PomeProposition 12.19.22.69BB70EA26D9 The threefold $X$ admits neither a pluriclosed Hermitian metric nor a closed Kähler current.
Proposition 12.19.2269BB70EA26D9
Proof. By proposition 12.19.18, the nonzero smooth semipositive $(2,2)$-form $\alpha \wedge \beta =\mathrm{dd}^{\mathrm{c}}(\omega /2)$ is $\mathrm{dd}^{\mathrm{c}}$-exact. If a Hermitian metric $\eta $ satisfied $\mathrm{dd}^{\mathrm{c}}\eta =0$, Stokes' theorem would give
\[ 0<\int _ X\eta \wedge \alpha \wedge \beta =\frac12\int _ X\eta \wedge \mathrm{dd}^{\mathrm{c}}\omega =\frac12\int _ X\omega \wedge \mathrm{dd}^{\mathrm{c}}\eta =0, \]
a contradiction. The first inequality holds because $\eta $ is positive in the vertical direction at every point.
Similarly, if a closed positive current $T$ satisfied $T\geq \varepsilon \omega $ for some $\varepsilon >0$, positivity and Stokes' theorem for currents would give
\[ 0<\varepsilon \int _ X\omega \wedge \alpha \wedge \beta \leq \int _ XT\wedge \alpha \wedge \beta =\frac12\int _ X\omega \wedge \mathrm{dd}^{\mathrm{c}}T=0. \]
Hence there is no closed Kähler current. $\square$
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