In the complex case we have by Stokes’ theorem

$$\int f\,dd^{c}g\wedge\theta_{1}\wedge...\wedge\theta_{n-1}=-\int df\wedge d^{c}g\wedge\theta_{1}\wedge...\wedge\theta_{n-1},$$

and negativity comes from that of the $(1,1)$-form $df\wedge d^{c}f$. Recall also that $df\wedge d^{c}f\wedge\omega^{n-1}=|df|_{\omega}^{2}\,\omega^{n}$ when $\omega$ is a Kähler form, so that $\left(-\int f\,dd^{c}f\wedge\omega^{n-1}\right)^{1/2}$ is the $L^{2}$-norm of the gradient of $f$.