Lemma 18.37.1.22D696DB398E Let $B$ be a connected compact complex manifold of dimension $q\geq 1$, let $A\to B$ be a holomorphic line bundle with a smooth Hermitian metric, let $v\in C(B)$, and let $\nu $ be a smooth positive probability volume on $B$. For every integer $m\geq 1$ let $W_ m\subseteq V_ m\subseteq H^0(B,A^ m)$ be subspaces, both with the norm $\| \cdot \| _{m,v,\nu }$, and put $d_ m=\dim V_ m-\dim W_ m$. Write $B_{V_ m,v}$, $B_{W_ m,v}$ for their kernels and $\beta _{V_ m,v}^{(q)}$, $\beta _{W_ m,v}^{(q)}$ for their kernel measures, defined as in section 18.30 with the space indicated by the subscript. Then
\[ \begin{gathered} B_{V_ m,v}-B_{W_ m,v}\geq 0,\\ \| \beta _{V_ m,v}^{(q)}-\beta _{W_ m,v}^{(q)}\| _{\mathrm{TV}} =\frac{q!d_ m}{m^ q}. \end{gathered} \]
The determinant functionals of the two spaces, each normalized to vanish at the zero weight, satisfy
\[ |\mathcal F_{V_ m}^{(q)}(v)-\mathcal F_{W_ m}^{(q)}(v)| \leq \frac{q!d_ m}{m^ q}\| v\| _\infty . \]
These statements include the case of zero-dimensional spaces.
Proof.
Extend an orthonormal basis of $W_ m$ to an orthonormal basis of $V_ m$ for the norm at the weight $v$. The difference of the kernels is the sum of the squared weighted evaluations of the $d_ m$ added basis vectors. It is nonnegative, and its integral against $\nu $ is $d_ m$, which gives the total variation formula. For the determinant functionals, apply the first-variation formula of proposition 18.34.1 to each of the two fixed spaces, and integrate the difference along $t\mapsto tv$, $0\leq t\leq 1$. By what we have just proved, at each $t$ the difference of the kernel measures is positive with total mass $q!d_ m/m^ q$, so its integral against $v$ is at most $q!d_ m\| v\| _\infty /m^ q$ in absolute value. Since both functionals vanish at zero, this proves the last assertion.
$\square$
Pome
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