PomeDefinition 9.11.2.76A75EB639CB An upper semicontinuous function $u:X\to [-\infty ,+\infty )$, not identically $-\infty $ on any nonempty open subset, is $\theta $-plurisubharmonic if it satisfies the following disk condition. For every holomorphic $f:\Delta \to X$, the function $u\circ f+a$ is subharmonic or identically $-\infty $ wherever $a$ is a smooth local potential for $f^*\theta $. The set of these functions is denoted by $\mathrm{PSH}(X,\theta )$.
9.11.1 Potentials and disk tests0753019C64C3
Since $X$ may be singular, $\theta $-plurisubharmonicity is defined by testing along holomorphic disks. Let $\Delta \subset \mathbb C$ be the unit disk. For a holomorphic map $f:\Delta \to X$, the pullback $f^*\theta $ is a smooth real $(1,1)$-form on $\Delta $ and therefore has smooth local potentials. Disk tests thus express the background inequality even when $\theta $ has no local potential on $X$.
Two choices of $a$ differ by a harmonic function, so the condition is independent of that choice. On every disk where $u\circ f\not\equiv -\infty $, it gives $\mathrm{dd}^{\mathrm{c}}(u\circ f)+f^*\theta \geq 0$ in the sense of distributions.
Lemma 9.11.3.E7163CA56764 Every member of $\mathrm{PSH}(X,\theta )$ is locally the sum of a plurisubharmonic function on $X$ and a smooth function. If $X$ is smooth, this definition agrees with definition 5.22.1.
Proof. In a local embedding, choose an ambient smooth strictly plurisubharmonic function $r$ with $\mathrm{dd}^{\mathrm{c}}r\geq \theta $ on a smaller neighborhood. For a disk $f$ and a local potential $a$ of $f^*\theta $, the function $r\circ f-a$ is subharmonic. Hence $(u+r)\circ f$ is subharmonic or identically $-\infty $. The Fornaess–Narasimhan criterion, together with the assumption that $u$ is not identically $-\infty $ on any open set, shows that $u+r$ is plurisubharmonic. The smooth case now follows from lemma 5.43.2. $\square$
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