PomeA1CA1768E392Example 13.27.1.A1CA1768E392 Let $\theta $ be the Fubini–Study form on $\mathbb P^2$, normalized by $\int _{\mathbb P^2}\theta ^2=1$. There are analytic ordinary models $p,q,r$ of positive lower volume such that
Proof. Write $|Z|^2=|Z_0|^2+|Z_1|^2+|Z_2|^2$ in homogeneous coordinates. Fix a rational $a\in (0,1/2)$ and put
With $H_ i=\{ Z_ i=0\} $, the corresponding currents are $a[H_0]+(1-a)\theta $, $a[H_1]+(1-a)\theta $ and $a[H_0]+a[H_1]+(1-2a)\theta $. The background is closed, so each singularity volume equals the non-pluripolar mass, which is the mass of the smooth part:
The maximum $\max (u,v)$ has the ordinary type of
To compute its volume, blow up the common point $[0:0:1]$ of $H_0$ and $H_1$. Denote the blowup by $\pi :Y\to \mathbb P^2$, its exceptional divisor by $E$, and the pencil map $[Z_0:Z_1]$ by $f:Y\to \mathbb P^1$. For the Fubini–Study form $\omega _1$ of mass one on $\mathbb P^1$, the Poincaré–Lelong formula gives
Here $\beta $ is smooth and semipositive. Also
The mixed integral is one because every fiber of the pencil maps onto a line. By lemma 10.17.3 and change of variables off $E$,
Since $w\leq u$ and $w\leq v$, (13.22.2) and the definition of $d_\theta $ give
Set $p=P_\theta [u]$, $q=P_\theta [v]$ and $r=P_\theta [w]$. By proposition 18.4.8, these models have the same analytic singularity types as $u$, $v$ and $w$, and their lower volumes are at least $(1-2a)^2$. By projection invariance (proposition 13.22.4), they have the same comparisons, and the difference between the two sides is $2a^2>0$. $\square$
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