  This paper divides into two parts. Let $(X,\omega)$ be a compact Hermitian
manifold. Firstly, if the Hermitian metric $\omega$ satisfies the assumption
that $\partial\overline{\partial}\omega^k=0$ for all $k$, we generalize the
volume of the cohomology class in the K\"{a}hler setting to the Hermitian
setting, and prove that the volume is always finite and the
Grauert-Riemenschneider type criterion holds true, which is a partial answer to
a conjecture posed by Boucksom. Secondly, we observe that if the anticanonical
bundle $K^{-1}_X$ is nef, then for any $\varepsilon>0$, there is a smooth
function $\phi_\varepsilon$ on $X$ such that
$\omega_\varepsilon:=\omega+i\partial\overline{\partial}\phi_\varepsilon>0$ and
Ricci$(\omega_\varepsilon)\geq-\varepsilon\omega_\varepsilon$. Furthermore, if
$\omega$ satisfies the assumption as above, we prove that for a
Harder-Narasimhan filtration of $T_X$ with respect to $\omega$, the slopes
$\mu_\omega(\mathcal{F}_i/\mathcal{F}_{i-1})\geq 0$ for all $i$, which
generalizes a result of Cao which plays a very important role in his studying
of the structures of K\"{a}hler manifolds.
