The limit and boundary characteristic classes in Borel-Moore motivic homology

2022/08/25 by Fangzhou Jin, Peng Sun, Jin, Fangzhou +3
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2208.11989

Abstract

We show that the zero-dimensional part of the pro-Chern-Schwarz-MacPherson class defined by Aluffi can be lifted to the zeroth Suslin homology. The proof uses the pro-characteristic class in the limit Borel-Moore motivic homology, which has a quadratic refinement in the limit Borel-Moore Milnor-Witt homology. In characteristic zero, this construction factors through the group of constructible functions, in a way compatible with the covariant functoriality; in positive characteristic this property fails, and we show that the failure can be measured by the boundary characteristic class in the boundary Borel-Moore motivic homology. We prove a push-forward formula for the boundary characteristic class, and conjecture it to agree with the Swan class defined by Kato-Saito.

Citations

• The characteristic cycle and the singular support of a constructible sheaf
• The characteristic class and ramification of an ℓ-adic étale sheaf

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