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Enriched categories of correspondences and characteristic classes of singular varieties. (English) Zbl 1477.14012

Let \(G_0\) for a complex algebraic variety denote its Grothendieck group of coherent sheaves. This invariant and many other homology theories are contravariantly functorial for all smooth morphisms and covariantly functorial for proper smooth morphisms. This article combines these two functorialities into a functoriality for correspondences. The naturality of this construction allows to show that transformations between two such homology theories that are natural for the two functorialities above remain natural for correspondences. This is applied to suitable Chern character transformations like the Todd class transformation of Baum, Fulton and MacPherson.
A correspondence from \(X\) to \(Y\) is a pair of maps \(b\colon M\to X\) and \(f\colon M \to Y\) for a third object \(M\) of the same kind. These correspondences are composed by forming a fibre product. This works because the pullback of a smooth morphisms is again smooth. For the more general local complete intersection morphisms, this is no longer the case. To allow also correspondences that contain such morphisms, the author considers a category that has zigzags of such correspondences as arrows.
In order for an invariant of complex algebraic varieties to be functorial for proper-smooth correspondences, it must be covariantly functorial for proper smooth maps and contravariantly functorial for smooth maps, and these two functorialities must be compatible in the sense that they satisfy a base change formula for fibre products. Given two such invariants a transformation between them that is natural for the co- and contravariant functorialities remains natural for correspondences. For the Todd class transformation, the contravariant functoriality must be changed by an extra Todd class factor. Then the Riemann-Roch Theorem ensures the required naturality. After this correction of the homology theory, the Todd class transformation becomes a functor on the category of proper-smooth correspondences. These invariants are also additive for disjoint unions, and this allows to prove a similar result where the set of correspondences with disjoint union is replaced by its group completion.
The article also studies the invariant that maps a variety \(X\) to a suitable set of equivalence classes of morphisms \(V\to X\), modulo the equivalence relation generated by isomorphism and additivity for a closed subvariety \(W\subseteq V\) and its complement. There is a natural transformation from this theory to homology with coefficients in the polynomial ring \(\mathbb{Q}[y]\), and this is also extended to suitable categories of correspondences.

MSC:

14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19E99 \(K\)-theory in geometry
14C40 Riemann-Roch theorems
18D20 Enriched categories (over closed or monoidal categories)
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
14F99 (Co)homology theory in algebraic geometry
55N99 Homology and cohomology theories in algebraic topology
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References:

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