Proposition 5.43.58BFC96F6E5F8

Proof. On a coordinate ball $V\subset X$, choose a smooth function $\rho $ with $\mathrm{dd}^{\mathrm{c}}\rho \geq \theta $. If $u\in \mathrm{PSH}(Y,\pi ^*\theta )$, then $u+\rho \circ \pi $ is plurisubharmonic on $\pi ^{-1}(V)$. Its restriction to a fiber is plurisubharmonic or identically $-\infty $, and since the fiber is compact and connected, the maximum principle shows that it is constant in either case. As $\rho \circ \pi $ is constant on fibers, $u=v\circ \pi $ for a uniquely defined function $v$ on $X$.

A holomorphic submersion admits local holomorphic sections. Composing $u+\rho \circ \pi $ with such a section shows that $v+\rho $ is plurisubharmonic or identically $-\infty $ on a smaller ball. The latter alternative would make $u=-\infty $ on a nonempty open set, which is impossible for a quasi-plurisubharmonic function. Hence $v$ is quasi-plurisubharmonic. Distributional differentiation in local product coordinates gives

Positivity of a pulled-back current of this form implies positivity on the base: test in product coordinates against products with a nonnegative compactly supported volume form of integral one in the fiber variables. It follows that $v\in \mathrm{PSH}(X,\theta )$.

Conversely, by lemma 5.43.2, the pullback of a $\theta $-plurisubharmonic function by $\pi $ is $\pi ^*\theta $-plurisubharmonic; in local product coordinates it is clearly not identically $-\infty $. The pullback map is injective because $\pi $ is surjective. $\square$