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Proposition 19.8.156400AAE4BA1E

Proposition 19.8.15.6400AAE4BA1E Let $\omega $ be a Hermitian metric on $X$ with $v_+(\omega )=+\infty $. There is a potential $U\in \mathrm{PSH}(X,\omega )$ such that, with $L=\mathcal O_ X$ and its constant metric,

\[ \dim \mathcal H_ m(U)=1\quad \hbox{for every }m\geq 1, \qquad \int _ X\mathrm{MA}^{\mathrm{ext}}_\omega (U)=+\infty . \]

All Bergman currents of $U$ are identically zero.

Proof. Take $U$ from theorem 13.14.18; its extended mass is infinite, and $\mathcal I(tU)=\mathcal O_ X$ for every $t>0$. Global holomorphic functions on $X$ are constant, and the constant function $1$ lies in $\mathcal H_ m(U)$ because $\mathcal I(mU)=\mathcal O_ X$ and $X$ is compact. Hence every weighted section space is one-dimensional, and its orthonormal basis consists of a nonzero constant. Each Bergman potential $\Phi _ m$ is therefore constant. Since the trivial metric has curvature zero, every Bergman current is zero. $\square$

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