Example 11.42.40B7F3ACBCCC4

Example 11.42.4.0B7F3ACBCCC4 On $\mathbb P^ n$, $n\geq 2$, let $\omega $ be the Fubini–Study form, normalized as in example 11.42.3, and let $f$ be a nonconstant smooth real function. Fix $0<t<1$ and put

Here $s_ D$ defines a hyperplane $D$ as in example 11.42.3, and $\theta +\mathrm{dd}^{\mathrm{c}}\varphi _ D=\beta +t[D]$, so every hyperplane has the same positive residual form $\beta $. This form is not closed: $\mathrm{d}\beta =\mathrm{e}^ f\mathrm{d}f\wedge \omega \not\equiv 0$, since $\mathrm{d}f$ is nonzero somewhere and wedging a one-form with $\omega ^{n-1}$ is injective. By theorem 11.41.1, the ordinary types of all $\varphi _ D$ have the same functional $B_\beta $ on bounded tests, hence the same supporting sets. At a smooth test, the common supporting probability is the Gauduchon adjoint probability of theorem 11.34.9.

If $v_ F$ is the common smooth residual solution, the normalized solutions are the functions $\varphi _ D+v_ F-\sup _ X(\varphi _ D+v_ F)$. Two distinct hyperplanes $D_1,D_2$ give different poles, different divisorial coefficients and, for sufficiently large $k$, different multiplier ideals $\mathcal O(-\lfloor kt\rfloor D_ i)$, by the calculation in normal coordinates of lemma 5.37.1. The potentials differ by more than a constant, since the difference of their complex Hessians is $t([D_1]-[D_2])$, and alternating between the two hyperplanes again produces two distinct $L^1$ limits. These conclusions come from the common residual form; for general nonclosed backgrounds we do not claim that the functional is determined by a mass.