Let (X, ω) be an n-dimensional compact Hermitian manifold with ω a pluri-closed Hermitian metric, i.e., ddcω = 0. Let \(\left\{\alpha \right\},\left\{\beta \right\} \in H_{BC}^{1,1}\left({X,\mathbb{R}} \right)\) be two nef classes, such that αn − nαn−1 · β > 0. In this short note, we prove that if there is a bounded quasi-plurisubharmonic potential ρ, such that α + ddcρ ≥ 0 in the weak sense of currents, then the class {α − β} contains a Kähler current. This gives a partial solution of Demailly’s transcendental Morse inequalities conjecture.
