Kollár's Package for Twisted Saito's S-sheaves

We generalize Kollár’s conjecture (including torsion freeness, injectivity theorem, vanishing theorem and decomposition theorem) to Saito’s $S$-sheaves twisted by a $\mathbb{Q}$-divisor. This gives a uniform treatment for various kinds of Kollár’s package in different topics in complex geometry. As a consequence we prove Kollár’s package of pluricanonical bundles twisted by a certain multiplier ideal sheaf. The method of the present paper is $L^2$-theoretic.