# Kodaira-Saito vanishing and applications

@article{Popa2014KodairaSaitoVA, title={Kodaira-Saito vanishing and applications}, author={Mihnea Cristian Popa}, journal={arXiv: Algebraic Geometry}, year={2014} }

The first part of the paper contains a detailed proof of M. Saito's generalization of the Kodaira vanishing theorem, following the original argument and with ample background, based on a lecture given at a Clay workshop on mixed Hodge modules. The second part contains some recent applications, and a Kawamata-Viehweg-type statement in the setting of mixed Hodge modules.

## 16 Citations

### Kodaira–Saito vanishing via Higgs bundles in positive characteristic

- MathematicsJournal für die reine und angewandte Mathematik (Crelles Journal)
- 2019

Abstract The goal of this paper is to give a new proof of a special case of the Kodaira–Saito vanishing theorem for a variation of Hodge structure on the complement of a divisor with normal…

### On Saito's vanishing theorem

- Mathematics
- 2014

We reprove Saito's vanishing theorem for mixed Hodge modules by the method of Esnault and Viehweg. The main idea is to exploit the strictness of direct images on certain branched coverings.

### An overview of Morihiko Saito's theory of mixed Hodge modules

- Mathematics, Philosophy
- 2014

After explaining the definition of pure and mixed Hodge modules on complex manifolds, we describe some of Saito's most important results and their proofs, and then discuss two simple applications of…

### Vanishing and injectivity theorems for Hodge modules

- Mathematics
- 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…

### Vanishing theorems for parabolic Higgs bundles

- MathematicsMathematical Research Letters
- 2019

This is a sequel to "Kodaira-Saito vanishing via Higgs bundles in positive characteristic" (arXiv:1611.09880). However, unlike the previous paper, all the arguments here are in characteristic zero.…

### Vanishing for Hodge ideals on toric varieties

- MathematicsMathematische Nachrichten
- 2019

In this article we construct a Koszul‐type resolution of the pth exterior power of the sheaf of holomorphic differential forms on smooth toric varieties and use this to prove a Nadel‐type vanishing…

### Weak positivity for Hodge modules

- Mathematics
- 2015

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.

### 𝒟-MODULES IN BIRATIONAL GEOMETRY

- MathematicsProceedings of the International Congress of Mathematicians (ICM 2018)
- 2019

It is well known that numerical quantities arising from the theory of D-modules are related to invariants of singularities in birational geometry. This paper surveys a deeper relationship between the…

### Logarithmic Kodaira dimension and zeros of holomorphic log-one-forms

- MathematicsMathematische Annalen
- 2020

In this paper, we prove that the zero-locus of any global holomorphic log-one-form on a projective log-smooth pair ( X ; D ) of log-general type must be non-empty.

### Logarithmic Kodaira dimension and zeros of holomorphic log-one-forms

- MathematicsMathematische Annalen
- 2020

In this paper, we prove that the zero-locus of any global holomorphic log-one-form on a projective log-smooth pair (X; D) of log-general type must be non-empty.

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We reprove Saito's vanishing theorem for mixed Hodge modules by the method of Esnault and Viehweg. The main idea is to exploit the strictness of direct images on certain branched coverings.

### An overview of Morihiko Saito's theory of mixed Hodge modules

- Mathematics, Philosophy
- 2014

After explaining the definition of pure and mixed Hodge modules on complex manifolds, we describe some of Saito's most important results and their proofs, and then discuss two simple applications of…

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We show that techniques inspired by Koll\'ar and Viehweg's study of weak positivity, combined with vanishing theorems for log-canonical pairs, lead to new consequences regarding generation and…

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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…

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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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Let V and W be non-singular projective varieties over the field of complex numbers C, n= dim (V) and m=dim (W). Let/: V---+W be a fibre space (this simply means that I is surjective with connected…

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We show that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point. The proof uses generic vanishing theory for Hodge modules on abelian…