Preface

Summary

These proceedings contain a selection of papers from the EAGER conference “Algebraic Cycles and Motives” that was held at the Lorentz Center in Leiden on the occasion of the 75th birthday of Professor J.P. Murre (Aug 30–Sept 3, 2004). The conference attracted many of the leading experts in the field as well as a number of young researchers. As the papers in this volume cover the main research topics and some interesting new developments, they should give a good indication of the present state of the subject. This volume contains sixteen research papers and six survey papers.

The theory of algebraic cycles deals with the study of subvarieties of a given projective algebraic variety X, starting with the free group Z^p (X) on irreducible subvarieties of X of codimension p. In order to make this very large group manageable, one puts a suitable equivalence relation on it, usually rational equivalence. The resulting Chow group CH^p (X) in general might still be very big. If X is a smooth variety, the intersection product makes the direct sum of all the Chow groups into a ring, the Chow ring CH* (X). Hitherto mysterious ring can be studied through its relation to cohomology, the first example of which is the cycle class map: every algebraic cycle defines a class in singular, de Rham, or l-adic cohomology. Ultimately this cohomological approach leads to the theory of motives and motivic cohomology developed by A.