Nearby cycles and characteristic classes of singular spaces

HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES

Journal of Topology and Analysis ◽

10.1142/s1793525310000239 ◽

2010 ◽

Vol 02 (01) ◽

pp. 1-55 ◽

Cited By ~ 42

Author(s):

JEAN-PAUL BRASSELET ◽

JÖRG SCHÜRMANN ◽

SHOJI YOKURA

Keyword(s):

Chern Class ◽

Blow Up ◽

Homology Class ◽

Grothendieck Group ◽

Characteristic Classes ◽

Chern Classes ◽

Singular Spaces ◽

Coherent Sheaves ◽

Roch Theorem ◽

Mixed Hodge Modules

In this paper we study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformationmCy: K0( var /X) → G0(X) ⊗ ℤ[y], which generalizes the total λ-class λy(T*X) of the cotangent bundle to singular spaces. Here K0( var /X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G0(X) is the Grothendieck group of coherent sheaves of [Formula: see text]-modules. A first construction of mCy is based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mCy is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito. We define a natural transformation Ty* : K0( var /X) → H*(X) ⊗ ℚ[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Ty* is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for y = -1), the Todd class transformation in the singular Riemann-Roch theorem of Baum–Fulton–MacPherson (for y = 0) and the L-class transformation of Cappell-Shaneson (for y = 1). We also explain the relation among the "stringy version" of our characteristic classes, the elliptic class of Borisov–Libgober and the stringy Chern classes of Aluffi and De Fernex–Lupercio–Nevins–Uribe. All our results can be extended to varieties over a base field k of characteristic 0.

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Singular Spaces, Characteristic Classes, and Intersection Homology

Annals of Mathematics ◽

10.2307/2944349 ◽

1991 ◽

Vol 134 (2) ◽

pp. 325 ◽

Cited By ~ 25

Author(s):

Sylvain E. Cappell ◽

Julius L. Shaneson

Keyword(s):

Characteristic Classes ◽

Intersection Homology ◽

Singular Spaces

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A Survey of Characteristic Classes of Singular Spaces

Singularity Theory ◽

10.1142/9789812707499_0037 ◽

2007 ◽

Cited By ~ 7

Author(s):

Jörg Schürmann ◽

Shoji Yokura

Keyword(s):

Characteristic Classes ◽

Singular Spaces

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On the Buchstaber Subring in MSp∗

Georgian Mathematical Journal ◽

10.1515/gmj.1998.401 ◽

1998 ◽

Vol 5 (5) ◽

pp. 401-414

Author(s):

M. Bakuradze

Keyword(s):

Tensor Product ◽

Characteristic Classes ◽

Generalized Quaternion ◽

Symplectic Cobordisms

Abstract A formula is given to calculate the last n number of symplectic characteristic classes of the tensor product of the vector Spin(3)- and Sp(n)-bundles through its first 2n number of characteristic classes and through characteristic classes of Sp(n)-bundle. An application of this formula is given in symplectic cobordisms and in rings of symplectic cobordisms of generalized quaternion groups.

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