A Survey of Characteristic Classes of Singular Spaces

Abstract

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.

Euler characteristics of algebraic varieties

“…The group F c (Y ) of all complex algebraically (respectively, analytically) constructible functions is defined as the direct limit of these F V (Y ). Then one has the following important group homomorphisms on F c (Y ) (e.g., see [10,11,12,13]):…”

Higher Todd Classes and Holomorphic Group Actions

“…Presumably there is some simultaneous generalization that applies to higher elliptic genera. Some useful references that discuss elliptic genera in the context of birational geometry are [47], [9], [10], [11], [13], [45]. (2) Are there examples that show that our vanishing theorem for higher Todd genera are false for finite π, if one does not rationalize?…”

“…5 This follows from the preprint [13] as well. Moreover, the precise integral statement that for smooth birational morphisms the derived pushforward of the structure sheaf is the structure sheaf, which is the key point in our proof, also follows from [45] 6 We do not distinguish between an elliptic operator and an elliptic complex, nor between a sheaf or a complex of sheaves.…”

Generating functions of orbifold Chern classes I: symmetric products

“…Futhermore, also related with stringy class invariants, a unified theory of (additive) characteristic classes for singular varieties, the Hirzebruch class (and the motivic Chern class), has appeared in Brasselet-Shürmann-Yokura [2] ( [17] for a survey). It is a natural transformation T y * from the relative Grothendieck ring 2. Review on equivariant Chern classes 2.1.…”