Example 12.10.304C03E42B184D

Proof. Let

The quadratic form is nonsingular, and $F$ is finite of degree two, with smooth ramification divisor $Z=\{ w=0\} $. Local branch coordinates show that $\mathrm dF$ has rank $n-1$ on $Z$ and $n$ elsewhere. Let $\omega $ be the restriction to $X$ of the normalized Fubini–Study form of $\mathbb P^{n+1}$, and let $A=F^*\omega _{\mathrm{FS}}$, where $\omega _{\mathrm{FS}}$ is the normalized Fubini–Study form of $\mathbb P^ n$. Then

is a smooth function on $X$ with $-\log 2\leq u\leq 0$, $\sup _ Xu=0$ and $A=\omega +\mathrm{dd}^{\mathrm{c}}u$. The change of variables for the branched cover gives $\int _ XA^ n=2$, and Stokes' theorem gives $V=\int _ X\omega ^ n=2$.

Set $\mathrm{d}V=\omega ^ n/V$, $f=A^ n/\mathrm{d}V$, $T=A^{n-1}/V$ and $\gamma =A/V^{1/(n-1)}$. Since $\omega $ is closed, theorem 12.10.23 gives $b_\omega (f)=V/\int _ Xf\, \mathrm{d}V=1$. Stokes' theorem gives $\mathrm{dd}^{\mathrm{c}}T=0$ and $\int _ X\omega \wedge T=1$, and computing the determinant density gives $D_{f^{1/n}}(T)=1$. The arithmetic–geometric mean inequality, which remains valid for semipositive $A$, bounds $D_{f^{1/n}}(Q)$ by $\int _ XA\wedge Q=1$ for every smooth strictly positive $Q\in \mathscr G_\omega $. Thus $T$ is a maximizer, strictly positive off the proper analytic set $Z$, and theorem 12.10.17 gives $\Phi _ f=u$. Its root $\gamma $ has rank $n-1\geq 2$ on $Z$ and is positive elsewhere, as required. $\square$