On a Verdier-type Riemann–Roch for Chern–Schwartz–MacPherson class

Yokura, Shoji

doi:10.1016/s0166-8641(98)00037-6

Cited by 35 publications

(29 citation statements)

References 11 publications

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“…/˝R of characteristic classes of singular varieties, which is any theory recalled in Section 2.2, we need to consider such a commutative diagram as above. One immediate answer for such a commutative diagram is the following Verdier-type Riemann-Roch theorem (see Schürmann [62] and Yokura [75]): 2.4.2 Theorem Let X; Y be complex algebraic varieties and f W X ! Y be a smooth morphism between them.…”

Section: Remarkmentioning

confidence: 99%

Characteristic classes of proalgebraic varieties and motivic measures

Yokura

2012

Algebr. Geom. Topol.

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Gromov initiated what he calls "symbolic algebraic geometry", in which he studied proalgebraic varieties. In this paper we formulate a general theory of characteristic classes of proalgebraic varieties as a natural transformation, which is a natural extension of the well-studied theories of characteristic classes of singular varieties. FultonMacPherson bivariant theory is a key tool for our formulation and our approach naturally leads us to the notion of motivic measure and also its generalization.

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Section: Remarkmentioning

confidence: 99%

Characteristic classes of proalgebraic varieties and motivic measures

Yokura

2012

Algebr. Geom. Topol.

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“…The third equality above follows from the so-called Verdier-Riemann-Roch theorem for Chern class [12,27,39].…”

Section: The Existence and Uniqueness Of Grothendieck Transformationsmentioning

confidence: 99%

On Grothendieck transformations in Fulton–MacPherson’s bivariant theory

Brasselet

Schürmann

Yokura

2007

Journal of Pure and Applied Algebra

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W. Fulton and R. MacPherson have introduced a notion unifying both covariant and contravariant theories, which they called a Bivariant Theory. A transformation between two bivariant theories is called a Grothendieck transformation. The Grothendieck transformation induces natural transformations for covariant theories and contravariant theories. In this paper we show some general uniqueness and existence theorems on Grothendieck transformations associated to given natural transformations of covariant theories. Our guiding or typical model is MacPherson's Chern class transformation c

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“…is the so-called Verdier-Riemann-Roch theorem for the smooth morphism f ′ and the Chern class transformation c * (compare [FM,Sch1,Y4]). …”

Section: With Y ′ An Oriented A-homology Manifold the Following Equalmentioning

confidence: 99%

A Survey of Characteristic Classes of Singular Spaces

Schürmann¹,

Yokura²

2007

Singularity Theory

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ABSTRACT. A theory of characteristic classes of vector bundles and smooth manifolds plays an important role in the theory of smooth manifolds. An investigation of reasonable notions of characteristic classes of singular spaces started since a systematic study of singular spaces such as singular algebraic varieties. We make a quick survey of characteristic classes of singular varieties, mainly focusing on the functorial aspects of some important ones such as the singular versions of the Chern class, the Todd class and the Thom-Hirzebruch's L-class. Then we explain our recent "motivic" characteristic classes, which in a sense unify these three different theories of characteristic classes. We also discuss bivariant versions of them and characteristic classes of proalgebraic varieties, which are related to the motivic measures/integrations. Finally we explain some recent work on "stringy" versions of these theories, together with some references for "equivariant" counterparts.

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On a Verdier-type Riemann–Roch for Chern–Schwartz–MacPherson class

Cited by 35 publications

References 11 publications

Characteristic classes of proalgebraic varieties and motivic measures

Characteristic classes of proalgebraic varieties and motivic measures

On Grothendieck transformations in Fulton–MacPherson’s bivariant theory

A Survey of Characteristic Classes of Singular Spaces

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