Enriched categories of correspondences and characteristic classes of singular varieties

Abstract

For the category $\mathscr V$ of complex algebraic varieties, the Grothendieck group of the commutative monoid of the isomorphism classes of correspondences $X \xleftarrow f M \xrightarrow g Y$ with proper morphism $f$ and smooth morphism $g$ (such a correspondence is called \emph{a proper-smooth correspondence}) gives rise to an enriched category $\mathscr Corr(\mathscr V)^+_{pro-sm}$ 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 (abbr. BFM-RR) and MacPherson's Chern class transformation and so on to this enriched category $\mathscr Corr(\mathscr V)^+_{pro-sm}$. In order to deal with local complete intersection (abbr. $\ell.c.i.$) morphism instead of smooth morphism, in a similar manner we consider an enriched category $\mathscr Zigzag(\mathscr V)^+_{pro-\ell.c.i.}$ of \emph{proper-$\ell.c.i.$} zigzags and extend BFM-RR to this enriched category $\mathscr Zigzag(\mathscr V)^+_{pro-\ell.c.i.}$. We also consider an enriched category $\mathscr M_{*,*}(\mathscr V)^+_{\otimes}$ of proper-smooth correspondences $(X \xleftarrow f M \xrightarrow g Y; E)$ equipped with complex vector bundle $E$ on $M$ (such a correspondence is called \emph{a cobordism bicycle of vector bundle}) and we extend BFM-RR to this enriched category $\mathscr M_{*,*}(\mathscr V)^+_{\otimes}$ as well.