Lemma 5.43.212585F177470

Proof. Suppose first that $\theta +\mathrm{dd}^{\mathrm{c}}u\geq 0$. On a coordinate ball, convolve $u$ and $\theta $ with a smooth nonnegative radial kernel. The resulting smooth functions $u_\epsilon $ satisfy

Write $u=p-r$ locally, with $p$ plurisubharmonic and $r$ smooth. The convolutions $p_\epsilon $ decrease to $p$. On a smaller disk on which $u\circ f$ is not identically $-\infty $, $p_\epsilon \circ f$ therefore converges in $L^1_{\mathrm{loc}}$ to $p\circ f$, while $r_\epsilon \circ f$ converges smoothly. Pulling back the smooth inequality and passing to the limit in the sense of distributions gives $\mathrm{dd}^{\mathrm{c}}(u\circ f)+f^*\theta \geq 0$. Since $u\circ f=p\circ f-r\circ f$ is the sum of a subharmonic and a smooth function, this distributional inequality makes $u\circ f+a$ subharmonic.

Conversely, test on affine complex lines in a coordinate ball. For a fixed nonzero constant direction $\xi $, local integrability of $u$ implies that the restriction of $u$ to almost every line parallel to $\xi $ is not identically $-\infty $. Fubini's theorem, applied to nonnegative smooth test functions, then gives positivity of the scalar distribution obtained by evaluating the Hermitian coefficient matrix of $\theta +\mathrm{dd}^{\mathrm{c}}u$ on $\xi $. Positivity for a countable dense set of constant directions proves positivity of the Hermitian matrix of coefficient measures, which is the current inequality. $\square$