HIRZEBRUCH CLASSES AND MOTIVIC CHERN CLASSES FOR SINGULAR SPACES

By Jean-Paul Brasselet, Joerg Schuermann, Shoji Yokura. 2005

178 citations recorded by the source corpus.

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 -modules. A first construction of mCy is based on resolution of singularities and a suitable "blow-up" relation, following the work of Du Bois, Guillen, 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2911d45199a65d956c6f4f58685ac9f52c06aca6

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Citations in the corpus

Hodge genera of algebraic varieties I (2006, methodology): “In particular, if X is a rational homology manifold, then QH X ∼ = IC H X in D b MHM(X ) and ITy(X ) = Ty(X ). Similarly, for any complex algebraic variety X, MHC∗(QH X ) is the motivic Chern class from [ 9 ].”
Characteristic classes of mixed Hodge modules (2009, methodology): “I also would like to thank S. Cappell, L. Maxim and S. Yokura for some discussions on the subject of this paper.”
The decomposition theorem, perverse sheaves and the topology of algebraic maps (2007, background): “In the case of complex algebraic varieties, one may also look at the MacPherson Chern classes [131], the Baum-Fulton-MacPherson Todd classes [6], the homology Hirzebruch classes [25, 37] and their associated Hodge-genera defined in terms of the mixed Hodge structures on the (intersection) cohomology”