PomeLemma 5.43.2.12585F177470 Let $\theta $ be a smooth real $(1,1)$-form on a complex manifold $M$, and let $u$ be quasi-plurisubharmonic on $M$. Then $\theta +\mathrm{dd}^{\mathrm{c}}u\geq 0$ if and only if the following condition holds: for every holomorphic map $f:\Delta \to M$ from the unit disk such that $u\circ f$ is not identically $-\infty $, and for every smooth function $a$ on an open subset of $\Delta $ with $\mathrm{dd}^{\mathrm{c}}a=f^*\theta $, the function $u\circ f+a$ is subharmonic there.
5.43.1 Holomorphic disks1F1A3CC0946E
A smooth background form need not have local $\mathrm{dd}^{\mathrm{c}}$-potentials, but its pullback to a disk does, since a real $(1,1)$-form on a disk is closed. Testing on holomorphic disks therefore characterizes the exact lower curvature bound in a way that is preserved by holomorphic maps.
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
\[ \mathrm{dd}^{\mathrm{c}}u_\epsilon \geq -\theta _\epsilon , \qquad \theta _\epsilon \longrightarrow \theta \quad \text{smoothly}. \]
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. $\square$
Consequently, holomorphic pullback preserves the inequality $\theta +\mathrm{dd}^{\mathrm{c}}u\geq 0$ wherever the pulled-back function is not identically $-\infty $: apply the disk criterion to the compositions with the given map. We also use descent of subharmonicity under the finite map $\rho (t)=t^ k$, $k\geq 1$, of the disk. If $h\circ \rho $ is subharmonic, local inverse branches of $\rho $ show that $h$ is subharmonic off zero. Moreover, $h$ is bounded above near zero, and its value at zero is its upper limit there, since
\[ h(0)=(h\circ \rho )(0) =\limsup _{t\to 0,\ t\ne 0}h(t^ k) =\limsup _{z\to 0,\ z\ne 0}h(z). \]
By the removable-singularity theorem, $h$ is subharmonic at zero as well.
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