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 $1$.
18.21 A polar metric in a background with unbounded mass49135DC5A99F
49135DC5A99F
Taking the Hopf threefold of theorem 23.1.1 as the auxiliary factor makes the ambient masses unbounded, while the complete metric on the product models is unchanged. Letting the coefficient of the Hopf factor vary over the curve product also makes the background nonclosed.
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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