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

TL;DR

This paper extends Kollár's conjecture to Saito's S-sheaves twisted by a rational divisor, providing a unified approach to key theorems in complex geometry using $L^2$ methods.

Contribution

It introduces a generalization of Kollár's package to twisted Saito's S-sheaves, including torsion freeness, injectivity, vanishing, and decomposition theorems.

Findings

Proves Kollár's package for pluricanonical bundles with multiplier ideal sheaves

Develops an $L^2$-theoretic method for the generalization

Provides a unified framework for various Kollár's theorems in complex geometry

Abstract

We generalize Koll\'ar'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 Koll\'ar's package in different topics in complex geometry. As a consequence we prove Koll\'ar's package of pluricanonical bundles twisted by a certain multiplier ideal sheaf. The method of the present paper is $L^2$-theoretic.