A survey of characteristic classes of singular spaces
arXiv:math/0511175
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.
58 pages
References in corpus (7)
- Equivariant Chern classes of singular algebraic varieties with group actions
- A generalized Verdier-type Riemann-Roch theorem for Chern-Schwartz-MacPherson classes
- A general construction of partial Grothendieck transformations
- Stringy Chern classes of singular varieties
- Chern classes of proalgebraic varieties and motivic measures
- Witt groups of sheaves on topological spaces
- Natural transformations from constructible functions to homology