Relevement de Cycles Algebriques et Homomorphismes Associes en Homologie d'Intersection

Barthel, Gottfried; Brasselet, Jean‐Paul; Fieseler, Karl-Heinz; Gabber, Ofer; Kaup, Ludger

doi:10.2307/2118630

Cited by 25 publications

(22 citation statements)

References 5 publications

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“…Indeed, as Goresky and MacPherson conjectured, all algebraic cycles, and in particular the homology Chern classes, lift rationally to intersection homology, by Barthel, Brasselet, Fieseler, Gabber, and Kaup [3]; but those lifts are not unique (see the comments on this problem in [3, p. 158]), and we will not use that approach.…”

Section: Statementsmentioning

confidence: 99%

Chern Numbers for Singular Varieties and Elliptic Homology

Totaro¹

2000

The Annals of Mathematics

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A fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones. Intersection homology, for example, directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety. Minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so.Here we use ideas from minimal model theory to define some characteristic numbers for singular varieties, generalizing the Chern numbers of a smooth variety. This was suggested by Goresky and MacPherson as a next natural problem after the definition of intersection homology [11]. We find that only a subspace of the Chern numbers can be defined for singular varieties. A convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex bordism, whereas a singular variety can at most have a fundamental class in a weaker homology theory, elliptic homology. We use this idea to give an algebro-geometric definition of elliptic homology: "complex bordism modulo flops equals elliptic homology."This paper was inspired by some questions asked by Jack Morava. The descriptions of elliptic homology given by Gerald Höhn [13] were also an important influence. Thanks to Dave Bayer, Mike Stillman, John Stembridge, Sheldon Katz, and Stein Stromme for their computer algebra programs Macaulay, SF, and Schubert, which helped in guessing the right answer. Contents 1. Statements 2. Weakly complex manifolds 3. The complex elliptic genus 4. Complex bordism modulo flops 5. Complex bordism modulo flops, continued 6. SU-bordism modulo flops 7. Saito's homology classes χ n−k i 8. The Chern numbers c k 1 χ n−k i 9. The twisted χ y genus

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Section: Statementsmentioning

confidence: 99%

Chern Numbers for Singular Varieties and Elliptic Homology

Totaro¹

2000

The Annals of Mathematics

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“…Even though intersection cohomology lacks functoriality with respect to algebraic maps (however, see [5]), the intersection cohomology groups of projective varieties enjoy the same properties of Hodge-Lefschetz-Poincaré-type as the singular cohomology of projective manifolds. Poincaré duality takes the form IH k (Y ) ≃ IH 2n−k (Y ) ∨ and follows formally from the canonical isomorphism IC Y ≃ IC ∨ Y stemming from Poincaré-Verdier duality; in particular, there is a non degenerate geometric intersection pairing…”

Section: Intersection Cohomologymentioning

confidence: 99%

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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“…In the case when X is quasi-projective, one can make distinguished choices which realize the summands as mixed Hodge substructures of a canonical mixed Hodge structure on IH * (X) (see [54,45] and §1. 9,5).…”

Section: Theorem 141 (Decomposition Theorem For Intersection Cohomomentioning

confidence: 99%

“…Even though intersection cohomology lacks functoriality with respect to algebraic maps (however, see [5] …”

Section: Perverse Sheavesmentioning

confidence: 99%

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Untitled

2009

Bull. Amer. Math. Soc.

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Abstract. 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 and indicate some important applications and examples.

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Relevement de Cycles Algebriques et Homomorphismes Associes en Homologie d'Intersection

Cited by 25 publications

References 5 publications

Chern Numbers for Singular Varieties and Elliptic Homology

Chern Numbers for Singular Varieties and Elliptic Homology

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

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