On the volume of a pseudo-effective class and semi-positive properties of the Harder-Narasimhan filtration on a compact Hermitian manifold

This paper divides into two parts. Let $(X,\omega )$ be a compact Hermitian manifold. Firstly, if the Hermitian metric $\omega$ satisfies the assumption that $\mathrm{\partial }\bar{\mathrm{\partial }}{\omega }^k=0$ 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 $K_X^{-1}$ is nef, then for any $\epsilon >0$, there is a smooth function ${\varphi }_{\epsilon }$ on $X$ such that ${\omega }_{\epsilon }:=\omega +i\mathrm{\partial }\bar{\mathrm{\partial }}{\varphi }_{\epsilon }>0$ and Ricci$({\omega }_{\epsilon })\ge -\epsilon {\omega }_{\epsilon }$. 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 ${\mu }_{\omega }({\mathcal{F}}_i\mathrm{/}{\mathcal{F}}_{i-1})\ge 0$ 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.