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We generalize Koll\'ar's conjecture (including torsion freeness, injectivitytheorem, vanishing theorem and decomposition theorem) to Saito's $S$-sheavestwisted by a $\mathbb{Q}$-divisor. This gives a uniform treatment for variouskinds of Koll\'ar's package in different topics in complex geometry. As aconsequence we prove Koll\'ar's package of pluricanonical bundles twisted by acertain multiplier ideal sheaf. The method of the present paper is$L^2$-theoretic.

Koll\'ar's Package for Twisted Saito's S-sheaves