PomeProposition 5.45.1.620B099907AB The curvature current of a singular Hermitian metric with locally integrable weights on a holomorphic line bundle is closed. In particular, a nonclosed form $\theta $ cannot itself be the curvature current of such a metric, and $\theta +\mathrm{dd}^{\mathrm{c}}u$ is not closed on any neighborhood of a point where $\mathrm{d}\theta \neq 0$.
5.45 Curvature of line bundles355A31F789F6
355A31F789F6
Let $M$ be a complex manifold and let $\theta $ be a smooth real $(1,1)$-form on $M$. To relate potentials to holomorphic sections, we ask when a current $\theta +\mathrm{dd}^{\mathrm{c}}u$, with $u$ locally integrable, can be the curvature current of a metric on a line bundle. The transition law for the local weights of such a metric forces a closedness condition on the background.
Proof. Let $h$ be a singular Hermitian metric on a holomorphic line bundle $L\to M$. Choose a local holomorphic frame $e$ of $L$ and write its squared norm as $|e|_ h^2=\mathrm{e}^{-\varphi }$; the curvature current is then locally $\mathrm{dd}^{\mathrm{c}}\varphi $. If $e'=ge$ is another frame, with $g$ holomorphic and nowhere zero, the new weight is $\varphi -\log |g|^2$. Since $\log |g|^2$ is pluriharmonic, these local currents agree on overlaps and glue to a global current, which is closed because $\mathrm{d}\mathrm{dd}^{\mathrm{c}}=0$. The final assertion follows from the identity
\[ \mathrm{d}(\theta +\mathrm{dd}^{\mathrm{c}}u)=\mathrm{d}\theta \]
of currents. $\square$
Accordingly, when we study sections subject to multiplier-ideal conditions, we specify both the line bundle, whose curvature is closed, and any auxiliary Hermitian form on $M$. An algebraic interpretation then requires relating the invariants of the line bundle to this analytic background.
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