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Let $(X,D)$ be a log-canonical (lc) pair, in which $X$ is a compact K\"ahlermanifold and $D$ is a reduced snc divisor, and let $F$ be a holomorphic linebundle on $X$ equipped with a smooth metric $h_F = e^{-\varphi_F}$. Via the useof the adjoint ideal sheaves (constructed from $\varphi_F$ and $D$) and theassociated residue morphisms, sections of $K_D \otimes \left. F\right|_D$ on$D$ (as well as those of $K_X \otimes D \otimes F$ on $X$) can be related tothe $F$-valued holomorphic top-forms on each lc center of $(X,D)$ by aninductive use of a certain residue exact sequence derived from the adjointideal sheaves. The theory of harmonic integrals is valid on each lc center(which is compact K\"ahler), so this provides a pathway to apply the techniquesin harmonic theory to the possibly singular K\"ahler space $D$. To illustratethe use of such apparatus in problems concerning lc pairs, we prove aKoll\'ar-type injectivity theorem for the cohomology on $D$ when $F$ issemi-positive. This in turn also solves the conjecture by Fujino on theinjectivity theorem for the compact K\"ahler lc pair $(X,D)$, providing analternative proof of a recent result by Cao and P\u{a}un.

An injectivity theorem on snc compact K\"ahler spaces: an application of  the theory of harmonic integrals on log-canonical centers via adjoint ideal  sheaves