Example 10.21.399FB104FBCD5

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.