PomeExample 13.24.10.E76D9C1B44B6 There is a nonclosed pluriclosed background on $\mathbb P^2$ with a nonanalytic ideal type satisfying 13.24.8. Its prescribed divisors are not contained in a single pencil, and both singularity volumes of its ideal model are at least $1/16$.
Example 13.24.10E76D9C1B44B6
Proof. In a coordinate ball choose a smooth real cutoff $\chi $ equal to one near the origin, and extend $\alpha =\chi |z_2|^2\mathrm dz_1$ by zero. Put
\[ \tau =\partial \overline\alpha +\bar\partial \alpha . \]
It is real and smooth, with $\mathrm{dd}^{\mathrm{c}}\tau =0$. Where $\chi =1$,
\[ \mathrm d\tau =\mathrm dz_2\wedge \mathrm d\bar z_2 \wedge (\mathrm dz_1-\mathrm d\bar z_1)\ne 0. \]
Choose $\epsilon >0$ small enough that $\theta =\kappa +\epsilon \tau \geq \kappa /2$. Set $A=1/4$, $a_ j=A2^{-j}$, $t_ j=2^{-j}$ and
\[ H_ j=\{ Z_0+t_ jZ_1+t_ j^2Z_2=0\} . \]
The Vandermonde determinant shows that any three defining linear forms are independent; hence the hyperplanes are not in one pencil. Here $R=\theta -A\kappa \geq \kappa /4$, so $\int _ XR^2\geq 1/16$. The preceding proposition gives the mass throughout the ordinary type and both singularity volumes. There are infinitely many positive base divisor coefficients. An analytic singularity on a compact manifold has only finitely many prime divisorial components, so no analytic potential has this ideal type. $\square$
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