Chow categories

  • J. Franke

Abstract

This paper arose from an attempt to solve some questions which were posed at the seminar of A. N. Parchin when Deligne’s program ([D]) was reviewed. These problems are related to hypothetical functorial and metrical versions of the Riemann-Roch-Hirzebruch theorem. One of the problems posed by Deligne is, for instance, the following construction:

Let a proper morphism of schemes XS of relative dimension n and a polynomial P(c i (E j )) of absolute degree n + 1 (where deg(c i) = i) in the Chern classes of vector bundles E 1, ... , E k be given. Construct a functor which to the vector bundles E j . on X associates a line bundle on S
$${I_{x/s}}P\left( {{c_i}\left( {{E_j}} \right)} \right)$$
(1)
which is an ‘incarnation’ of ∫x/s P(c i (E j )) CH 1 (S). The functor (1) should be equipped with some natural transformations which correspond to well-known equalities between Chern classes (cf. [D, 2.1]). Further steps in Deligne’s program. are to equip the line bundles (1) with metrics, to prove a functorial version of the Riemann-Roch-Hirzebruch formula which provides an isomorphism between the determinant det(R p *(F)) of the cohomology of a vector bundle F and a certain line bundle of type (1); and (finally) to compare the metric on the right side of the Riemann-Roch isomorphism and the Quillen metric on the determinant of the cohomology.

Keywords

Vector Bundle Line Bundle Commutative Diagram Natural Transformation Follow Diagram Commute 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Kluwer Academic Publishers. Printed in the Netherlands 1990

Authors and Affiliations

  • J. Franke
    • 1
    • 2
  1. 1.Universität JenaJenaGermany
  2. 2.Karl-Weierstraß-Institut für MathematikGermany

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