Proposition 14.19.3856B19584AE9E

Proof. Integrating the equation gives $\int _ X\omega _ u^3=6c_ f I_ f$. Both the inequality $c_ fI_ f\geq 1$ and (14.19.39) therefore follow from (23.1.39), whose left-hand side is nonnegative.

Suppose $c_ f I_ f=1$. By proposition 23.1.38, $u=\pi ^*q$ for a bounded $\omega _{\mathrm{FS}}$-plurisubharmonic function $q$; local holomorphic sections of $\pi $ show that $q$ is continuous. Thus $u\circ \tau _ a=u$ for every $a\in E$. Since bounded products are natural under biholomorphisms and $\omega $ is invariant under the translations, it follows that

As $f$ is defined only almost everywhere, we use this invariance as an identity of measures.

By Fubini's theorem, the fiber average $Af$ is defined almost everywhere. It equals $\pi ^*g$ for a measurable function $g$ on the base, because local trivializations differ by translations of the Haar variable. Jensen's inequality gives

For every continuous real test function $\chi $ on $X$, the invariance of the measure gives $\int _ X\chi f\, \mathrm{d}\nu =\int _ X(\chi \circ \tau _ a)f\, \mathrm{d}\nu $. Integrating in $a$, and using Fubini's theorem together with the invariance of Haar measure under $a\mapsto -a$, we obtain $\int _ X\chi f\, \mathrm{d}\nu =\int _ X\chi Af\, \mathrm{d}\nu $. Hence $f=Af=\pi ^*g$ almost everywhere, with $g\geq 0$.

Conversely, suppose $f=\pi ^*g$ almost everywhere. Disintegration gives $I_ f=\int _{\mathbb P^2}g\omega _{\mathrm{FS}}^2$. Applying theorem 14.19.2 on the base, we find a continuous $\omega _{\mathrm{FS}}$-plurisubharmonic function $q$ and a constant $k>0$ with $(\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q)^2=kg\, \omega _{\mathrm{FS}}^2$. Bounded decreasing regularization and Stokes' theorem give $\int _{\mathbb P^2}(\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q)^2=1$, so $k=I_ f^{-1}$.

Put $u=\pi ^*q$. To compute its bounded product, take smooth strictly $\omega _{\mathrm{FS}}$-plurisubharmonic functions $q_ j\downarrow q$ by theorem 5.32.1. Their pullbacks decrease to $u$ and are strictly $\omega $-plurisubharmonic. Since $\alpha ^3=\beta ^2=0$, the smooth cube identity gives, for each $j$,

In a local trivialization, the right-hand side is $6\mathrm{d}A_ E$ times the base measure $(\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q_ j)^2$: the horizontal components of $\beta $ vanish against the base top-degree form, and its vertical restriction is $2\mathrm{d}A_ E$. Continuity of bounded products on the base and on $X$, tested in these trivializations and patched by a partition of unity, therefore gives

Uniqueness of the constant implies $c_ f=I_ f^{-1}$. Every continuous solution then has mass $6$, so the mass characterization in proposition 23.1.38 makes it a pullback from the base, and the descended potential is continuous by local sections, as above. $\square$