Mixed Hodge modules

Saito, Morihiko

doi:10.3792/pjaa.62.360

Cited by 159 publications

(339 citation statements)

References 4 publications

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“…Most of results mentioned above have been obtained earlier and in greater generality by M. Saito in [156,157] by the use of mixed Hodge modules. While our approach uses heavily the theory of perverse sheaves, it ultimately rests on classical and mixed Hodge theory.…”

Section: A Proof Via Classical Hodge Theorymentioning

confidence: 88%

“…The work of M. Saito [156,157] settled these issues completely with the use of mixed Hodge modules. The reader interested in the precise statements and generalizations is referred to Saito's papers (for brief summaries, see [70] and §3.2).…”

Section: The Decomposition Theorem and Mixed Hodge Structuresmentioning

confidence: 99%

“…The hard Lefschetz theorem for the intersection cohomology of projective varieties is proved in [9]. In the 1980's, M. Saito ( [156,157]) proved that in the projective case these groups admit a pure Hodge structure (i.e. a (p, q)-Hodge decomposition), re-proved that they satisfy the hard Lefschetz theorem and proved the Hodge-Riemann bilinear relations.…”

Section: Historical and Other Remarksmentioning

confidence: 99%

“…All these questions have been answered in the work of M. Saito [156,157] in the 1980's. The case of IC X (i.e.…”

Section: Historical and Other Remarksmentioning

confidence: 99%

“…The extended RiemannHilbert correspondence assigns to the pair (M, W ) a filtered perverse sheaf (P, W ) and this data extends to a functor of t-categories In the case when Y is a point, the category MHM(pt) is naturally equivalent to the category of graded polarizable rational mixed Hodge structures (cf. [157], p.319); here "graded" means that one has polarizations on the graded pieces of the weight filtration. At the end of the day, the W and F filtrations produce two filtrations on the cohomology and on the cohomology with compact supports of a complex in the image of r and give rise to mixed Hodge structures compatible with the usual operations.…”

Section: Saito's Approach Via Mixed Hodge Modulesmentioning

confidence: 99%

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The decomposition theorem, perverse sheaves and the topology of algebraic maps

Andrea

Migliorini

2009

Bull. Amer. Math. Soc.

162

152

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We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem, indicate some important applications and examples.

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Section: A Proof Via Classical Hodge Theorymentioning

confidence: 88%

Section: The Decomposition Theorem and Mixed Hodge Structuresmentioning

confidence: 99%

Section: Historical and Other Remarksmentioning

confidence: 99%

“…All these questions have been answered in the work of M. Saito [156,157] in the 1980's. The case of IC X (i.e.…”

Section: Historical and Other Remarksmentioning

confidence: 99%

Section: Saito's Approach Via Mixed Hodge Modulesmentioning

confidence: 99%

See 3 more Smart Citations

The decomposition theorem, perverse sheaves and the topology of algebraic maps

Andrea

Migliorini

2009

Bull. Amer. Math. Soc.

162

152

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Hodge genera of algebraic varieties I

Cappell

Maxim

Shaneson

2007

Comm Pure Appl Math

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The aim of this paper is to study the behavior of Hodge-theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized families of) global invariants of a complex algebraic variety X to such invariants of singularities of proper algebraic maps defined on X . Such formulae severely constrain, both topologically and analytically, the singularities of complex maps, even between smooth varieties. Similar results were announced by the first and third author in [13,32].

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Equivariant Characteristic Classes of Singular Complex Algebraic Varieties

Cappell

Maxim

Schürmann³

et al. 2012

Comm Pure Appl Math

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Abstract. Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet-Schürmann-Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, BaumFulton-MacPherson Todd classes, and Goresky-MacPherson L-classes, respectively). In this note we define equivariant analogues of these classes for singular quasi-projective varieties acted upon by a finite group of algebraic automorphisms, and show how these can be used to calculate the homology Hirzebruch classes of global quotient varieties. We also compute the new classes in the context of monodromy problems, e.g., for varieties that fiber equivariantly (in the complex topology) over a connected algebraic manifold. As another application, we discuss Atiyah-Meyer type formulae for twisted Hirzebruch classes of global orbifolds.

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Mixed Hodge modules

Cited by 159 publications

References 4 publications

The decomposition theorem, perverse sheaves and the topology of algebraic maps

The decomposition theorem, perverse sheaves and the topology of algebraic maps

Hodge genera of algebraic varieties I

Equivariant Characteristic Classes of Singular Complex Algebraic Varieties

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