Example 11.34.141B8C3827F8D7

Proof. Take the Hopf surface $Y=(\mathbb C^2\setminus \{ 0\} )/\langle z\mapsto 2z\rangle $ and its unit-volume metric $\kappa =c|z|^{-2}\mathrm{dd}^{\mathrm{c}}|z|^2$, where $c>0$ is chosen so that $\int _ Y\kappa ^2=1$. Put $s=\log |z|^2$, $\alpha =\mathrm{dd}^{\mathrm{c}}s$ and $\beta =\mathrm ds\wedge \mathrm d^{\mathrm c}s$, so that $\kappa =c(\alpha +\beta )$. The differentiation in (23.1.3) applies in this dimension. Here $\alpha $ has rank one, so $\mathrm{dd}^{\mathrm{c}}\kappa =-c\alpha ^2=0$ and $\nu _\kappa =\kappa ^2$. Polar coordinates identify $Y$ with $S^3\times S^1$, hence $H^2(Y,\mathbb R)=0$. A closed positive form would be exact and have positive square integral, contradicting Stokes. Thus $Y$ is non-Kähler, and so is $\mathbb P^1\times Y$, by restriction.

Let $\tau \in [-1,1]$ be the height on $\mathbb P^1$, set $b=2+\tau $, and put

Since $0\leq f\leq 1$, this is a positive continuous function of mean $a$. Along a normal segment to the equator, $|\tau |$ is comparable to the distance $r$. The oscillation of $A$ is therefore bounded below by a positive multiple of $1/\log (e/r)$. Its Dini integral $\int _0^{r_0}\mathrm dr/(r\log (e/r))$ diverges, and this modulus is larger than every $r^\gamma $, $\gamma >0$, as $r\downarrow 0$. The logarithm of $A$ has the same lower bound, because $A$ is uniformly positive and bounded. The smooth term $2\log b$ contributes only $O(r)$, so $F=\log (3Ab^2)$ has neither regularity. The mixed component $\mathrm db\wedge \kappa $ shows that $\omega =aw+b\kappa $ is nonclosed. By proposition 11.34.13, its supporting probability is $(A/a)w\wedge \kappa ^2$. $\square$