  In this paper, we introduce a coherent subsheaf of Saito's $S$-sheaf, which
is a combination of the $S$-sheaf and the multiplier ideal sheaf. We construct
its $L^2$-Dolbeault resolution, which generalizes MacPherson's conjecture on
the $L^2$ resolution of the Grauert-Riemenschneider sheaf. We also prove
various vanishing theorems for the $S$-sheaf (Saito's vanishing theorem,
Kawamata-Viehweg vanishing theorem and some new ones like Nadel vanishing
theorem) transcendentally. Finally, we discuss some applications of our results
on the relative version of Fujita's conjecture (e.g. Kawamata's conjecture).
