# Vanishing and injectivity theorems for Hodge modules

@article{Wu2015VanishingAI, title={Vanishing and injectivity theorems for Hodge modules}, author={Lei Wu}, journal={arXiv: Algebraic Geometry}, year={2015} }

We prove a surjectivity theorem for the Deligne canonical extension of a polarizable variation of Hodge structure with quasi-unipotent monodromy at infinity along the lines of Esnault-Viehweg. We deduce from it several injectivity theorems and vanishing theorems for pure Hodge modules. We also give an inductive proof of Kawamata-Viehweg vanishing for the lowest graded piece of the Hodge filtration of a pure Hodge module using mixed Hodge modules of nearby cycles.

## 10 Citations

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### Weak positivity for Hodge modules

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We prove the weak positivity of the kernels of Kodaira-Spencer- type maps for pure Hodge module extensions of generically defined variations of Hodge structure.

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

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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…

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