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

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).

36 pages. Some parts are rewritten without changing the main resutls. Comments are welcomed

References in corpus (1)

• Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules