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

TL;DR

This paper develops an $L^2$-Dolbeault resolution for a new coherent subsheaf combining Saito's $S$-sheaf and multiplier ideals, leading to vanishing theorems and applications to Fujita's conjecture.

Contribution

It introduces a novel $L^2$-Dolbeault resolution for a combined sheaf, extending MacPherson's conjecture and proving new vanishing theorems with applications to Fujita's conjecture.

Findings

Constructed an $L^2$-Dolbeault resolution for the combined sheaf.

Proved Saito's, Kawamata-Viehweg, and Nadel vanishing theorems.

Applied results to the relative version of Fujita's conjecture.

Abstract

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).