Lemma 23.1.33264893FAE204

Proof. Choose $n-2$ horizontal angular vectors, one horizontal radial vector and the vertical vector $\partial _ w$. In the normalization of the proof of lemma 23.1.7, each angular entry is $A=(\mu +V_ x)/\mu $, and the radial–vertical block is

This follows by differentiating $s=w+\bar w+\log (1+\mathrm e^ x)$: the radial part of $\partial \bar\partial x$ is zero, whereas its angular part contributes $V_ x/\mu $. The Schur complement of the last entry gives the stated positivity criterion. At $\xi =0$ the same endpoint argument as in dimension three applies.

Set $d=n-1$. The radial Fubini–Study probability density is $d\mu ^{d-1}\mathrm d\mu =d\mu ^ d(1-\mu )\mathrm dx$; it follows from the polar-coordinate determinant of $\mathrm{dd}^{\mathrm{c}}\log (1+|\xi |^2)$ and its integral is one. The background block $K_0$ has determinant one. Using the fiber mass two, therefore,

For $n=3$ this is $12\mu ^2(1-\mu )A\det K_ V$. Comparing the two expressions proves (23.1.34). $\square$