Motivic characteristic classes for singular schemes

TL;DR

This paper develops new motivic characteristic classes for singular algebraic varieties, extending classical invariants and formulas to the singular setting within motivic homotopy theory.

Contribution

It introduces motivic characteristic classes for singular schemes, refining existing classes and extending the motivic Gauss-Bonnet formula to non-smooth varieties.

Findings

Constructed motivic characteristic classes for singular schemes.

Extended the motivic Gauss-Bonnet formula to non-smooth varieties.

Refined existing classes in motivic homotopy theory.

Abstract

We construct characteristic classes for singular algebraic varieties in motivic Borel-Moore homology, extending the motivic Euler class of the tangent bundle defined for smooth varieties. The two classes we define refine, in the setting of motivic homotopy theory, the top degree of the class constructed by Brasselet-Schuermann-Yokura in G-theory, and the top degree of the pro-CSM class constructed by Aluffi in pro-Chow groups. We deduce an extension of the motivic Gauss-Bonnet formula to non-smooth proper varieties.