Corollary 23.2.66BF1CE00DF8A7

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 $\mathcal C$. $\square$