Proposition 5.43.747C6B39BA71A

Proof. The critical set is a proper compact analytic subset of $X$. It has finitely many irreducible components, and $p$ is constant on each: its differential vanishes on the smooth locus of that component, and continuity extends the constant to the component. Thus the set $D\subset C$ of critical values is finite.

Let $U\in \mathrm{PSH}(X,p^*\beta )$. On a base coordinate disk choose a smooth $a$ with $\mathrm{dd}^{\mathrm{c}}a=\beta $. Over $C\setminus D$, the function $U+a\circ p$ restricts to a plurisubharmonic function, or the constant $-\infty $, on each compact connected fiber. The maximum principle makes it constant there. Local holomorphic sections then give a function $v$ with $U=v\circ p$ and $a+v$ subharmonic. This is the local argument of proposition 5.43.5; it uses compactness of the fibers, not compactness of the punctured total space.

The function $v$ is bounded above by $\sup _ XU$. The removable singularity theorem extends $a+v$ subharmonically across the finitely many punctures. The extensions glue, giving $v\in \mathrm{PSH}(C,\beta )$. Its pullback is $p^*\beta $-plurisubharmonic by lemma 5.43.2. It agrees with $U$ off the proper analytic set $p^{-1}(D)$, hence almost everywhere and everywhere by lemma 5.24.3.

Conversely pullback preserves the curvature inequality by the disk criterion. It is not identically $-\infty $, because $p$ is surjective. Surjectivity also proves uniqueness and preserves the supremum and every inequality $v\leq w+c$. $\square$