18.36.4 The Hopf equilibrium measure16C99F55F2A7

On the Hopf manifold, the quotient equilibrium measure is a Hermitian Monge–Ampère measure: the vertical part of the Hopf metric supplies the missing fiber direction. The constant relating the two measures combines the fiber probability with a coefficient of the top power.

Use the Hopf manifold $X_ n$, $n\geq 2$, its projection $\pi :X_ n\to \mathbb P^{n-1}$, the Fubini–Study form $\sigma $ on $\mathbb P^{n-1}$ with $\int \sigma ^{n-1}=1$, and the forms $\alpha =\pi ^*\sigma $, $\zeta $ and $\omega =\alpha +\zeta $ of section 21.14; recall that $\zeta ^2=0$ and that $\zeta $ has integral two on each fiber. Take $A=\mathcal O(1)$ with its Fubini–Study metric, so that $\theta =\sigma $, and keep the selecting potential $\psi $ and the weighted data $(K,w,\mu )$ above. On each elliptic fiber put $\nu _ y=\zeta |_{X_ y}/2$. The lift of a base measure $\tau $ is then

in the fiber-integration notation of section 21.14. Since $E\in \mathrm{PSH}(\mathbb P^{n-1},\sigma )$, theorem 21.14.4 gives $\mathcal L(\mathrm{MA}_\sigma (E))=(2n)^{-1}\mathrm{MA}^{\mathrm{ext}}_\omega (\pi ^*E)$, and theorem 18.36.3 becomes

The factor $1/(2n)$ combines the fiber probability $1/2$ with the coefficient $n$ of $\alpha ^{n-1}\wedge \zeta $ in $\omega ^ n=(\alpha +\zeta )^ n$. The closed curvature form $\alpha $ alone gives nothing: by theorem 21.14.4, descended potentials have zero Monge–Ampère measure with respect to $\alpha $, and the nonzero limit comes from the vertical form $\zeta $. For $n=3$ this gives a nonzero limit, with the normalization $2/m^2$, on a manifold without the bounded mass property (theorem 23.1.1).

For a positive integer $k$, every $k\alpha $-plurisubharmonic function on $X_ n$ is pulled back from $\mathbb P^{n-1}$, by theorem 21.14.2. An $\omega $-plurisubharmonic function, by contrast, can have an isolated pole, and then it does not descend. Its selected sections still descend, so the pole is recorded by a vanishing condition on the base. The next two sections determine this condition and its effect on the kernels.