Lemma 10.7.11.F8C4EC9BBA24 Let $L$ and $T$ be holomorphic line bundles on $X$ with smooth quadratic metrics $h_0$ and $h_ T$, and let $\mathrm dV$ be a smooth positive volume form on $X$. For an integer $k\geq 1$ and a quasi-plurisubharmonic function $\varphi $ on $X$, the space
\[ \mathcal H_ k=\left\{ s\in H^0(X,L^ k\otimes T): \int _ X|s|_{h_0^ k\otimes h_ T}^2\mathrm e^{-k\varphi }\, \mathrm dV<\infty \right\} \]
is a finite-dimensional Hilbert space for the displayed squared norm $\| s\| _ k^2$. If $\mathcal H_ k\ne \{ 0\} $ and $s_1,\ldots ,s_{d_ k}$ is an orthonormal basis, then
E4884F8C8C74
10.7.12
\begin{equation} \varphi _ k=\frac1k\log \sup _{\| s\| _ k\leq 1}|s|_{h_0^ k\otimes h_ T}^2 =\frac1k\log \sum _{i=1}^{d_ k}|s_ i|_{h_0^ k\otimes h_ T}^2. \end{equation}
Writing $\theta =c_1(L,h_0)$ and $\theta _ T=c_1(T,h_ T)$, the function $\varphi _ k$ belongs to $\mathrm{PSH}(X,\theta +k^{-1}\theta _ T)$ and has gentle analytic singularities with coefficient $1/k$. Except for the gentle analytic singularities, which are defined only on compact Kähler manifolds (definition 6.14.12), these assertions hold more generally on every compact complex manifold $X'$ of positive dimension in place of $X$.
Proof.
By lemma 10.6.2(3), $H^0(X,L^ k\otimes T)$ and its linear subspace $\mathcal H_ k$ are finite dimensional. The weighted integral defines an inner product on $\mathcal H_ k$: the function $\varphi $ is finite almost everywhere, so the weight is positive almost everywhere, and a section of norm zero vanishes almost everywhere and hence identically. A finite-dimensional inner product space is complete.
At a point $y$, with the coefficients $c_ i$ of $s=\sum _ ic_ is_ i$, the Cauchy–Schwarz inequality gives $|s(y)|^2\leq \sum _ i|c_ i|^2\sum _ i|s_ i(y)|^2$, with equality when the $c_ i$ are proportional to the complex conjugates of the values of the $s_ i$ at $y$ in a local frame. This proves the equality in (10.7.12). In local frames with $|e|_{h_0}^2=\mathrm e^{-p_0}$ and $|e_ T|_{h_ T}^2=\mathrm e^{-p_ T}$, write $s_ i=f_ ie^ k\otimes e_ T$. Then
\[ \varphi _ k=\frac1k\log \sum _ i|f_ i|^2-p_0-\frac{p_ T}k, \]
and $\theta +k^{-1}\theta _ T+\mathrm{dd}^{\mathrm{c}}\varphi _ k=k^{-1}\mathrm{dd}^{\mathrm{c}}\log \sum _ i|f_ i|^2\geq 0$ by proposition 5.10.3. Hence $\varphi _ k\in \mathrm{PSH}(X,\theta +k^{-1}\theta _ T)$, with analytic singularities of type $(1/k,\mathfrak b)$, where $\mathfrak b$ is the ideal sheaf generated locally by the $f_ i$. This ideal sheaf does not depend on the frames, because a change of frames multiplies all $f_ i$ by the same nowhere vanishing holomorphic function, and it is coherent by lemma 6.8.4. Moreover $\mathrm e^{k\varphi _ k}=\sum _ i|s_ i|^2$ is smooth on $X$. The local expressions for $\varphi _ k$ above are presentations with the smooth remainders $-p_0-p_ T/k$, so proposition 6.14.8 shows that on every log resolution $\pi $ of $\mathfrak b$, with $\pi ^*\mathfrak b=\mathcal O(-D)$, locally $\varphi _ k\circ \pi $ is $k^{-1}\log |s_ D|^2$ plus a smooth function. These are the conditions of definition 6.14.12.
Apart from this appeal to definition 6.14.12, the proof uses only that $X$ is a compact complex manifold of positive dimension, since the last assertion of lemma 10.6.2 gives its part (3) in this generality; hence it also proves the last assertion of the lemma.
$\square$
Pome
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