Example 23.2.721EE117A3B0B0

Proof. Consider the holomorphic action of the additive group $\mathbb C$ on

The action is free, since $\mathrm{e}^ t=\mathrm{e}^{\mathrm i t}=1$ forces $t=0$. It is also proper. Indeed, if both $(z,w)$ and $t\cdot (z,w)$ range in compact subsets of $\widetilde X$, then the norms of the four vectors $z$, $w$, $\mathrm e^ tz$ and $\mathrm e^{\mathrm it}w$ in $\mathbb C^2$ are bounded above and away from zero. Writing $t=a+\mathrm i b$, the ratios of these norms bound $\mathrm{e}^ a$ and $\mathrm{e}^{-b}$ above and away from zero, so $a$ and $b$ stay bounded. A free proper holomorphic action has local holomorphic slices, and its quotient $X$ is a Hausdorff complex manifold of dimension three.

Every orbit meets $S^3\times S^3$ exactly once, at the image of $(z,w)$ under the group element with

These expressions depend smoothly on $(z,w)$, so the slice $S^3\times S^3$ identifies the quotient with $S^3\times S^3$ as a smooth manifold. In particular, $X$ is compact and connected, and

It follows that $V_ X=K_ X=W_ X=0$ and $\operatorname {LG}(X)=\{ 0\} $; the latter cone is nonempty by Gauduchon's theorem and lies in the zero group $H^5_{\mathrm{dR}}(X,\mathbb R)$.

The map $(z,w)\mapsto ([z],[w])$ is invariant under the action and descends to a holomorphic map

It is a submersion, because the map on $\widetilde X$ is a submersion and the orbits of the action lie in its fibers. Let $\eta $ be the sum of the Fubini–Study forms on the two factors. The smooth form

is closed, semipositive and of rank two everywhere. It is nonzero and de Rham exact, since $H^2_{\mathrm{dR}}(X,\mathbb R)=0$. By proposition 23.2.69, $X$ admits no strongly Gauduchon metric.

The Bott–Chern class of $\alpha $ is nonzero: otherwise $\alpha =\mathrm{dd}^{\mathrm{c}}f$ for a smooth real function $f$, and the integral of $\alpha $ against any Gauduchon form would vanish, contradicting $\alpha \geq 0$ and $\alpha \ne 0$. The class is pseudoeffective, since $\alpha $ is a positive representative. Finally, $j_ X$ has zero domain, so this class is not in its image. $\square$