Pome51F37429633651F374296336
Demailly's approximation of plurisubharmonic functions by functions with analytic singularities is built from Bergman kernels of weighted spaces of holomorphic functions. In this section we recall the local construction and its estimates. Fix a bounded pseudoconvex domain $\Omega \subset \mathbb C^ n$ and a plurisubharmonic function $\psi $ on $\Omega $. Write $\mathrm{d}\lambda $ for Euclidean volume and $B(x,r)$ for the Euclidean ball of radius $r>0$ about $x$. For a positive integer $m$, define the weighted Hilbert space of holomorphic functions
Choose an orthonormal basis $(g_{m,j})_ j$ of $\mathcal H_ m$ and define $\beta _ m=m^{-1}\log \sum _ j|g_{m,j}|^2$. The sum is the squared norm of point evaluation,
and is therefore independent of the basis. By the local finite generation part of Bergman regularization, $\beta _ m$ has analytic type $(1/m,\mathcal I(m\psi ))$ on relatively compact open subsets of $\Omega $. This follows from the generation of the multiplier ideal by weighted $L^2$ holomorphic functions and from local $L^2$ division; see [Proposition 5.7 and Section 14.A, DemAM11]. We use this local statement, together with the point-extension estimate in the proof below, as a standard input.
Proposition 5.35.1.DBEB8E0E0318 There are constants $C_1,C_2$ independent of $m$ such that, for $B(x,r)\Subset \Omega $,
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5.35.2
Consequently $\beta _ m\to \psi $ pointwise and in $L^1_{\mathrm{loc}}$, and
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5.35.3
Proof. For $\| f\| _ m\leq 1$, the submean inequality gives
Taking the supremum over the unit ball of $\mathcal H_ m$ and then the logarithm proves the upper bound.
For the lower bound, suppose that $\psi (x)>-\infty $. The local Ohsawa–Takegoshi point-extension theorem gives a holomorphic $f$ with $f(x)=1$ and
where the constant is independent of the plurisubharmonic weight and of $m$. After normalizing $f$ in $\mathcal H_ m$, we see that the squared norm of point evaluation is at least $C^{-1}\mathrm{e}^{m\psi (x)}$, which proves the lower bound. At a pole of $\psi $ the lower bound holds trivially.
Choosing $r=1/m$ in the upper bound and using upper semicontinuity of $\psi $ proves pointwise convergence. On a fixed relatively compact set, the same estimates bound $\beta _ m$ above by a constant and below by $\psi -C_1/m$. Since $\psi $ is locally integrable, dominated convergence also gives convergence in $L^1_{\mathrm{loc}}$.
For the Lelong estimate, take suprema over $B(x,r)$ in the two inequalities, bounding the right-hand side of the upper one by a supremum over $B(x,2r)$. Divide by $\log r^2<0$ and let $r\downarrow 0$. After the division, the term $-(2n/m)\log r$ contributes $-n/m$, while the constant terms disappear. This proves the two inequalities in (5.35.3); since Lelong numbers are nonnegative, the lower bound may be replaced by its maximum with zero. $\square$
In [DemAM11], the local weight is $\mathrm{e}^{-2m\phi }$ and Lelong numbers are normalized by $\log |z|$. With $\psi =2\phi $, these conventions give exactly the formulas above; in particular, the loss is $n/m$ for the weight $\mathrm{e}^{-m\psi }$ and the normalization by $\log |z|^2$.
The same construction and estimates hold for every real $m>0$. The coefficient $1/m$ of the resulting logarithmic presentation is rational exactly when $m$ is rational, and we speak of an analytic type only in that case. The global sequence constructed below is indexed by integers.
For a quasi-plurisubharmonic function $u$, the generators of $\mathcal I(mu)$ also control the multiplier ideals of $u$ at nearby weights; a small loss in the weight compensates for the finite Bergman index.
Lemma 5.35.4.FE03F7C266DD Let $u$ be quasi-plurisubharmonic, let $m>0$, and let $a_ m$ be a function which locally satisfies $a_ m=m^{-1}\log \sum _ k|g_ k|^2+O(1)$, where the $g_ k$ generate $\mathcal I(mu)$. For $t>0$ and $s>1$,
Moreover, $u\preceq a_ m$ locally, and hence $\mathcal I(tu)\subseteq \mathcal I(ta_ m)$ for every $t>0$.
Proof. On a sufficiently small coordinate neighborhood, choose generators $g_1,\ldots ,g_ N$ of $\mathcal I(mu)$ and put $B=(\sum _ k|g_ k|^2)\mathrm{e}^{-mu}$. After shrinking the neighborhood, $B$ is integrable and $a_ m=m^{-1}\log \sum _ k|g_ k|^2+O(1)$ with a bounded remainder. If $f\in \mathcal I(sta_ m)$, shrink once more so that $f$ is bounded and $A=|f|^2\mathrm{e}^{-sta_ m}$ is integrable. Then, almost everywhere,
Hölder's inequality with exponents $s$ and $s/(s-1)$ applies because $st/(m(s-1))\leq 1$, and it shows that the left side is locally integrable, also in the case of equality in the bound for $m$. Finally, the local Bergman lower estimate of proposition 5.35.1 gives $u\preceq a_ m$ locally; bounded smooth corrections affect neither this comparison nor any of the ideals. $\square$
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