Enriched categories of correspondences and characteristic classes of singular varieties
Volume 253 / 2021
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 a proper morphism $f$ and a smooth morphism $g$ (such a correspondence is called a proper-smooth correspondence) gives rise to an enriched category $\mathscr C\!{\mathit orr}(\mathscr V)^+_{\mathit{pro}\hbox{-}\mathit{sm}}$ 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 ($\ell .c.i.$) morphisms instead of smooth morphisms, in a similar manner we consider an enriched category $\mathscr Z\!\mathit {igzag}(\mathscr V)^+_{\mathit {pro}\hbox {-}\ell .c.i.}$ of proper-$\ell .c.i.$ zigzags and extend BFM–RR to this category. 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 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.
