Example 10.21.109F3252D2DC77

Example 10.21.10.9F3252D2DC77 On a polydisk with coordinates $(z,w)\in \mathbb C\times \mathbb C^{n-1}$, $n\geq 2$, set $P(z,w)=|z|^{2/n}(1+|w|^2)$. Off $z=0$, its Hessian block in the $w$-directions is $|z|^{2/n}I$ and its normal Schur complement is $n^{-2}|z|^{2/n-2}$, so its Hessian determinant is the positive constant $n^{-2}$. Since $P$ is bounded, it extends across $z=0$ as a plurisubharmonic function whose Monge–Ampère measure has no additional mass on the divisor. Yet $P$ is not $C^1$ along that divisor.

This local example extends to a compact Kähler model on $\mathbb P^ n$. On an affine chart with coordinates $y=(z,w)$, the Fubini–Study form is $\omega _{\mathrm{FS}}=\mathrm{dd}^{\mathrm{c}}\log (1+|y|^2)$; for constants $\lambda >0$ and $B$, put $H=\lambda \log (1+|y|^2)$ and $Q=H-B$. Choose nested radii and the parameters so that $Q>P+2\epsilon $ on an outer collar of a ball and $Q<P-2\epsilon $ on an inner ball. A smooth convex regularized maximum $U=M_\epsilon (P,Q)$ then equals $P$ on the inner ball and $Q$ on the collar. Put $v=U-H$ on the ball and $v=-B$ outside it, so that $\lambda \omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}v$ equals $\mathrm{dd}^{\mathrm{c}}U$ on the ball and $\lambda \omega _{\mathrm{FS}}$ outside. This current is positive and smooth off $\{ z=0\} $; on the gluing annulus its density is at most $C|z|^{2/n-2}$, so it belongs globally to $L^ p$ for every $1<p<n/(n-1)$. On the inner ball the density is smooth and strictly positive, but the potential is not $C^1$. There the metric is comparable to $|z|^{2/n-2}|\mathrm{d} z|^2+|z|^{2/n}|\mathrm{d} w|^2$, and the distance to the divisor is of order $|z|^{1/n}$.

The construction also works on a Hopf manifold. Choose a global Hermitian form equal to $\mathrm{dd}^{\mathrm{c}}(\lambda |y|^2)$ on a coordinate ball and equal to a fixed nonclosed Hermitian form outside a slightly larger ball, using a convex interpolation of positive forms. Carry out the same regularized maximum with $H=\lambda |y|^2$ on the inner ball, and set $v=-B$ outside it. The background is still nonclosed on the outer region, and the local potential and density are unchanged. As in the projective case, the density is controlled globally only in $L^ p$.