Motivic Milnor classes

Details Motivic Milnor classes Motivic Milnor classes

2009-06-29

arxiv.org/abs/0906.5200v2

Journal of Singularities, Vol. 1 (2010), 39 - 59

Mathematics

Algebraic Geometry

18 pages, some revision was made with more references

Scientific paper

The Milnor class is a generalization of the Milnor number, defined as the difference (up to sign) of Chern--Schwartz--MacPherson's class and Fulton--Johnson's canonical Chern class of a local complete intersection variety in a smooth variety. In this paper we introduce a "motivic" Grothendieck group $K^{\mathcal Prop}_{\ell.c.i}(\mathcal V/X \to S)$ and natural transformations from this Grothendieck group to the homology theory. We capture the Milnor class, more generally Hirzebruch--Milnor class, as a special value of a distinguished element under these natural transformations. We also show a Verdier-type Riemann--Roch formula for our motivic Hirzebruch-Milnor class. We use Fulton--MacPherson's bivariant theory and the motivic Hirzebruch class.

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