Proposition 12.17.14119B81D2AF8F

Proof. Every orbit contains a unique point with $|z|=|w|=1$: the required parameter is determined by

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

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. $\square$