Hirzebruch Classes and Motivic Chern Classes for Singular Spaces

Abstract

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 transformation mCy: K 0 (var/X) → G 0 (X) ⊗ Z[y], which generalizes the total λ-class λy(T * X) of the cotangent bundle to singular spaces. Here K 0 (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 G 0 (X) is the Grothendieck group of coherent sheaves of O X -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 * : K 0 (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 1 2 J.-P. Brasselet, J. Schürmann & S. Yokura 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.We introduce characteristic homology class transformations mC y and T y * on K 0 (var/X) related by the following commutative diagram:by taking fiberwise the (topological) Euler characteristic with compact support. Note that the homomorphisms e and mC 0 are surjective. Remark 0.1. Note that in the algebraic context the Chern-Schwartz-MacPherson transformation c * of [44] is defined only on spaces X embeddable into a smooth space (e.g. for quasi-projective varieties). But using the technique of "Chow envelopes" as in [31, Sec. 18.3], this transformation can uniquely be extended to all (reduced) separated schemes of finite type over spec(k). 4 J.-P. Brasselet, J. Schürmann & S. YokuraFor X a compact complex algebraic variety, we can also construct the following commutative diagram of natural transformations:4) with L * the homology L-class transformation of Cappell-Shaneson [21] (as reformulated by Yokura [81]). Here Ω(X) is the Abelian group of cobordism classes of self-dual constructible complexes. To make Ω(·) covariant functorial (correcting [81]), we use the definition of a "cobordism" given by Youssin [89] in the more general context of triangulated categories with duality. Note that the homology L-class transformation of Cappell-Shaneson [21] is defined o...

Equivariant Characteristic Classes of Singular Complex Algebraic Varieties

“…Moreover, for a subgroup H of G with g ∈ H, these transformations commute with the obvious restriction functors Res G H . Also, by construction, MHT y * (id) is the complexified version of the transformation defined in [BSY,Sc09]. Finally, (52) and (63) yield the following multiplicativity property:…”

“…var/X), for any subgroup H < G. As in [BSY,Sc09], there is a natural transformation (as explained in the Appendix) χ G Hdg : K G 0 (var/X) → K 0 (MHM G (X)) to the Grothendieck group of G-equivariant mixed Hodge modules, mapping [id X ] to the class of the constant Hodge module [Q H X ]. The motivic Atiyah-Singer class transformation…”

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

“…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 groups. The papers [35,36,37] provide Hodgetheoretic applications of the above topological stratified multiplicative formulae.…”

Untitled

“…In the case of complex algebraic varieties, one may also look at the MacPherson Chern classes [132], 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 groups. The papers [35,36,37] provide Hodge-theoretic applications of the above topological stratified multiplicative formulae.…”