Kodaira-Saito vanishing and applications

@article{Popa2014KodairaSaitoVA,
  title={Kodaira-Saito vanishing and applications},
  author={Mihnea Cristian Popa},
  journal={arXiv: Algebraic Geometry},
  year={2014}
}
  • M. Popa
  • Published 11 July 2014
  • Mathematics
  • arXiv: Algebraic Geometry
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. 
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16 Citations
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16 Citations

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References

SHOWING 1-10 OF 29 REFERENCES

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.

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…

On direct images of pluricanonical bundles

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…

Vanishing and injectivity theorems for Hodge modules

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…

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

Abstract We extend most of the results of generic vanishing theory to bundles of holomorphic forms and rank-one local systems, and more generally to certain coherent sheaves of Hodge-theoretic origin…

GV-sheaves, Fourier-Mukai transform, and generic vanishing

We prove a formal criterion for generic vanishing, in the sense originated by Green and Lazarsfeld and pursued further by Hacon, but in the context of an arbitrary Fourier-Mukai correspondence. For…

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.

Weak Positivity and the Additivity of the Kodaira Dimension for Certain Fibre Spaces

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…

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…

Kodaira dimension and zeros of holomorphic one-forms

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…