Proposition 5.20.17.2DDBBF280789 Let $(L,h)$ be a Hermitian line bundle on $X$ and set $\theta =c_1(L,h)$. Let $(T,h_ T)$ be a Hermitian line bundle on $X$. Assume that $\varphi \in \mathrm{PSH}(X,\theta )$ is a potential with analytic singularities such that $\theta _{\varphi }$ is a Kähler current. Fix a Kähler form $\omega $ on $X$. For each $k\geq 1$, we let
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5.20.18
\begin{equation} \varphi _ k \coloneqq \frac{1}{k} \log \sup _{\substack {s \in \mathrm{H}^0(X,L^ k \otimes T)\\ \int _ X |s|^2_{h^ k \otimes h_ T}\mathrm{e}^{-k\varphi } \omega ^ n \leq 1}} |s|^2_{h^ k \otimes h_ T}. \end{equation}
Then for all sufficiently large $k$,
\[ \varphi \preceq \varphi _ k\preceq \alpha _ k \varphi , \]
where $\alpha _ k\in (0,1)$ is an increasing sequence with limit $1$.
Proof.
First we prove $\varphi \preceq \varphi _ k$. Since $\theta _{\varphi }$ is a Kähler current, there is $\delta >0$ such that $\varphi \in \mathrm{PSH}(X,\theta -\delta \omega )$. Let $x\in X$ with $\varphi (x)>-\infty $. Applying theorem 5.20.16 to the point $Y=\{ x\} $, for all sufficiently large $k$ and every unit vector $e_ x\in (L^ k\otimes T)_ x$, there is a section $\tilde{s}\in \mathrm{H}^0(X,L^ k\otimes T\otimes \mathcal{I}(k\varphi ))$ with $\tilde{s}(x)=e_ x$ and
\[ \int _ X |\tilde{s}|^2_{h^ k\otimes h_ T}\mathrm{e}^{-k\varphi }\omega ^ n\leq C\mathrm{e}^{-k\varphi (x)}. \]
After rescaling $\tilde{s}$, the definition of $\varphi _ k$ gives
\[ \varphi _ k(x)\geq \varphi (x)-\frac{\log C}{k}. \]
This proves $\varphi \preceq \varphi _ k$.
The other part of the inequality follows from 5.18.15 and (5.18.17).
More precisely, if $s_1,\ldots ,s_ N$ is an orthonormal basis of $\mathrm{H}^0(X,L^ k\otimes T\otimes \mathcal{I}(k\varphi ))$, with respect to the $L^2$-norm:
\[ s\mapsto \left( \int _ X |s|^2_{h^ k\otimes h_ T}\mathrm{e}^{-k\varphi }\, \omega ^ n \right)^{1/2}, \]
then by (5.18.17),
\[ s_ i\in \mathrm{H}^0\left(X,L^ k\otimes T\otimes \mathcal{I}_{\infty }\left((k-c^{-1}n)_+\varphi \right)\right), \]
where $c>0$ depends only on $\varphi $. So there is $C_ k>0$ so that
\[ \sum _{i=1}^ N |s_ i|^2_{h^ k\otimes h_ T}\mathrm{e}^{-(k-c^{-1}n)_+\varphi }\leq C_ k\quad \textup{quasi-everywhere}. \]
In other words,
\[ \varphi _ k\preceq \frac{1}{k}\left( k-c^{-1}n \right)_+ \varphi . \]
It suffices to take $\alpha _ k=(1-k^{-1}c^{-1}n)_+$.
$\square$
Pome
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