Lemma 23.2.5670F067DD267F

Proof. If $j_ X([\lambda ])=0$, write $\mathrm d^{\mathrm c}\lambda =\mathrm{dd}^{\mathrm{c}}f$ for a smooth real function $f$. Since $\mathrm d^{\mathrm c}\mathrm{d}=-\mathrm{d}\mathrm d^{\mathrm c}$, the form $\lambda +\mathrm{d}f$ is both $\mathrm{d}$-closed and $\mathrm d^{\mathrm c}$-closed. Since the two equations separate the $\partial $- and $\bar\partial $-derivatives, its $(1,0)$-part is holomorphic and closed. Thus $\lambda +\mathrm{d}f$ is the real part of the closed holomorphic form $2(\lambda +\mathrm{d}f)^{1,0}$. Conversely, the real part of a closed holomorphic one-form is closed for both operators, so its image under $j_ X$ is zero.

By proposition 10.11.2, the pseudoeffective Bott–Chern cone contains no line. Thus $j_ X(\xi )$ and $-j_ X(\xi )$ are both pseudoeffective exactly when $j_ X(\xi )=0$, which is the identity $\mathcal C_ X\cap (-\mathcal C_ X)=K_ X$. $\square$