Theorem 9.11.5F4C06D21AA94

Proof. Pullback preserves the disk inequalities. Every nonempty open subset of $Y$ meets the locus where $\pi $ is biholomorphic, so a pullback is not identically $-\infty $ on any nonempty open set. Injectivity follows from surjectivity of $\pi $.

For the converse, take $v\in \mathrm{PSH}(Y,\pi ^*\theta )$. Near a fiber, choose the local correction $r$ on $X$ used above. The function $v+r\circ \pi $ is plurisubharmonic on the inverse image of that neighborhood, since

Its restriction to each compact irreducible component of the fiber is constant, possibly $-\infty $, by the maximum principle on a resolution of that component. Connectedness of the fiber makes these constants equal: intersecting components have the same value, and the finite intersection graph of the components is connected. Since $r\circ \pi $ is constant on the fiber, so is $v$.

Thus $v=u\circ \pi $ for a unique function $u$ on $X$. This function is upper semicontinuous: for each real $c$, the closed set $\{ v\geq c\} $ has closed image under the proper map $\pi $, and that image is $\{ u\geq c\} $. If $u$ were identically $-\infty $ on a nonempty open set, the same would hold for $v$ on its inverse image. Hence $u$ is not identically $-\infty $ on any nonempty open set.

It remains to verify the disk condition for $u$ with the background $\theta $ itself. Let $f:\Delta \to X$ be a holomorphic disk with $u\circ f\not\equiv -\infty $. Near a given point of $\Delta $, take a curve in the fiber product $\Delta \times _ XY$ dominating the disk. Its normalization gives, after shrinking, a finite cover $\rho (t)=t^ k$ and a holomorphic lift $f':\Delta \to Y$ with $\pi \circ f'=f\circ \rho $. Choose a smooth local function $a$ on the original disk with $\mathrm{dd}^{\mathrm{c}}a=f^*\theta $. The function

is subharmonic by lemma 5.43.2. Subharmonicity descends through $\rho $ by the finite-disk argument following that lemma. Thus $u\circ f+a$ is subharmonic near every point, which proves that pullback is a bijection.

Since $\pi $ is surjective, pullback preserves suprema, uniform bounds and inequalities up to additive constants; this proves the remaining assertions. $\square$