Example 10.21.11554ACF84A13B

Example 10.21.11.554ACF84A13B Let $\pi :X=\operatorname {Bl}_0\mathbb P^ n\to \mathbb P^ n$ be the blowup at the origin of an affine chart with coordinates $y$, let $E$ be its exceptional divisor, and write $s=|y|^2$. For every $\gamma ,c>0$, one can choose $\lambda >0$ so that the background $\theta =\pi ^*(\lambda \omega _{\mathrm{FS}})+c\Theta _ E$ admits a potential $u=cL+q\circ \pi $ with bounded Hölder remainder $q$, hence of minimal type, whose total local potential $\lambda \log (1+s)+q$ equals $As^\gamma +B$ near the origin, for constants $A>0$ and $B\in \mathbb R$. To construct $q$, choose a smooth function $m$ with $m(s)>0$ and $m'(s)>0$, equal to $A\gamma s^\gamma $ near $s=0$ and to $\lambda s/(1+s)$ outside a slightly larger interval; taking $A$ small puts the values at the two ends in the right order. Match the positive derivative germs on narrow collars, fill the remaining positive deficit of the integral with a positive interior derivative and a bump, and integrate $U'=m/s$, choosing the constant so that $U(s)=\lambda \log (1+s)$ for large $s$. Set $q(y)=U(|y|^2)-\lambda \log (1+|y|^2)$ on the affine chart and extend it by zero outside. Since $m$ and $m'$ are positive, positivity holds throughout the compact interpolation region.

On the blowup, in coordinates with $y=(z,zw)$, the total local potential of $u$ is $A|z|^{2\gamma }(1+|w|^2)^\gamma +B$. The density of $u$ is $|z|^{2\ell }G$ with $G$ smooth and positive and $\ell =n\gamma -1>-1$, since the Hessian determinant of $A|y|^{2\gamma }$ is $A^ n\gamma ^{n+1}s^{n(\gamma -1)}$ and the squared Jacobian of the blowup is $|z|^{2(n-1)}$. The metric off $E$ is comparable to

the distance to $E$ is of order $|z|^\gamma $, and the metric completion collapses $E$ to a point. If $\ell $ is a nonnegative integer, the density is smooth. At $\ell =0$ the density is smooth and strictly positive, but the remainder has exact normal Hölder exponent $2/n$ for $n>2$, is Lipschitz but not $C^1$ for $n=2$, and the normal coefficient of the metric is unbounded. For $-1<\ell <0$, the density belongs to $L^ p$ exactly when $p\ell >-1$. Taking $\ell =n$ gives a smooth density with a zero of order $2n\geq 2(n-1)$, while the residual Hessian remains nonsmooth; this proves the insufficiency asserted in proposition 10.21.9. A small nonclosed perturbation of the background supported away from the origin leaves these local calculations unchanged, and gives the same examples in a nonclosed background, with the Monge–Ampère measure of the perturbed potential as prescribed measure.