Example 10.17.8ED565B280C89

Example 10.17.8.ED565B280C89 Let $k\geq 1$, let $Y\simeq \mathbb P^ k$ be a hyperplane in $X=\mathbb P^{k+1}$, and let another hyperplane $H_ X\subset X$ cut out $H=H_ X\cap Y$. Let $\omega _{\mathrm{FS}}$ be the Fubini–Study form, normalized by $\int _ Y\omega _{\mathrm{FS}}^ k=1$, and let $s_{H_ X}$ be a section of $\mathcal O(1)$ defining $H_ X$, measured with the Fubini–Study metric. For $c>0$ and $d\geq 0$, take $\alpha _0=(c+d)\omega _{\mathrm{FS}}$ on $X$ and $u=c\log |s_{H_ X}|^2$. The restriction $v=u|_ Y$ is not identically $-\infty $, and

Thus for $d=0$ the trace mass is zero, although the restricted invariant is positive. The divisor $c[H]$ is a nonzero part of the trace current, but the non-pluripolar top product discards it.