PomeExample 18.25.1.6CA9B76E77A0 There is a connected compact complex threefold $Z$ with a surjective holomorphic submersion $p:Z\to \mathbb P^1$ whose fibers are scalar Hopf surfaces. The manifold $Z$ does not have the bounded mass property. For the pullback $\theta =p^*\omega _{\mathrm{FS}}$ of the Fubini–Study form of area one, the entire ordinary model space has the complete geodesic polar metric of 18.22.3, with diameter two. Its ideal subspace is the analytic closure, with the unique nearest-point, $1$-Lipschitz ideal retraction. All potentials in this background have zero extended Monge–Ampère measure, yet it contains analytic ideal models at polar distance one.
18.25 Hopf surface fibers and unbounded ambient massB094CC4C0076
B094CC4C0076
The complete model space of a curve submersion can occur on a manifold whose ambient Hermitian masses are unbounded. Blowing up the elliptic base locus of the projection $[z_0:z_1]$ on the scalar Hopf threefold turns this projection into a submersion with Hopf surface fibers. The rank-one background then retains only the curve singularities, while the smooth potentials of the Hopf threefold still force unbounded mass in a positive ambient background.
Proof. Put $\lambda =\mathrm e^{-1}$ and consider the vector bundle
\[ \mathcal V=\mathcal O_{\mathbb P^1}(-1) \oplus \mathcal O_{\mathbb P^1}. \]
Let $\mathcal V^\times $ be its total space with the zero section removed, and define
\[ Z=\mathcal V^\times /\langle v\mapsto \lambda v\rangle . \]
The bundle projection descends to a holomorphic map $p:Z\to \mathbb P^1$. A Hermitian norm on the bundle satisfies $\| \lambda ^ kv\| =\lambda ^ k\| v\| $. The action is free and proper: on compact subsets of $\mathcal V^\times $ the norms have positive lower and finite upper bounds, so only finitely many powers can move one such subset to meet another. Thus the quotient is a complex threefold. The annulus bundle $\{ \lambda \leq \| v\| \leq 1\} $ is compact over $\mathbb P^1$ and meets every orbit, so $Z$ is compact. Bundle charts identify $p^{-1}(U)$ with
\[ U\times \bigl ((\mathbb C^2\setminus \{ 0\} )/ \langle w\mapsto \lambda w\rangle \bigr ). \]
Hence $p$ is a holomorphic submersion with connected compact Hopf surface fibers, and $Z$ is connected.
The tautological inclusion into $\mathbb C^2$ identifies $\mathcal V^\times $ with the incidence manifold
\[ \left\{ ([s:t],z)\in \mathbb P^1\times (\mathbb C^3\setminus \{ 0\} ):\ t z_0=s z_1\right\} . \]
The last coordinate $z_2$ comes from the trivial summand. This is the blowup of $\mathbb C^3\setminus \{ 0\} $ along $L=\{ z_0=z_1=0,\ z_2\ne 0\} $: in the chart $s\ne 0$, write $a=t/s$ and $z_1=az_0$, while in the chart $t\ne 0$ write $b=s/t$ and $z_0=bz_1$. These are the two usual blowup charts of the ideal $(z_0,z_1)$. The scalar contraction preserves the incidence relation and commutes with projection to $z$. It therefore descends to a modification
\[ \mu :Z\longrightarrow H=(\mathbb C^3\setminus \{ 0\} )/ \langle z\mapsto \lambda z\rangle . \]
The map is an isomorphism away from the elliptic curve $L/\langle \lambda \rangle $. Local quotient charts show that it is its blowup; in particular it has degree one.
Take the Gauduchon form $\omega $ and smooth potentials $V_ j$ on $H$ from theorem 23.1.1. Since $Z$ is compact, a sufficiently large multiple $\Omega $ of any Hermitian form satisfies $\Omega -\mu ^*\omega >0$. Put $W_ j=V_ j\circ \mu $. Then
\[ \Omega +\mathrm{dd}^{\mathrm{c}}W_ j =\Omega -\mu ^*\omega +\mu ^*(\omega +\mathrm{dd}^{\mathrm{c}}V_ j)>0. \]
Positivity of smooth mixed products and the degree-one change of variables off the exceptional divisor give
\[ \begin{aligned} \int _ Z(\Omega +\mathrm{dd}^{\mathrm{c}}W_ j)^3 & \geq \int _ Z\mu ^*(\omega +\mathrm{dd}^{\mathrm{c}}V_ j)^3\\ & =\int _ H(\omega +\mathrm{dd}^{\mathrm{c}}V_ j)^3\longrightarrow +\infty . \end{aligned} \]
The exceptional divisor and its image have zero measure for smooth top-degree forms, so they do not affect the equality. Every $W_ j$ is smooth and bounded, so $v_+(\Omega )=+\infty $. By proposition 13.14.15, $Z$ does not have the bounded mass property.
Apply 18.22.1, 18.22.3 to $p$ and the area-one form $\omega _{\mathrm{FS}}$. This gives all the assertions about the full model space. To see the stated distance explicitly, fix $x\in \mathbb P^1$ and let $g_ x$ be its normalized Green model, with $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}g_ x=\delta _ x$. The models $0$ and $g_ x\circ p$ are ordinary and ideal, and the latter has a logarithmic pole of coefficient one along the smooth reduced fiber $p^{-1}(x)$. Thus both have analytic singularities. Their base polar measures are $0$ and $\delta _ x$, so their polar distance is one, although their top-degree measures vanish. $\square$
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