The multiplier ideals of $mu$ determine which sections of $L^ m$ have finite weighted $L^2$ norm. Evaluation at points of the resulting Hilbert space defines a Bergman kernel, and its extremal characterization turns comparisons of norms into comparisons of the associated potentials.
For every positive integer $m$, set
\[ \mathcal H_ m(u)=H^0(X,L^ m\otimes \mathcal I(mu)),\qquad \| s\| _{m,u}^2=\int _ X|s|_{h^ m}^2\mathrm{e}^{-mu}\, \mathrm{d}V. \]
The integrand is defined off the pole set $\{ u=-\infty \} $, which has volume zero. A section $s\in H^0(X,L^ m)$ belongs to $\mathcal H_ m(u)$ if and only if $\| s\| _{m,u}<\infty $. Indeed, if the local coefficients of $s$ lie in $\mathcal I(mu)$, the integrand is locally integrable, and a finite covering of the compact manifold $X$ gives a finite integral; conversely, a finite integral gives local integrability on every coordinate ball. Thus $\mathcal H_ m(u)$, a subspace of the finite-dimensional space $H^0(X,L^ m)$, is a Hilbert space for this norm. Write $N_ m=\dim \mathcal H_ m(u)$.
Let $v\in \mathrm{QPSH}(X)$. If $u\preceq _{\mathcal I}v$, that is, $\mathcal I(ku)\subseteq \mathcal I(kv)$ for every real $k>0$ (section 5.33), then $\mathcal H_ m(u)\subseteq \mathcal H_ m(v)$ for every $m$. In particular, weights of the same ordinary singularity type (definition 16.7.1) have the same section spaces, by proposition 5.33.2. When the two weights differ by a bounded function, their Hilbert norms are also comparable. Below, $\| s\| _{m,v,\mathrm{d}V'}$ denotes the norm defined with $v$ and $\mathrm{d}V'$ in place of $u$ and $\mathrm{d}V$.
Proposition 19.8.2.F2503E6C53F2 Let $v\in \mathrm{QPSH}(X)$, let $D,E\geq 0$, and let $F\in C^\infty (X,\mathbb R)$. Suppose that $|u-v|\leq D$ outside the union of the pole sets of $u$ and $v$, and that $\mathrm{d}V'=\mathrm{e}^ F \mathrm{d}V$ with $|F|\leq E$. Then $u$ and $v$ have the same section spaces, and for every positive integer $m$ and every $s$ in the common space $\mathcal H_ m(u)=\mathcal H_ m(v)$,
\[ \mathrm{e}^{-mD-E}\| s\| _{m,u}^2 \leq \| s\| _{m,v,\mathrm{d}V'}^2 \leq \mathrm{e}^{mD+E}\| s\| _{m,u}^2. \]
Proof.
Off the pole sets, the integrand of $\| s\| _{m,v,\mathrm{d}V'}^2$ is that of $\| s\| _{m,u}^2$ multiplied by $\mathrm{e}^{-m(v-u)+F}$, a function with values between $\mathrm{e}^{-mD-E}$ and $\mathrm{e}^{mD+E}$. Hence one norm is finite if and only if the other is, so the section spaces coincide, and integration gives both estimates.
$\square$
Fix $m\geq 1$ and choose an orthonormal basis $s_1,\ldots ,s_{N_ m}$ of $\mathcal H_ m(u)$. Define $K_{m,u}$ on $X$ and $B_{m,u}$ off the pole set by
\[ K_{m,u}(x)=\sum _{j=1}^{N_ m}|s_ j(x)|_{h^ m}^2, \qquad B_{m,u}(x)=K_{m,u}(x)\mathrm{e}^{-mu(x)}, \]
and set $B_{m,u}=0$ on the pole set. When $N_ m=0$, both functions are zero. Since two orthonormal bases differ by a unitary matrix, $K_{m,u}$ does not depend on the choice of basis. We call $\Phi _ m=m^{-1}\log K_{m,u}$ the Bergman potential; when $N_ m>0$, $\beta +\mathrm{dd}^{\mathrm{c}}\Phi _ m$ is its Bergman current. For $N_ m=0$, we have $\Phi _ m\equiv -\infty $.
Proposition 19.8.3.9E7A6E9D52A6 If $N_ m>0$, then
\[ K_{m,u}(x)=\sup _{0\neq s\in \mathcal H_ m(u)} \frac{|s(x)|_{h^ m}^2}{\| s\| _{m,u}^2}, \qquad \int _ X B_{m,u}\, \mathrm{d}V=N_ m. \]
Moreover, the Bergman potential $\Phi _ m=m^{-1}\log K_{m,u}$ is not identically $-\infty $, has analytic singularities, and satisfies
\[ \beta +\mathrm{dd}^{\mathrm{c}}\Phi _ m\geq 0. \]
In particular, the Bergman current $\beta +\mathrm{dd}^{\mathrm{c}}\Phi _ m$ is a closed positive current.
Proof.
Write a section as $s=\sum _ ja_ js_ j$ and trivialize the fiber of $L^ m$ at $x$ by a unit vector. By the Cauchy–Schwarz inequality, $|s(x)|_{h^ m}^2\leq \| s\| _{m,u}^2K_{m,u}(x)$, with equality when the coefficient vector $(a_ j)$ is proportional to the complex conjugate of $(s_ j(x))$. If $K_{m,u}(x)=0$, both sides of the extremal formula vanish. This proves the extremal formula. Integrating the finite sum defining $B_{m,u}$ gives $\sum _ j\| s_ j\| _{m,u}^2=N_ m$.
Locally $s_ j=f_ j e^{\otimes m}$ with holomorphic functions $f_ j$, so
\[ \Phi _ m=\frac1m\log \sum _ j|f_ j|^2-\chi , \qquad \beta +\mathrm{dd}^{\mathrm{c}}\Phi _ m=\frac1m\mathrm{dd}^{\mathrm{c}}\log \sum _ j|f_ j|^2\geq 0. \]
Since $X$ is connected, a nonzero section cannot vanish identically on an open set; hence the sum $\sum _ j|f_ j|^2$ does not vanish identically. The local expression therefore shows that $\Phi _ m$ is a quasi-plurisubharmonic function with analytic singularities, not identically $-\infty $. It also shows directly that $\beta +\mathrm{dd}^{\mathrm{c}}\Phi _ m$ is closed.
$\square$
Write $\Phi _ m(u,\mathrm{d}V)$ when the dependence on the weight and the volume form matters. Under the hypotheses of proposition 19.8.2, and provided that the common section space is nonzero, the extremal formula gives
\[ |\Phi _ m(u,\mathrm{d}V)-\Phi _ m(v,\mathrm{d}V')|\leq D+E/m \]
outside the common base locus. Both potentials equal $-\infty $ on that locus. In particular, the two potentials have the same ordinary singularity type, and for a fixed weight a change of the smooth volume form changes the Bergman potential by $O(m^{-1})$.
A simultaneous change of metric and weight leaves the weighted norm unchanged. Let $a\in C^\infty (X,\mathbb R)$, replace the metric by $h'=h\mathrm{e}^{-a}$, so that $|\cdot |_{h'}^2=|\cdot |_ h^2\mathrm{e}^{-a}$, and replace the weight by $u'=u-a$. Then $h'^ m\mathrm{e}^{-mu'}=h^ m\mathrm{e}^{-mu}$, so the Hilbert space, its norm and the density $B_{m,u}\, \mathrm{d}V$ are unchanged, while
\[ \beta '=\beta +\mathrm{dd}^{\mathrm{c}}a,\qquad \Phi '_ m=\Phi _ m-a. \]
Consequently the Bergman current $\beta +\mathrm{dd}^{\mathrm{c}}\Phi _ m$ does not change.
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