$L^2$-Dolbeault resolution of the lowest Hodge piece of a Hodge module

Let $X$ be a complex space and $M$ a pure Hodge module with strict support $X$. The purpose of this paper is to introduce a coherent subsheaf $S(M,\varphi)$ of M. Saito's $S(M)$ which is a combination of $S(M)$ and the multiplier ideal sheaf $\mathscr{I}(\varphi)$ while constructing a resolution of $S(M,\varphi)$ by differential forms with certain $L^2$-boundary conditions. This could be viewed as a wide generalization of MacPherson's conjecture on the $L^2$-representation of the Grauert-Riemenschneider sheaf. As applications, various vanishing theorems for $S(M)$ (Saito's vanishing, Kawamata-Viehweg vanishing and some new ones like Nadel vanishing, partial vanishing) are proved via standard differential geometrical arguments.