$\mathrm{\partial }\bar{\mathrm{\partial }}$-lemmas and a conjecture of O. Fujino

It contains the proof of a very general $\mathrm{\partial }\bar{\mathrm{\partial }}$-lemma, together with a decomposition theorem for currents with values in a (singular) Hermitian line bundle. As a corollary, we establish the Kähler version on an injectivity theorem due to O. Fujino in the projective case.