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Lemma 5.18.53B54149D4222

Lemma 5.18.5.3B54149D4222 Suppose that $X$ is a compact Kähler manifold, $\theta $ is a closed smooth real $(1,1)$-form on $X$, and $\varphi \in \mathrm{PSH}(X,\theta )$. Suppose that $\varphi $ has log singularities along an effective $\mathbb {Q}$-divisor $D$ on $X$. Then the cohomology class $\{ \theta \} -\{ D\} $ is nef.

Moreover, if in addition $\theta _{\varphi }$ is a Kähler current2, then the cohomology class $\{ \theta \} -\{ D\} $ is Kähler.

Proof. The current $\theta _{\varphi }-[D]$ has bounded local potentials. Therefore, the class $\{ \theta \} -\{ D\} $ is nef.

If $\theta _{\varphi }$ is a Kähler current, say $\theta _{\varphi }\geq \omega $ for a Kähler form $\omega $, then $\theta _{\varphi }-[D]-\omega $ is positive and has bounded local potentials, and therefore represents a nef class. We conclude that $\{ \theta \} -\{ D\} $ is Kähler. $\square$

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