A Hermitian metric $\Omega $ is locally conformally Kähler, or LCK, if $\mathrm d\Omega =\vartheta \wedge \Omega $ for a closed real one-form $\vartheta $, its Lee form; such metrics are studied in section 23.2. The quotients of the preceding subsection resemble Hopf manifolds. A second family shows that the curvature criterion also applies on manifolds that carry no LCK metric at all.
Let $\tau =A+\mathrm iB_0$, where $A<0$ and $B_0>0$, and let $\mathbb C$ act holomorphically on $(\mathbb C^2\setminus \{ 0\} )^2$ by
\[ t\cdot (z,w)=(\mathrm e^ t z,\mathrm e^{\tau t}w). \]
Write $X_\tau $ for the quotient and put $r_1=\log |z|^2$, $r_2=\log |w|^2$. The map $(z,w)\mapsto ([z],[w])$ descends to a projection $\pi :X_\tau \to \mathbb P^1\times \mathbb P^1$; let $\omega _1,\omega _2$ be the pullbacks to the base of the Fubini–Study forms of the two factors, normalized to have integral one.
Proposition 12.17.14.119B81D2AF8F The quotient $X_\tau $ is a compact complex threefold diffeomorphic to $S^3\times S^3$, and $\pi $ is a principal elliptic fibration over $\mathbb P^1\times \mathbb P^1$. The form $\eta =\tau \partial r_1-\partial r_2$ descends to $X_\tau $, and for
\[ g=\pi ^*(\omega _1+\omega _2)+s\mathrm i\eta \wedge \bar\eta , \qquad s>0, \]
one has
D04D662546BA
12.17.15
\begin{equation} v_-(g)=48\pi ^2sB_0. \end{equation}
Every Hermitian form $h\geq c g$, with $c>0$, satisfies
\[ \sup _{T\in \mathscr G_ h}\| k_ T\| _{L^{3/2}(\mathrm dV)} \leq c^{-1}(48\pi ^2sB_0)^{-1/3}. \]
There is no balanced, pluriclosed or LCK metric on $X_\tau $.
Proof.
Every orbit contains a unique point with $|z|=|w|=1$: the required parameter is determined by
\[ \operatorname {Re}t=-r_1/2,\qquad \operatorname {Im}t=(r_2-Ar_1)/(2B_0). \]
These smooth formulas identify $(\mathbb C^2\setminus \{ 0\} )^2$ with $\mathbb C\times (S^3\times S^3)$, so the action is free and proper, and $X_\tau $ is diffeomorphic to $S^3\times S^3$. To obtain holomorphic charts, choose a nonzero coordinate $z_ k$ of $z$ and a branch of its logarithm, and use the action to normalize $z_ k=1$; the functions $z_ l/z_ k$, $l\ne k$, and $w_ jz_ k^{-\tau }$ are then local holomorphic coordinates on the quotient. Changing the branch of the logarithm accounts for the periods along the fibers. Thus $X_\tau $ is a compact complex threefold.
The invariant form $\eta $ vanishes on the generator $z\partial _ z+\tau w\partial _ w$ of the action, so it descends. On an elliptic fiber it equals $\mathrm d\chi $, where $\chi =\tau \log v_1-\log v_2$ in terms of fiber coordinates $(v_1z,v_2w)$; its period lattice is generated by $2\pi \mathrm i\tau $ and $-2\pi \mathrm i$. Hence $A_ E=8\pi ^2B_0$. Direct differentiation gives
\[ \begin{aligned} F& =2\pi \mathrm i(\tau \omega _1-\omega _2),& \alpha & =-2\pi B_0\omega _1,\\ \beta & =2\pi (A\omega _1-\omega _2),& Q& =-8\pi ^2A\, \omega _1\wedge \omega _2>0. \end{aligned} \]
Since $\int _ B(\omega _1+\omega _2)^2=2$, theorem 12.17.7 gives the lower mass, the endpoint bound and the absence of pluriclosed metrics.
Moreover $\mathrm d\operatorname {Re}\eta =-2\pi B_0\pi ^*\omega _1$. This nonzero semipositive exact form excludes every balanced metric by the Stokes argument at the end of the preceding proof, and hence also every Kähler metric. The quotient is simply connected, so the Lee form of a hypothetical LCK metric would be exact, equal to $\mathrm df$; multiplying the metric by $\mathrm e^{-f}$ would then give a Kähler metric, a contradiction. In particular $X_\tau $ carries no Vaisman metric, that is, no LCK metric with parallel Lee form.
$\square$
The curvature criterion controls bounded potentials and the full set of normalized Gauduchon currents in these threefolds. Curvature pairs of changing sign, primitive curvature pairs and higher-dimensional analogues are not covered by this argument.
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