Example 15.22.70362BBF11A0B

Example 15.22.7.0362BBF11A0B Take $X=\mathbb P^ n$, $h=\lambda \omega _{\mathrm{FS}}$, $\int _ X\omega _{\mathrm{FS}}^ n=1$, and $0<c<\lambda $. Put

Its pole is $P=[1:0:\cdots :0]$. On the point blowup $\pi :Y\to X$, let $q:Y\to \mathbb P^{n-1}$ be the direction map. In affine blowup coordinates $x=(z,zw)$, Poincaré–Lelong gives

The residual form $\beta =(\lambda -c)\pi ^*\omega _{\mathrm{FS}}+cq^*\omega _{\mathrm{FS}}$ is positive: $\pi ^*\omega _{\mathrm{FS}}$ is positive in the directions normal to $E$, and $q^*\omega _{\mathrm{FS}}$ in the directions tangent to $E$. Write $H=\pi ^*[\omega _{\mathrm{FS}}]$. Since $H|_ E=0$ and $E^ n=(-1)^{n-1}$, its mass is

Hence, for every nonzero nonnegative $f\in L^ p$ with $\int _ Xfh^ n=\lambda ^ n-c^ n$, theorem 15.22.2 gives $\kappa =1$ and a solution of $\mathrm{MA}^{\mathrm{ext}}_ h(u)=fh^ n$ with the nonminimal type $u=\psi _ c+O(1)$.

For an explicit solution, choose a smooth function $v$ on $Y$ with $v|_ E$ nonconstant, small enough that $S=\beta +\mathrm{dd}^{\mathrm{c}}v$ is still positive, and put $u=\psi _ c+v\circ \pi ^{-1}$ off $P$. The density $f$ of its measure, given by $f\circ \pi =S^ n/\pi ^*h^ n$ off $E$, lies in every $L^ p$ with $1<p<n/(n-1)$: $\pi ^*h^ n$ vanishes to order $|z|^{2(n-1)}$ along $E$, so the local integral of $f^ p$ is finite precisely when $(n-1)(p-1)<1$. Different directions of approach to $P$ give different limits of $u-\psi _ c$. The divisor $c[E]$ of the current upstairs leaves no divisor downstairs, since $E$ is contracted to a point. On $X\setminus \{ P\} $, where $u$ is smooth and strictly $h$-plurisubharmonic, the Hermitian metric $h+\mathrm{dd}^{\mathrm{c}}u$ is isometric to $(Y\setminus E,S)$. Its metric completion is $Y$ with the metric $S$, so the whole divisor $E$ is restored: removing the smooth real codimension-two submanifold $E$ does not change the infimum of path lengths in $Y$. Finally, if $h$ is replaced by a sufficiently small nonclosed perturbation, both $h$ and the residual form remain positive, so the existence part of theorem 15.22.2 still applies.