Pome197B18655594If zero belongs to the Lee–Gauduchon cone, then the image of $j_ X$ contains no nonzero pseudoeffective class; this is the case when $X$ carries a balanced metric. On the Hopf threefold, by contrast, the Lee form spans a ray of classes with pseudoeffective image under $j_ X$, and the Lee–Gauduchon cone is an open half-line.
Corollary 23.2.66.BF1CE00DF8A7 The following conditions are equivalent:
A balanced metric implies these conditions. On a manifold in Fujiki's class $\mathcal C$, one has $W_ X=\{ 0\} $ and $\operatorname {LG}(X)=\{ 0\} $.
Proof. An open cone in a finite-dimensional vector space that contains zero is the entire space, since scaling a neighborhood of zero reaches every point. The two dual descriptions in theorem 23.2.60 now prove the equivalences, including the case $W_ X=\{ 0\} $. If $b$ is a balanced metric, the form $b^{n-1}$ is closed and of pure type, hence also $\mathrm d^{\mathrm c}$-closed. Thus $b$ is Gauduchon and gives the element $[\mathrm d^{\mathrm c}(b^{n-1})]=0$ of $\operatorname {LG}(X)$.
On a compact Kähler manifold the $\partial \bar\partial $-lemma gives $j_ X=0$, since every $\kappa _\lambda $ is closed and de Rham exact. Thus $K_ X=V_ X$ and $W_ X=\{ 0\} $, and since $\operatorname {LG}(X)$ is nonempty, $\operatorname {LG}(X)=\{ 0\} $. By theorem 23.2.64, this conclusion extends to Fujiki's class $\mathcal C$. $\square$
Consequently, a nonzero pseudoeffective class in the image of $j_ X$ obstructs balanced metrics on every compact complex manifold bimeromorphic to $X$. For LCK manifolds, 23.2.12 would provide such a class.
For the Hopf threefold of section 23.1, with the Lee form $\lambda $ of proposition 23.2.24, the covering description identifies the underlying manifold with $S^1\times S^5$, so $V_ X=\mathbb R[\lambda ]$. The class $j_ X([\lambda ])=\{ \alpha \} $ is nonzero by (23.2.8), and it is pseudoeffective because $\alpha \geq 0$. Its negative multiples are not pseudoeffective, since their pairing with the Gauduchon metric $\omega $ is negative. Thus
The same manifold has unbounded Monge–Ampère masses by theorem 23.1.1. The positive ray detected here controls pairings with cohomology classes, which are linear and do not bound those nonlinear masses.
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