PomeExample 18.21.1.00A4B8521863 There is a compact complex sixfold with a smooth positive background $\Theta $ that is not closed and has infinite upper mass $v_+(\Theta )=+\infty $ in the sense of definition 13.14.14, while the locus of separated product models of proposition 18.20.2, along a divisor of weight one, has a complete geodesic polar metric of diameter $4$. The ideal subspace is closed and is the metric closure of the models with analytic singularities, and ideal projection is the unique nearest-point retraction onto it and is $1$-Lipschitz. The upper-volume comparison vanishes on this locus. The locus contains distinct ideal models with analytic singularities at polar distance $1$.
Example 18.21.100A4B8521863
Proof. Take $Y=C_1\times C_2$ with $C_1=C_2=\mathbb P^1$ and normalized Fubini–Study forms $\omega _ i$ of area one, and write $\pi _ i:Y\to C_ i$ for the projections. Let $(T,\omega _ T,V_ j)$ be the Hopf threefold, its Gauduchon form and the smooth potentials of theorem 23.1.1. Take $Z=\mathbb P^1$ with its normalized Fubini–Study form $\kappa $, a point $D$ and its normalized potential $\psi $ satisfying $\kappa +\mathrm{dd}^{\mathrm{c}}\psi =[D]$. On $M=Y\times T\times Z$, write $p,s,r$ for the projections. Choose a positive nonconstant smooth function $h$ on $Y$ and put
\[ \Theta =p^*(\pi _1^*\omega _1+\pi _2^*\omega _2) +(h\circ p)s^*\omega _ T+r^*\kappa , \qquad \Psi =\psi \circ r. \]
This form is strictly positive. The component $\mathrm dh\wedge \omega _ T$ of its exterior derivative, with one direction in $Y$ and two in $T$, is nonzero and cannot be canceled by the component with three directions in $T$. Thus $\Theta $ is not closed.
Put $c=\min _ Yh>0$. The smooth functions $W_ j=cV_ j\circ s$ are strictly $\Theta $-plurisubharmonic, since their current in the $T$-directions is
\[ h\omega _ T+c\mathrm{dd}^{\mathrm{c}}V_ j =c(\omega _ T+\mathrm{dd}^{\mathrm{c}}V_ j)+(h-c)\omega _ T>0. \]
Only the part of degree three in the $T$-directions contributes to the top-degree product. The three curve forms have integral one, so positivity of smooth mixed products and Fubini's theorem give
\[ \begin{aligned} \int _ M(\Theta +\mathrm{dd}^{\mathrm{c}}W_ j)^6 & =120\int _ Y\pi _1^*\omega _1\wedge \pi _2^*\omega _2 \int _ T(h(y)\omega _ T+c\mathrm{dd}^{\mathrm{c}}V_ j)^3\\ & \geq 120c^3\int _ T(\omega _ T+\mathrm{dd}^{\mathrm{c}}V_ j)^3 \longrightarrow +\infty . \end{aligned} \]
The factor $120=6!/3!$ counts the ways of choosing one factor for each curve direction and three for $T$. Each $W_ j$ is bounded, so definition 13.14.14 gives $v_+(\Theta )=+\infty $, and the bounded mass property fails.
Apply proposition 18.20.2 to this product, with both curve areas equal to one. On the locus of potentials $\Psi +q_1\circ \pi _1\circ p+q_2\circ \pi _2\circ p$, where each $q_ i$ ranges over the ordinary $\omega _ i$-models with supremum zero, it gives a complete metric $\rho $ of diameter $4$, the analytic closure of the ideal subspace, the nearest-point retraction and the vanishing of the upper-volume comparison. These conclusions use no bound on the ambient masses.
For $x\in C_1$, let $g_ x$ be the normalized Green potential of the point $x$, with $\omega _1+\mathrm{dd}^{\mathrm{c}}g_ x=[x]$. By lemma 18.20.1, the potentials
\[ \Phi _0=\Psi ,\qquad \Phi _ x=\Psi +g_ x\circ \pi _1\circ p \]
are ordinary and ideal models, and their divisorial poles with coefficient one give analytic singularities. Their tuples of polar curvatures are $(0,0)$ and $(\delta _ x,0)$, so $\rho (\Phi _0,\Phi _ x)=\| \delta _ x\| _{\mathrm{TV}}=1$. Nevertheless, by the same lemma, every representative of either ordinary type has zero extended mass; in particular, all their upper and lower singularity volumes are zero. $\square$
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