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18.7.4 Dyadic regularization3A07CE3E627A

Demailly regularization provides the common ideal bound, since its approximants lie above $u$, and the Bergman estimate gives convergence of each divisorial Lelong number. Along dyadic indices, subadditivity of multiplier ideals also orders the ordinary singularity types of the approximants, which makes their envelopes decrease pointwise. This is the mechanism used in Theorem 18.1.2.

Apply theorem 5.36.1 to $u$, with $\gamma =-\theta $, along the indices $m_ j=2^ j$, starting at a sufficiently large index. Denote the resulting potentials by $a_ j$, and choose numbers $\epsilon _ j>0$ decreasing to zero such that

E93CF05C13A8 18.7.5
\begin{equation} a_ j\in \mathrm{PSH}(X,\theta +\epsilon _ jh),\qquad a_ j\geq u,\qquad a_ j\text{ has type }(1/m_ j,\mathcal I(m_ ju)). \end{equation}

These approximants satisfy $a_ j\to u$ in $L^1$. Their values need not decrease, but their ordinary singularity types do: the subadditivity property in (5.37.6) gives

\[ \mathcal I(2m_ ju)\subseteq \mathcal I(m_ ju)^2, \qquad a_{j+1}\preceq a_ j, \]

where the ordinary comparison follows by taking logarithms of local generators, with a constant that may depend on $j$.

Theorem 18.7.6.EA60F3EAF248 For every $u\in \mathrm{PSH}(X,\theta )$ and every choice in (18.7.5), put

\[ q_ j=P_{\theta +\epsilon _ jh}[a_ j]. \]

Then $q_ j$ has analytic singularities, and

\[ q_ j=P_{\theta +\epsilon _ jh}[a_ j]_{\mathcal I} \searrow P_\theta [u]_{\mathcal I} \]

pointwise and in $L^1$. The limit is independent of the choice of the approximants $a_ j$, of the error bounds $\epsilon _ j$ and of the metric $h$.

Proof. By proposition 18.4.8, $q_ j=P_{\theta +\epsilon _ jh}[a_ j]_{\mathcal I}\sim a_ j$. Since the backgrounds decrease and $a_{j+1}\preceq a_ j$, the function $q_{j+1}$ is a competitor for $P_{\theta +\epsilon _ jh}[a_ j]$. Hence $q_{j+1}\leq q_ j$.

The inequality $a_ j\geq u$ gives $u\preceq _{\mathcal I}a_ j$. For every local divisor $E$, lemma 5.38.9 gives

\[ \nu _ E(a_ j)=\frac1{m_ j}\operatorname {ord}_ E\mathcal I(m_ ju) \longrightarrow \nu _ E(u). \]

By proposition 18.7.1, $q_ j\to p:=P_\theta [u]_{\mathcal I}$ in $L^1$, and $p\leq q_ j\leq 0$. The decreasing limit of the $q_ j$ is therefore bounded below by $p$; it is a $\theta $-plurisubharmonic function that agrees with $p$ almost everywhere, hence everywhere. This identifies the limit in terms of $\theta $ and the multiplier-ideal class of $u$ alone, which proves the asserted independence. $\square$

Corollary 18.7.7.E941DFE18F1D For the sequence above, the following are equivalent:

\[ P_\theta [u]=P_\theta [u]_{\mathcal I},\qquad q_ j\longrightarrow P_\theta [u]\text{ in }L^1, \qquad q_ j\searrow P_\theta [u]\text{ pointwise}. \]

They hold for one choice in (18.7.5) if and only if they hold for every such choice. For the fixed volume form $\mathrm{d}V$,

\[ \lim _ j\int _ X(q_ j-P_\theta [u])\, \mathrm{d}V =\int _ X(P_\theta [u]_{\mathcal I}-P_\theta [u])\, \mathrm{d}V\geq 0, \]

and the vanishing of this limit is a further equivalent condition.

Proof. The theorem identifies both limits of the $q_ j$ as $P_\theta [u]_{\mathcal I}$ for every such choice, and its $L^1$ convergence gives the integral formula. By proposition 18.4.4, the limiting difference is nonnegative. Its integral therefore vanishes exactly when the two envelopes agree almost everywhere, hence everywhere by lemma 5.24.3. This proves all the equivalences. $\square$

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