In this paper, we prove a \(\partial \overline{\partial }\) -type lemma on compact Kähler manifolds for logarithmic differential forms valued in the dual of a certain pseudo-effective line bundle, thereby confirming a conjecture proposed by Wan. We then derive several applications, including strengthened results by Esnault–Viehweg on the degeneracy of the spectral sequence at the \(E_1\) -stage for projective manifolds associated with the logarithmic de Rham complex, as well as by Katzarkov–Kontsevich–Pantev on the unobstructed locally trivial deformations of a projective generalized log Calabi–Yau pair with some weights, both of which are extended to the broader context of compact Kähler manifolds. Furthermore, we establish the Kähler version of an injectivity theorem originally formulated by Ambro in the algebraic setting. Notably, while Fujino previously addressed the Kähler case, our proof takes a different approach by avoiding the reliance on mixed Hodge structures for cohomology with compact support.
