Lemma 9.9.283D4170FB2B3

Proof. We prove (1) and the first assertion of (2) together. Let $u$ be locally plurisubharmonic on $X$, let $F\colon Y\to X$ be holomorphic, and let $y\in Y$. Choose a local model of a neighborhood of $F(y)$, with ambient coordinates $w$ and a ball $V$ around $F(y)$ on which $u$ is the restriction of a function $v$ that is plurisubharmonic or identically $-\infty $, and an arbitrary local model of a neighborhood of $y$, with ambient coordinates $z$. By the definition of holomorphic maps and of $\mathcal O_ Y$, the components of $F$ in the $w$-model extend holomorphically to a neighborhood of $y$ in the $z$-ambient space, and their joint extension $G$, being continuous with $G(y)=F(y)$, maps a smaller ball $V'$ around $y$ into $V$. By proposition 5.9.3, $v\circ G$ is plurisubharmonic or identically $-\infty $ on $V'$, and its restriction to $Y\cap V'$ is $u\circ F$. Hence $u\circ F$ is locally plurisubharmonic, with respect to every local model of $Y$. For $Y=X$ and $F$ the identity, with two local models at the same point, this is (1).

If $Y$ is a one-dimensional complex manifold, its holomorphic charts are local models. By what we have just proved, $u\circ F$ is therefore, in local coordinates near each point of $Y$, plurisubharmonic or identically $-\infty $, that is, subharmonic in the sense of definition 3.4.2. Subharmonicity is a local property, so $u\circ F$ is subharmonic in every local coordinate, and on the whole disc if $Y$ is a disc.

(3) Holomorphic charts are local models. So by (1), every point of $X$ has a coordinate ball on which a locally plurisubharmonic function $u$ is plurisubharmonic or identically $-\infty $, and by lemma 5.9.4, $u$ is plurisubharmonic or identically $-\infty $ on each connected component of $X$. Conversely, a function with this property is locally plurisubharmonic, with the charts as local models, since its expressions in charts with connected domains are plurisubharmonic or identically $-\infty $. The irreducible components of the manifold $X$ are its connected components, by corollary 9.3.16. Hence $\mathrm{PSH}(X)$, in the sense of definition 9.9.1, consists of the functions that are plurisubharmonic on each connected component of $X$, which are the plurisubharmonic functions in the sense of section 5.9, since plurisubharmonicity is a local property there. $\square$