PomeProposition 23.2.11.26993B1C8C74 Suppose that $X$ carries a strictly LCK metric with Lee form $\lambda $ and that $\{ \kappa _\lambda \} $ is pseudoeffective. Then $X$ admits no balanced metric.
Proposition 23.2.1126993B1C8C74
Proof. Let $T=\kappa _\lambda +\mathrm{dd}^{\mathrm{c}}u\geq 0$ be a closed positive current representing the class. It is nonzero, since the class is nonzero, and it is de Rham exact. If a Hermitian metric $b$ were balanced, then $\mathrm{d}(b^{n-1})=0$, and Stokes' theorem for currents would give
\[ \int _ XT\wedge b^{n-1}=0. \]
This is impossible, since the mass of a nonzero positive $(1,1)$-current against a smooth strictly positive $(n-1,n-1)$-form is strictly positive. $\square$
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