L2-Dolbeault resolution of the lowest Hodge piece of a Hodge module

In this paper, we introduce a coherent subsheaf of Saito's S-sheaf, which combines the S-sheaf and the multiplier ideal sheaf. We construct its L2-Dolbeault resolution, which generalizes MacPherson's conjecture on the L2 resolution of the Grauert-Riemenschneider sheaf. We also prove various transcendentally-based vanishing theorems for the S-sheaf, including Saito's vanishing theorem, Kawamata-Viehweg vanishing theorem, and new ones such as the Nadel vanishing theorem. Finally, we discuss the applications of our results on the relative version of Fujita's conjecture, such as Kawamata's conjecture.