Proposition 16.29.20D9BF9081FC38

Proof. First, an $E$-invariant $q\in \mathrm{PSH}(X,bh)$ descends to an upper semicontinuous function $w$ on $B$, since $\pi $ is proper and $q$ is constant on its fibers. In a product chart, Fubini gives $w\in L^1_{\mathrm{loc}}$, and

Lift a smooth base vector to a vector annihilated by $\partial s$. Its pairing with $\beta $ is zero. Testing the positive current above against these lifts and a nonnegative smooth fiber density of integral one proves $b\kappa +\mathrm{dd}^{\mathrm{c}}w\geq 0$ distributionally. A local potential of $b\kappa $ gives its canonical plurisubharmonic representative. Its pullback agrees almost everywhere with $q$, hence everywhere by lemma 5.24.3. Thus $w\in \mathrm{PSH}(B,b\kappa )$. Conversely, every such $w$ lifts to a $bh$-plurisubharmonic function by the displayed identity.

Now set $q=P_{bh}[\pi ^*v]_{\mathcal I}$. Translation by any $g\in E$ preserves its complete competitor family: it preserves the background, the obstacle zero, and the ideals of the invariant input. The envelope is their greatest member, so $q\circ g=q$. Descend $q=\pi ^*w$ as above. By lemma 5.37.7, its ideal equivalence to $\pi ^*v$ implies $w\sim _{\mathcal I}v$: a germ pulled back from the base is weighted-integrable in a product chart if and only if the base germ is, by Fubini. Hence $w$ is a base competitor. Every base competitor also lifts to an ambient competitor by the same lemma and the current identity. Taking the greatest members gives equality. $\square$