Example 20.4.3174A0526E4030

Example 20.4.31.74A0526E4030 Let $Y=\mathbb P^2$ with homogeneous coordinates $[t_0:t_1:t_2]$, let $\sigma $ be the Fubini–Study form normalized by $\int _ Y\sigma ^2=1$, and choose $a\in \mathbb Q_{>0}$. Put

and take the ambient manifold to be $Y$ itself. Then $u\in \mathrm{PSH}(Y,a\sigma )$. It has analytic singularities, with its only pole at $p=[1:0:0]$, and zero generic Lelong number on $Y$. By proposition 20.3.6, its trace data are given by the generic Lelong numbers of its pullbacks: $\tau _{Y,u}(E)=\nu (u\circ \pi ,E)$ for every prime divisor $E$ on a smooth compact modification $\pi $ of $Y$. Since $u$ has no divisorial poles on $Y$, we have $S_ Y=0$, whose canonical ordinary type is that of a smooth function. If $\pi :\widetilde Y\to Y$ is the blowup at $p$ with exceptional curve $F$, then

Indeed, on either standard chart of the blowup, the numerator is the squared modulus of the exceptional coordinate times a smooth positive function, so the pole order is $a$ along $F$ and zero along every other prime divisor. Thus the trace data are realized exactly by $u$, although $E_\pi \neq 0$. The exceptional difference does not obstruct exact realization; it prevents the identification of the realizing potential with the canonical ordinary type of the divisorial part on the original model, which here is smooth.

The same phenomenon occurs on a manifold without the bounded mass property. Let $(W,\omega _ W)$ be a compact Hermitian manifold of dimension $k$ that fails the bounded mass property, such as the Hopf threefold of theorem 23.1.1, and pull $u$ back to $Y\times W$; take both the ambient manifold and the submanifold to be $Y\times W$. The pole set of the pullback is $\{ p\} \times W$, of codimension two, and the modification $\widetilde Y\times W\to Y\times W$ has exceptional coefficient $a$ along $F\times W$. Analytic restriction again shows that the trace data are realized exactly. Let $\operatorname {pr}_ Y$ and $\operatorname {pr}_ W$ be the two product projections.

Choose smooth $\omega _ W$-plurisubharmonic functions $H_ j$ whose masses tend to infinity. For the Hermitian form $\Omega =\operatorname {pr}_ Y^*\sigma +\operatorname {pr}_ W^*\omega _ W$, the pullbacks of the $H_ j$ have masses

by the binomial expansion of smooth forms and the dimensions of the two factors. Hence the product also fails the bounded mass property. The pulled-back potential belongs to $\mathrm{PSH}(Y\times W,C\Omega )$ for every $C\geq \max \{ a,1\} $, since its curvature is bounded below by $-a\operatorname {pr}_ Y^*\sigma $. Scaling the smooth functions $H_ j$ by $C$ shows that $C\Omega $ also has unbounded mass.