Algebraic Cycles And Motives Volume 1

This 2007 book is a self-contained account of the subject of algebraic cycles and motives.

'Motives' were introduced in the mid-1960s by Grothendieck to explain the analogies among the various cohomology theories for algebraic varieties, and to play the role of the missing rational cohomology. This work contains the texts of the lectures presented at the AMS-IMS-SIAM Joint Summer Research Conference on Motives, held in Seattle, in 1991.

A self-contained account of the subject of algebraic cycles and motives as it stands.

This book presents a coherent suite of computational tools for the study of group cohomology algebraic cycles.

The relations that could or should exist between algebraic cycles, algebraic K-theory, and the cohomology of - possibly singular - varieties, are the topic of investigation of this book. The author proceeds in an axiomatic way, combining the concepts of twisted Poincaré duality theories, weights, and tensor categories. One thus arrives at generalizations to arbitrary varieties of the Hodge and Tate conjectures to explicit conjectures on l-adic Chern characters for global fields and to certain counterexamples for more general fields. It is to be hoped that these relations ions will in due course be explained by a suitable tensor category of mixed motives. An approximation to this is constructed in the setting of absolute Hodge cycles, by extending this theory to arbitrary varieties. The book can serve both as a guide for the researcher, and as an introduction to these ideas for the non-expert, provided (s)he knows or is willing to learn about K-theory and the standard cohomology theories of algebraic varieties.

Spencer Bloch's 1979 Duke lectures, a milestone in modern mathematics, have been out of print almost since their first publication in 1980, yet they have remained influential and are still the best place to learn the guiding philosophy of algebraic cycles and motives. This edition, now professionally typeset, has a new preface by the author giving his perspective on developments in the field over the past 30 years. The theory of algebraic cycles encompasses such central problems in mathematics as the Hodge conjecture and the Bloch–Kato conjecture on special values of zeta functions. The book begins with Mumford's example showing that the Chow group of zero-cycles on an algebraic variety can be infinite-dimensional, and explains how Hodge theory and algebraic K-theory give new insights into this and other phenomena.

The subject of algebraic cycles has its roots in the study of divisors, extending as far back as the nineteenth century. Since then, and in particular in recent years, algebraic cycles have made a significant impact on many fields of mathematics, among them number theory, algebraic geometry, and mathematical physics. The present volume contains articles on all of the above aspects of algebraic cycles. It also contains a mixture of both research papers and expository articles, so that it would be of interest to both experts and beginners in the field.

Varieties with very little transcendental cohomology by D. Arapura $\mathcal{E}$-factors for the period determinants of curves by A. Beilinson Hodge cohomology of invertible sheaves by H. Esnault and A. Ogus Arithmetic intersection theory on Deligne-Mumford stacks by H. Gillet Notes on the biextension of Chow groups by S. Gorchinskiy Demonstration geometrique du theoreme de Lang-Neron et formules de Shioda-Tate by B. Kahn Surjectivity of the cycle map for Chow motives by S.-i. Kimura On codimension two subvarieties in hypersurfaces by N. M. Kumar, A. P. Rao, and G. V. Ravindra Smooth motives by M. Levine Cycles on varieties over subfields of $\mathbb{C}$ and cubic equivalence by J. D. Lewis Euler characteristics and special values of zeta-functions by S. Lichtenbaum Local Galois symbols on $E\times E$ by J. Murre and D. Ramakrishnan Semiregularity and Abelian varieties by V. K. Murty Chern classes, $K$-theory and Landweber exactness over nonregular base schemes by N. Naumann, M. Spitzweck, and P. A. Ostvaer Adams operations and motivic reduced powers by V. Snaith Chow forms, Chow quotients and quivers with superpotential by J. Stienstra

A self-contained account of the subject of algebraic cycles and motives as it stands.