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}
}
  • Lei Wu
  • Published 5 May 2015
  • Mathematics
  • arXiv: Algebraic Geometry
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. 
View PDF on arXiv
10 Citations
Highly Influential Citations
1
Background Citations
3
Methods Citations
4
Results Citations
2

10 Citations

Vanishing and Injectivity for Hodge Modules and R-Divisors

  • Lei Wu
  • Mathematics
    Michigan Mathematical Journal
  • 2021
We prove the injectivity and vanishing theorem for R-Hodge modules and R-divisors over projective varieties, extending the results for rational Hodge modules and integral divisors in \cite{Wu15}. In…

Kodaira-type Vanishings via Non-abelian Hodge Theory

. In this paper, we use non-abelian Hodge Theory to study Kodaira type vanishings and its generalizations. In particular, we generalize Saito van- ishing using Mixed Twistor D -modules. We also…

Injectivity theorem for pseudo-effective line bundles and its applications

We formulate and establish a generalization of Kollár’s injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár’s torsion-freeness,…

Kodaira–Saito vanishing via Higgs bundles in positive characteristic

  • D. Arapura
  • Mathematics
    Journal 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…

Weak positivity for Hodge modules

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.

Positivity for Hodge modules and geometric applications

  • M. Popa
  • Mathematics
    Algebraic Geometry: Salt Lake City 2015
  • 2018
This is a survey of vanishing and positivity theorems for Hodge modules, and their recent applications to birational and complex geometry, expanding on my lecture at the 2015 AMS Summer Institute.

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

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…

Kodaira-Saito vanishing and applications

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…

Hyperbolicity for Log Smooth Families with Maximal Variation

We prove that the base space of a log smooth family of log canonical pairs of log general type is of log general type as well as algebraically degenerate, when the family admits a relative good…

Vanishing theorems for projective morphisms between complex analytic spaces

. We discuss vanishing theorems for projective morphisms between complex analytics spaces and some related results. They will play a crucial role in the minimal model theory for projective morphisms…

References

SHOWING 1-10 OF 28 REFERENCES

Some remarks on the semi-positivity theorems

We show that the dualizing sheaves of reduced simple normal crossings pairs have a canonical weight ltration in a compatible way with the one on the corresponding mixed Hodge modules by calculating…

HIGHER DIRECT IMAGES OF DUALIZING SHEAVES OF

Let / : X - > S be a Lagrangian fibration between protective varieties. We prove that R% Ox ¥ Q^ if S is smooth. Suppose that X is an irreducible symplectic manifold or a certain moduli space of…

Mixed Hodge modules

Introduction 221 § 1. Relative Monodromy Filtration 227 §2. Mixed Hodge Modules on Complex Spaces (2. a) Vanishing Cycle Functors and Specializations (Divisor Case) 236 (2.b) Extensions over Locally…

Variation of mixed Hodge structure. I

Around 1970, Griffiths introduced the notion of a variation of Hodge structure on a complex manifold S (see [17, w 2]). It constitutes the axiomatization of the features possessed by the local…

D-Modules, Perverse Sheaves, and Representation Theory

D-Modules and Perverse Sheaves.- Preliminary Notions.- Coherent D-Modules.- Holonomic D-Modules.- Analytic D-Modules and the de Rham Functor.- Theory of Meromorphic Connections.- Regular Holonomic…

Shafarevich Maps and Automorphic Forms

This text studies various geometric properties and algebraic invariants of smooth projective varieties with infinite fundamental groups. This approach allows for much interplay between methods of…

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

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…

Kodaira-Saito vanishing and applications

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…

On Saito's vanishing theorem

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.

Characterization of abelian varieties

Abstract.We prove that any smooth complex projective variety X with plurigenera P1(X)=P2(X)=1 and irregularity q(X)=dim(X) is birational to an abelian variety.