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.
$\square$
Pome
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