On characteristic classes of complete intersections

Yokura, Shoji

doi:10.1090/conm/241/03645

Cited by 16 publications

(14 citation statements)

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“…In the case we consider here, where Z is a hypersurface, this uses the virtual tangent bundle τ (Z) of Z, which plays the role of the tangent bundle. This virtual bundle is by definition: [2,9,10,38,27]) The total Milnor class of Z is defined by:…”

Section: Milnor Classesmentioning

confidence: 99%

Lê cycles and Milnor classes

2013

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The purpose of this work is to establish a link between the theory of Chern classes for singular varieties and the geometry of the varieties in question. Namely, we show that if Z is a hypersurface in a compact complex manifold, defined by the complex analytic space of zeroes of a reduced non-zero holomorphic section of a very ample line bundle, then its Milnor classes, regarded as elements in the Chow group of Z, determine the global Lê cycles of Z; and viceversa: The Lê cycles determine the Milnor classes. Morally this implies, among other things, that the Milnor classes determine the topology of the local Milnor fibres at each point of Z, and the geometry of the local Milnor fibres determines the corresponding Milnor classes.

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Section: Milnor Classesmentioning

confidence: 99%

Lê cycles and Milnor classes

2013

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“…By [Sch,Prop. 5.21], this formula specializes at y = −1 to a formula for the Chern classes, which was conjectured by S. Yokura [Yo2], and was proved by A. Parusiński and P. Pragacz [PaPr] (where m = 1).…”

Section: Introductionmentioning

confidence: 86%

Hirzebruch–Milnor Classes and Steenbrink Spectra of Certain Projective Hypersurfaces

Maxim

Saito

Schürmann

2016

Arbeitstagung Bonn 2013

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Abstract. We show that the Hirzebruch-Milnor class of a projective hypersurface, which gives the difference between the Hirzebruch class and the virtual one, can be calculated by using the Steenbrink spectra of local defining functions of the hypersurface if certain good conditions are satisfied, e.g. in the case of projective hyperplane arrangements, where we can give a more explicit formula. This is a natural continuation of our previous paper on the Hirzebruch-Milnor classes of complete intersections.

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“…If cl * = c * is the Chern class transformation, the problem amounts to comparing the Fulton-Johnson class c F J * (X) := c vir * (X) (e.g., see [16,17]) with the homology Chern class c * (X) of MacPherson. The difference between these two classes is measured by the so-called Milnor class M * (X) of X, which is studied in many references like [1,6,7,8,27,29,34,35,45]. This is a homology class supported on the singular locus of X, and for a global hypersurface it was computed in [29] (see also [34,35,45,27]) as a weighted sum in the Chern-MacPherson classes of closures of singular strata of X, the weights depending only on the normal information to the strata.…”

Section: Functorial Characteristic Classes Of Singular Spacesmentioning

confidence: 99%

“…A natural problem in complex geometry is the relation between invariants of a singular complex hypersurface X (like Euler characteristic and Hodge numbers) and the geometry of the singularities of the hypersurface (like the local Milnor fibrations). For the Euler characteristic this is for example a special case of the difference between the Fulton-and MacPherson-Chern classes of X, whose differences are the now well studied Milnor classes of X ( [1,6,7,8,27,29,34,35,45]). Their degrees are related to Donaldson-Thomas invariants of the singular locus ( [3]).…”

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Nearby cycles and characteristic classes of singular spaces

Schürmann

2012

Singularities in Geometry and Topology

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In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach and Kagoshima. We explain the relation between nearby cycles for constructible functions or sheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodge modules, and the specialization of characteristic classes of singular spaces like the Chern-, Todd-, Hirzebruch-and motivic Chernclasses. As an application we get a description of the differences between the corresponding virtual and functorial characteristic classes of complex hypersurfaces in terms of vanishing cycles related to the singularities of the hypersurface. at conferences in Strasbourg, Oberwolfach and Kagoshima. Here I would like to thank the organizers for the invitation to these conferences. I also would like to thank Sylvain Cappell, Laurentiu Maxim and Shoji Yokura for the discussions on our joint work related to the subject of this paper. Contents 1. Virtual classes of local complete intersections 2 2. Functorial characteristic classes of singular spaces 7 3. Nearby and vanishing cycles 12 References 22 1. Virtual classes of local complete intersectionsRecall that we are working in the complex algebraic context. A characteristic class cl * of (complex algebraic) vector bundles over X is a map cl * : V ect(X) → H * (X) ⊗ R from the set V ect(X) of isomorphism classes of complex algebraic vector bundles over X to some cohomology theory H * (X) ⊗ R with a coefficient ring R, which is compatible with pullbacks. Here we use as a cohomology theoryH 2 * (X, Z) , the usual cohomology in even degrees. CH * (X), the operational Chow cohomology of [16]. K 0 (X), the Grothendieck group of vector bundles.We also assume that cl * is multiplicative, i.e. cl * (V ) = cl * (V ′ ) ∪ cl * (V ′′ ) for any short exact sequence

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On characteristic classes of complete intersections

Cited by 16 publications

References 0 publications

Lê cycles and Milnor classes

Lê cycles and Milnor classes

Hirzebruch–Milnor Classes and Steenbrink Spectra of Certain Projective Hypersurfaces

Nearby cycles and characteristic classes of singular spaces

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