Proposition 5.9.3F6423BCE3BB1

Proof. The statement for open $D$ follows by applying the statement for domains to each connected component of $D$. If $n=0$, then $\Omega $ is a point, $u$ is a real constant, and so is $u\circ F$. If $m=0$, then $D$ is a point and $u\circ F$ is a single value in $[-\infty ,\infty )$; it is either $-\infty $ or a real number, that is, a plurisubharmonic function on the point. Now let $m,n\geq 1$. By theorem 5.9.1, $u$ is the canonical subharmonic representative of a locally integrable function with $\mathrm{dd}^{\mathrm{c}}u\geq 0$, so proposition 5.4.4 applies: $u\circ F$ is either identically $-\infty $, or locally integrable and subharmonic, in particular upper semicontinuous. In the second case, the restriction of $u\circ F$ to a connected component of an affine complex-line section of $D$ is the composition of $u$ with a holomorphic map from an open subset of $\mathbb {C}$, so it is subharmonic or identically $-\infty $ by the same proposition. Hence $u\circ F$ is plurisubharmonic. $\square$