Proposition 23.1.38E42306969D71

Proof. First suppose that $u$ is smooth and $\omega +\mathrm{dd}^{\mathrm{c}}u>0$. Averaging the positive forms $\tau _ a^*(\omega +\mathrm{dd}^{\mathrm{c}}u)$ gives $\omega +\mathrm{dd}^{\mathrm{c}}Au>0$. Since $Au$ is smooth and constant on the fibers, $Au=\pi ^*v$ for a smooth function $v$ on $\mathbb P^2$. At each point, the kernel of $\partial s$ maps isomorphically onto the tangent space of the base, and $\beta $ vanishes on it. Restricting $\omega +\mathrm{dd}^{\mathrm{c}}Au=\pi ^*(\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}v)+\beta $ to this kernel therefore shows that $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}v>0$.

For a smooth function $f:E\to \mathbb R$, the nonconstant Fourier characters in (23.1.11), indexed by $(p,q)\in \mathbb Z^2\setminus \{ (0,0)\} $, have eigenvalues $\pi ^2p^2+q^2/4\geq 1/4$ for $-\partial _ w\partial _{\bar w}$. Parseval's identity therefore gives

Applying this inequality on every fiber, integrating against $\omega _{\mathrm{FS}}^2$ and using the mass identity (23.1.5), we obtain (23.1.39).

For bounded $u$, choose uniformly bounded smooth functions $u_ j\downarrow u$ with $\omega +\mathrm{dd}^{\mathrm{c}}u_ j>0$ by theorem 5.32.1. Their averages decrease pointwise to $Au$ by dominated convergence. The corresponding smooth base potentials $v_ j$ decrease to a bounded $\omega _{\mathrm{FS}}$-plurisubharmonic function $v$, so $Au=\pi ^*v$. Dominated convergence gives $u_ j\to u$ and $Au_ j\to Au$ in $L^2(X,\nu )$, while bounded Bedford–Taylor continuity gives convergence of the total masses. Thus (23.1.39) passes to $u$.

If the mass is $6$, the inequality gives $u=Au$ almost everywhere. Since both functions are bounded and quasi-plurisubharmonic, lemma 5.24.4 gives equality everywhere. Conversely, if $u=\pi ^*v$ with bounded $v\in \mathrm{PSH}(\mathbb P^2,\omega _{\mathrm{FS}})$, choose smooth $v_ j\downarrow v$ with $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}v_ j>0$ by theorem 5.32.1. The pullbacks decrease to $u$ and are uniformly bounded. They are strictly $\omega $-plurisubharmonic because the base forms are positive and $\beta $ is positive on the vertical line. Their vertical derivatives vanish, so (23.1.5) gives mass $6$ for each of them. Bounded Bedford–Taylor continuity gives the same mass for $u$.

It remains to prove that the constant $2/3$ is optimal. Choose a nonzero real $\rho \in C_ c^\infty (U_0)$. For $\delta \in \mathbb R$, the function

extends by zero to a smooth function on $X$. For sufficiently small nonzero real $\delta $, it satisfies $\omega +\mathrm{dd}^{\mathrm{c}}u>0$. Its fiber average is zero, and its only fiber frequencies are $(p,q)=(0,\pm 1)$, with eigenvalue $1/4$. Hence equality holds in the fiber inequality, and therefore in (23.1.39), with nonzero left-hand side. No smaller coefficient can hold. $\square$