A note on Demailly's transcendental Morse inequalities conjecture

Abstract

Let $(X,ω)$ be an $n$-dimensional compact Hermitian manifold with $ω$ a pluriclosed Hermitian metric, i.e. $dd^cω=0$. Let $\{α\},\{β\}\in H^{1,1}_{BC}(X,\mathbb R)$ be two nef classes, such that $α^n-nα^{n-1}\cdotβ>0$. In this short note, we prove that if there is a bounded quasi-plurisubharmonic potential $ρ$, such that $α+dd^cρ\geq 0$ 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.