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FIELDS INSTITUTE COMMUNICATIONS Motives and Algebraic Cycles FIELDS INSTITUTE COMMUNICATIONS THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES Motives and Algebraic Cycles A Celebration in Honour of Spencer J. Bloch Rob de Jeu James D. Lewis Editors American Mathematical Society The Fields Institute for Research in Mathematical Sciences Motives and Algebraic Cycles http://dx.doi.org/10.1090/fic/056 FIELDS INSTITUTE COMMUNICATIONS THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES Motives and Algebraic Cycles A Celebration in Honour of Spencer J. Bloch Rob de Jeu James D. Lewis Editors American Mathematical Society Providence, Rhode Island The Fields Institute for Research in Mathematical Sciences Toronto, Ontario The Fields Institute for Research in Mathematical Sciences The Fields Institute is a center for mathematical research, located in Toronto, Canada. Our mission is to provide a supportive and stimulating environment for mathematics research, innovation and education. The Institute is supported by the Ontario Ministry of Training, Colleges and Universities, the Natural Sciences and Engineering Research Council of Canada, and seven Ontario universities (Carleton, McMaster, Ottawa, Toronto, Waterloo, Western Ontario, and York). In addition there are several affiliated universities and corporate sponsors in both Canada and the United States. Fields Institute Editorial Board: Carl R. Riehm (Managing Editor), Juris Steprans (Acting Director of the Institute), Matthias Neufang (Interim Deputy Director), James G. Arthur (Toronto), Kenneth R. Davidson (Waterloo), Lisa Jeffrey (Toronto), Barbara Lee Keyfitz (Ohio State), Thomas S. Salisbury (York), Noriko Yui (Queen’s). 2000 Mathematics Subject Classification. Primary 11-XX, 14-XX, 16-XX, 19-XX, 55-XX. Library of Congress Cataloging-in-Publication Data Motives and algebraic cycles : a celebration in honour of Spencer J. Bloch / Rob de Jeu, James D. Lewis, editors. p. cm. — (Fields Institute Communications ; v. 56) Includes bibliographical references. ISBN 978-0-8218-4494-6 (alk. paper) 1. Algebraic cycles. 2. Motives (Mathematics). I. Bloch, Spencer. II. Jeu, Rob de, 1964– III. Lewis, James Dominic, 1953– QA564.M683 2009 512.66–dc22 2009023440 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. This publication was prepared by the Fields Institute. http://www.fields.utoronto.ca Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 Contents Introduction vii Acknowledgments ix Speakers and Talks xi Varieties with very Little Transcendental Cohomology 1 Donu Arapura E-Factors for the Period Determinants of Curves 15 Alexander Beilinson Hodge Cohomology of Invertible Sheaves 83 Hel´ ene` Esnault and Arthur Ogus Arithmetic Intersection Theory on Deligne-Mumford Stacks 93 Henri Gillet Notes on the Biextension of Chow Groups 111 Sergey Gorchinskiy D´emonstration G´eom´etrique du Th´eor`eme de Lang-N´eron et Formules de Shioda-Tate 149 Bruno Kahn Surjectivity of the Cycle Map for Chow Motives 157 Shun-ichi Kimura On Codimension Two Subvarieties in Hypersurfaces 167 N. Mohan Kumar, A. P. Rao, and G. V. Ravindra Smooth Motives 175 Marc Levine Cycles on Varieties over Subfields of C and Cubic Equivalence 233 James D. Lewis Euler Characteristics and Special Values of Zeta-Functions 249 Stephen Lichtenbaum v vi Contents Local Galois Symbols on E × E 257 Jacob Murre and Dinakar Ramakrishnan Semiregularity and Abelian Varieties 293 V. Kumar Murty Chern Classes, K-Theory and Landweber Exactness over Nonregular Base Schemes 307 Niko Naumann, Markus Spitzweck, and Paul Arne Østvær Adams Operations and Motivic Reduced Powers 319 Victor Snaith Chow Forms, Chow Quotients and Quivers with Superpotential 327 Jan Stienstra Introduction Spencer J. Bloch, one of the world’s leading mathematicians, has had a sub- stantial impact on, in particular, algebraic K-theory, algebraic cycles and motives. This conference and subsequent proceedings served as a tribute to and a celebra- tion of his work and a dedication to his mathematical heritage. Among those who participated in this conference were his collaborators, former students, a majority of those who have benefited from his trail blazing work, as well as recent post-docs and students. The atmosphere of this conference could be accurately described as “electric”. The feeling one had among participants and speakers alike, was that of being identified as “adopted” students of Spencer Bloch in that they had read a rather large proportion of his work and were strongly influenced by him. His abil- ity to connect physics with the subject of motives and degenerating mixed Hodge structures is very typical of his style of doing mathematics. One instance of this, earlier on, was his insight into how algebraic K-theory could be useful to algebraic geometry. This in turn revolutionized the subject of algebraic cycles. Virtually all the talks would begin with a few words on how Spencer influenced their research (sometimes rather humourously), followed by a presentation of recent developments in the field of motives and algebraic cycles. The agreeable environment at Fields, specifically the excellent support and facilities of the Fields Institute, together with the backdrop of a bustling major city, added to the upbeat mood of this conference. Many of the participants voiced their appreciation of this fantastic conference, one which they are unlikely to ever forget! Summary of works in this volume. This is a volume comprised of a number of independent research articles. In particular, these papers give a snapshot on the evolving nature of the subject of motives and algebraic cycles, written by leading ex- perts in the field. A breezy summary of the flavour of these articles goes as follows. Motivated by the celebrated Hodge conjecture, D. Arapura’s paper concerns a non- negative integral invariant of a complex smooth projective variety X, which is zero precisely when the cohomology of X is generated by algebraic cycles. In general it gives a measure of the amount of transcendental cohomology of X, or alternatively can be rather loosely thought of as measuring the complexity of the motive of X. A. Beilinson’s article, which addresses an earlier question of P. Deligne, provides an -factorization of the determinant of the period isomorphism associated to a holonomic D-module on a compact complex curve. The paper of H. Esnault and A. Ogus deals with a conjecture of Pink and Roessler on the nature of Hodge coho- mology for a smooth projective scheme over an algebraically closed field, twisted by an invertible sheaf. H. Gillet’s paper deals with extending arithmetic intersection theory to the case of Deligne-Mumford stacks. Such an extension is important since many computations involving arithmetic intersection theory are carried over mod- ular varieties that are best viewed as stacks. S. Gorchinskiy’s article discusses four vii viii Introduction approaches to the biextension of Chow groups and their equivalences, including an explicit construction given by S. Bloch. B. Kahn provides a new proof of the classi- cal finiteness theorem of Lang and N´eron on abelian varieties. The proof is different from the traditional one which uses heights. Rather, it has a cycle-theoretic flavour employing a famous correspondence trick used by S. Bloch. The paper of Shun-ichi Kimura concerns the following. For a smooth projective variety X over C,Uwe Jannsen proved that if the cycle map cl : CH∗(X; Q) → H∗(X, Q) is surjective, then it is actually bijective. Kimura generalizes this result to Chow motives. The work of N. Mohan Kumar, A. P. Rao, G. V. Ravindra revisits an earlier question of P. Griffiths and J. Harris of a Noether-Lefschetz nature, by providing the existence of a large class of counterexamples, which subsumes C. Voisin˜os earlier counterex- amples. M. Levine’s paper concerns the construction of DG categories of smooth motives over a smooth k-scheme S (k a field) essentially of finite type. This in turn gives a well-behaved triangulated category of mixed motives over S generated by smooth and projective S-schemes. J. D. Lewis’ contribution involves constructing infinite rank subspaces of cycles with prescribed transcendence degree (over Q)in graded pieces of a certain candidate Bloch-Beilinson filtration for smooth projec- tive varieties defined over subfields of C, complementing some earlier works of C. Schoen and P. Griffiths, M. Green, and K. Paranjape. S. Lichtenbaum’s article provides some evidence for the philosophy that all special values of arithmetic zeta and L-functions are given by Euler characteristics. Regarding the work of J. P. Murre and D. Ramakrishnan, the Galois symbol on elliptic curves is closely related to the cycle map of a product of elliptic curves.
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