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 partial derivative(partial derivative) over bar omega(k) = 0 for all k, we generalize the volume of the cohomology class in the Kahler 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 phi(epsilon) on X such that omega(epsilon) := omega + i partial derivative(partial derivative) over bar phi(epsilon) and Ricci(omega(epsilon)) >= - epsilon omega(epsilon). Furthermore, if omega satisfies the assumption as above, we prove that for a Harder Narasimhan filtration of Tx with respect to omega, the slopes mu(omega) (Fi/Fi-1) are nonnegative for all i; this generalizes a result of Cao which plays an important role in his study of the structures of Kahler manifolds.SCI(E)ARTICLEwangzw@amss.ac.cn141-5811