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

### Volume 117 / 2016

#### Abstract

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 Kähler 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 $\varepsilon \gt 0$, there is a smooth function $\phi _\varepsilon $ on $X$ such that $\omega _\varepsilon :=\omega +i\partial \overline {\partial }\phi _\varepsilon \gt 0$ 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 $\mu _\omega (\mathcal {F}_i/\mathcal {F}_{i-1})$ are nonnegative for all $i$; this generalizes a result of Cao which plays an important role in his study of the structures of Kähler manifolds.