambda _ j\omega , \]
where $C>0$ is a fixed constant and the continuous functions $\lambda _ j$ decrease pointwise to $\nu (T,\cdot )$. If all Lelong numbers of $T$ vanish, Dini's theorem shows that $\lambda _ j\to 0$ uniformly, so the loss of positivity becomes uniformly small. Analytic regularization instead gives representatives $S_ j$ with analytic singularities and
\[ S_ j\geq \gamma -\varepsilon _ j\omega ,\qquad \varepsilon _ j\downarrow 0,\qquad \nu (S_ j,x)\leq \nu (T,x). \]
Thus analytic regularization preserves strict positivity up to a small loss and creates no positive Lelong number at a point where the Lelong number of $T$ vanishes.
For arguments on a modification, the bounded remainder in an analytic presentation is sometimes insufficient. The following version of regularization gives a smooth remainder after resolution.
Theorem 5.36.5 (Demailly regularization).88038BD901A0 Let $\alpha $ be a smooth real $(1,1)$-form and let $u\in \mathrm{PSH}(X,\alpha )$. There are numbers $\delta _ j\downarrow 0$ and functions $\widetilde u_ j\in \mathrm{PSH}(X,\alpha +\delta _ j\omega )$ decreasing to $u$ with the following properties. Each $\widetilde u_ j$ is smooth outside an analytic pole set, and there is a modification $\pi _ j:Y_ j\to X$, obtained by a finite sequence of blowups with smooth centers, such that locally on $Y_ j$
\[ \pi _ j^*\widetilde u_ j=\sum _\ell a_\ell \log |s_\ell |^2+b, \qquad a_\ell >0,\quad b\in C^\infty . \]
Here the $s_\ell $ are local holomorphic defining functions of divisors, the $a_\ell >0$ are real numbers, and $b$ is a smooth real function. The divisors may be taken to have simple normal crossings.
This classical theorem is applied to the closed current $\mathrm{dd}^{\mathrm{c}}u$ with the smooth lower bound $-\alpha $; see section 25.3 for its source. The lower bound need not be closed, and $X$ need not be Kähler. What matters here is the smooth remainder $b$ after resolution; an analytic type, whose remainder is merely bounded, would not suffice. The sequence $(\widetilde u_ j)$ need not be the multiplier-ideal sequence of theorem 5.36.1, and we do not claim that the conclusions of the two theorems hold for one and the same sequence. If $\theta +\mathrm{dd}^{\mathrm{c}}u\geq \varepsilon \omega $ for some $\varepsilon >0$, applying the theorem with $\alpha =\theta -\varepsilon \omega $ and discarding the terms with $\delta _ j\geq \varepsilon $ gives approximants with $\theta +\mathrm{dd}^{\mathrm{c}}\widetilde u_ j\geq (\varepsilon -\delta _ j)\omega >0$; thus strict positivity is preserved.
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