PomeLemma 12.16.1.744D35F9BE16 Suppose a smooth real $(n-1,n-1)$-form $T$ satisfies
\[ \int _ X \mathrm{d}f\wedge \mathrm d^{\mathrm c}f\wedge T\geq 0 \quad \hbox{for every smooth real }f. \]
Then $T\geq 0$.
Lemma 12.16.1744D35F9BE16
Proof. Suppose instead that $\mathrm i\alpha \wedge \bar\alpha \wedge T<0$ at a point $x_0$ for some $(1,0)$-covector $\alpha $. Every $(1,0)$-covector is the $(1,0)$-part of a real covector, so we can choose a real linear function $\ell $ in local real coordinates around $x_0$ whose differential has $(1,0)$-part $\alpha $ at $x_0$. Then $\mathrm{d}\ell \wedge \mathrm d^{\mathrm c}\ell $ is a positive multiple of $\mathrm i\alpha \wedge \bar\alpha $ at $x_0$, and after shrinking the coordinate ball, $\mathrm{d}\ell \wedge \mathrm d^{\mathrm c}\ell \wedge T$ is strictly negative on it. Choose a nonzero smooth cutoff function $\chi $ supported in this ball and, for positive integers $k$, set
\[ f_ k=k^{-1}\chi \sin (k\ell ), \]
extended by zero to $X$. Since $\mathrm{d}\ell $ is a constant nonzero covector in these coordinates,
\[ \mathrm{d}f_ k\wedge \mathrm d^{\mathrm c}f_ k =\chi ^2\cos ^2(k\ell )\, \mathrm{d}\ell \wedge \mathrm d^{\mathrm c}\ell +O(k^{-1}), \]
where $k$ times the remainder has uniformly bounded coefficients. In $\cos ^2(k\ell )=(1+\cos (2k\ell ))/2$, the oscillatory part contributes integrals tending to zero, as one sees by integrating by parts in a coordinate direction in which $\ell $ has nonzero derivative. Hence
\[ \lim _{k\to \infty }\int _ X\mathrm{d}f_ k\wedge \mathrm d^{\mathrm c}f_ k\wedge T =\frac12\int _ X\chi ^2\mathrm{d}\ell \wedge \mathrm d^{\mathrm c}\ell \wedge T<0, \]
which contradicts the hypothesis. $\square$
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