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$L^2$-Dolbeault resolution of the lowest Hodge piece of a Hodge module (2104.04905v3)
Published 11 Apr 2021 in math.AG and math.CV
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 $L2$-Dolbeault resolution, which generalizes MacPherson's conjecture on the $L2$ 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).