Enriched categories of correspondences and characteristic classes of singular varieties

Abstract

For the category $\mathcal{V}$ of complex algebraic varieties, the Grothendieck group of the commutative monoid of the isomorphism classes of correspondences $X\stackrel{f}{\leftarrow }M\stackrel{g}{\to }Y$ with proper morphism $f$ and smooth morphism $g$ (such a correspondence is called \emph{a proper-smooth correspondence}) gives rise to an enriched category $\mathcal{C}orr(\mathcal{V}{)}_{pro-sm}^{+}$ 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 $\mathcal{C}orr(\mathcal{V}{)}_{pro-sm}^{+}$. 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 $\mathcal{Z}igzag(\mathcal{V}{)}_{pro-\ell .c.i.}^{+}$ of \emph{proper-$\ell .c.i.$} zigzags and extend BFM-RR to this enriched category $\mathcal{Z}igzag(\mathcal{V}{)}_{pro-\ell .c.i.}^{+}$. We also consider an enriched category ${\mathcal{M}}_{\ast ,\ast }(\mathcal{V}{)}_{\otimes }^{+}$ of proper-smooth correspondences $(X\stackrel{f}{\leftarrow }M\stackrel{g}{\to }Y;E)$ 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 ${\mathcal{M}}_{\ast ,\ast }(\mathcal{V}{)}_{\otimes }^{+}$ as well.

35 pages, comments are welcome; to appear in Fundamenta Mathematicae

References in corpus (3)

• Algebraic cobordism
• Derived algebraic cobordism
• Cobordism bicycles of vector bundles

Cited by in corpus (1)

• Cobordism bicycles of vector bundles