10.21 Density zeros and metric degenerationAE84FAD03138

Example 10.21.3.99FB104FBCD5 Let $n\geq 2$, $q>1$ and let $k\geq 2$ be an integer. Let $H_ q=(\mathbb C^ n\setminus \{ 0\} )/(Z\sim qZ)$ be the Hopf manifold, and on the covering space put $R=\sum _ j|Z_ j|^2$, $S=\sum _ j|Z_ j|^{2k}$,

These forms and functions are invariant under $Z\mapsto qZ$, so they descend to $H_ q$; the form $h_0$ descends to every Hopf manifold $H_{q'}$. The coordinate power map $F_ k([Z])=[(Z_1^ k,\ldots ,Z_ n^ k)]$ defines a finite surjective map $H_ q\to H_{q^ k}$: indeed $F_ k(qZ)=q^ kF_ k(Z)$, and the choices of roots are finite modulo scaling in the source. The form $\mathrm{dd}^{\mathrm{c}}\log R$ is positive transverse to the complex Euler direction, while $\partial \log S$ is nonzero on that direction; hence $\theta _ k>0$. Direct differentiation gives

The background $\theta _ k$ is not closed, since $\mathrm{d}\theta _ k=F_ k^*(\mathrm{d} h_0)$ is not identically zero. Along a coordinate divisor $D=\{ Z_ j=0\} $, at a generic point and in local coordinates $(z,w)$ with $D=\{ z=0\} $, the determinant vanishes to order $2(k-1)$, the metric $F_ k^*h_0$ is comparable to $|z|^{2(k-1)}|\mathrm{d} z|^2+|\mathrm{d} w|^2$, and the distance to $D$ is of order $|z|^ k$, as in proposition 10.21.1.

A small logarithmic potential along $D$ gives a strict barrier. Choose a smooth metric on $\mathcal O(D)$ with curvature form $\Theta _ D$ and canonical section $s_ D$. For small $c>0$, set $h_ c=\theta _ k+c\Theta _ D$ and $U_ c=c\log |s_ D|^2+u_ k$. Then $h_ c>0$ and

By theorem 13.14.3, the extended Monge–Ampère measure of $U_ c$ in the background $h_ c$ is the singular measure $(F_ k^*h_0)^ n$. The ordinary singularity type of $U_ c$ is exactly that of $c\log |s_ D|^2$, which is not minimal, while its metric off $D$ is the same smooth metric $F_ k^*h_0$. Finally, $H_ q\cong S^{2n-1}\times S^1$ has $H^2(H_ q;\mathbb R)=0$. A Kähler form would therefore be exact, and Stokes' theorem would make its positive top-degree integral zero; thus $H_ q$ is not Kähler.

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 $L^ p$.

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.