Limits of Chow groups, and a new construction of Chern-Schwartz-MacPherson classes

Abstract

We define an `enriched' notion of Chow groups for algebraic varieties, agreeing with the conventional notion for complete varieties, but enjoying a functorial push-forward for arbitrary maps. This tool allows us to glue intersection-theoretic information across elements of a stratification of a variety; we illustrate this operation by giving a direct construction of Chern-Schwartz-MacPherson classes of singular varieties, providing a new proof of an old (and long since settled) conjecture of Deligne and Grothendieck.

arXiv comment23 pages, final version. Dedicated to Robert MacPherson on the occasion of his 60th birthday