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

It contains the proof of a very general $\partial\bar\partial$-lemma, together with a decomposition theorem for currents with values in a (singular) Hermitian line bundle. As a corollary, we establish the K\"ahler version on an injectivity theorem due to O. Fujino in the projective case.
Comment: Comments welcome! We have decided to split into two parts our previous work arXiv:2012.05063. This is the second part