Motivic Milnor classes

Journal of Singularities
volume 1 (2010), 39-59

Received 10 September 2009. Received in revised form 5 January 2010.

DOI: 10.5427/jsing.2010.1c

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Abstract:

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 and natural transformations from this Grothendieck group to the homology theory. We capture the Milnor class, more generally Milnor--Hirzebruch 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 Milnor--Hirzebruch class. We use Fulton--MacPherson's bivariant theory and the motivic Hirzebruch class.

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Author(s) information:

Shoji Yokura
Department of Mathematics and Computer Science
Faculty of Science
Kagoshima University
21-35 Korimoto 1-chome
Kagoshima 890-0065, Japan
email: yokura@sci.kagoshima-u.ac.jp