Enriched categories of correspondences and characteristic classes of singular varieties

Abstract

For the category V of complex algebraic varieties, the Grothendieck group of the commutative monoid of the isomorphism classes of correspondences X f ← − M g − → Y with a proper morphism f and a smooth morphism g (such a correspondence is called a proper-smooth correspondence) gives rise to an enriched category Corr(V) + pro-sm of proper-smooth correspondences. In this paper we extend the well-known theories of characteristic classes of singular varieties such as Baum-Fulton-MacPherson's Riemann-Roch transformation (abbr. BFM-RR), MacPherson's Chern class transformation etc. to this enriched category. In order to deal with local complete intersection (.c.i.) morphisms instead of smooth morphisms, in a similar manner we consider an enriched category Zigzag(V) + pro-.c.i. of proper-.c.i. zigzags and extend BFM-RR to this category. We also consider an enriched category M * , * (V) + ⊗ of proper-smooth correspondences (X f ← − M g − → Y ; E) equipped with a complex vector bundle E on M (such a correspondence is called a cobordism bicycle of a vector bundle) and we extend BFM-RR to this enriched category as well.

A bi-variant algebraic cobordism via correspondences

“…This is a continuation of our previous works [20] and [21] and also partially motivated by a recent book [11] by D. Gaitsgory and N. Rozenblyum.…”

“…In[21] we consider such a chain of length n of correspondence, called a n-correspondence, in the case when a vert morphism is a proper morphism and a horiz morphism is a local complete intersection morphism. There we consider the classical category, not the category of derived schemes, thus a local complete intersection morphism is not closed under pull-back.…”