Ko{\l}odziej-Tosatti's conjecture on compact Hermitian manifold with bounded mass property

Abstract

In this note, we show a conjecture of Ko{\l}odziej-Tosatti about Morse-type integrals in nef $(1,1)$ classes on compact Hermitian manifold with bounded mass property. As a consequence, we give positive answers to Demailly-P\u{a}un's conjecture and Tosatti-Weinkove's conjecture when compact Hermitian manifold with bounded mass property.