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. They study the Galois symbol for an elliptic curve E over a local field F with good ordinary reduction. In particular, they provide a class of examples of surfaces (self products of such elliptic curves), for which the cycle map is injective on the the Albanese kernel modulo p,wherep is the characteristic of the residue field, even when the p-torsion of E is not F -rational. K. Murty’s article deals with the semiregularity map defined by Bloch, and semiregu- larity in the context of abelian varieties. The paper by N. Naumann, M. Spitzweck, and P. A. Østvær uses the motivic Landweber exact functor theorem to deduce that the Bott inverted infinite projective space is homotopy algebraic K-theory. V. Snaith’s paper draws attention to a phenomena in motivic stable homotopy in which Adams operations should control reduced power operations in motivic co- homology. J. Stienstra’s paper recasts the correspondence between 3-dimensional toric Calabi-Yau singularities and quivers with superpotential in the setting of an (N −2)-dimensional abelian algebraic group acting on a linear space CN .Heshows how the quiver with superpotential gives a simple explicit description of the Chow forms of the closures of the orbits in the projective space PN−1. Acknowledgments

The editors are extremely grateful to the Fields Institute, the Clay Mathe- matics Institute, the National Science Foundation (NSF), as well as the organizers connected with the thematic program on geometric aspects of homotopy theory of schemes, for including this conference in their program, and for providing some NSF funds for participants from the USA. A special thanks goes to Rick Jardine, both logistically and financially, for his unwavering support in making this conference possible. We also wish to thank the contributors to this volume, and the referees, for doing a splendid job, particularly under pressing time constraints. Finally, it is a pleasure to acknowledge the excellent support provided by the staff of the Fields Institute both before and during the conference, as well as the efforts beyond the call of duty by Debbie Iscoe for the preparation of these proceedings.

The Editors, Rob de Jeu James D. Lewis

ix

Speakers and Talks

Alexander A. Beilinson (Chicago), Towards a motivic descent Spencer Bloch (Chicago), Mixed Hodge structurers and motives in physics H´el`ene Esnault (Essen), Remarks and questions on coniveau Eric M. Friedlander (Northwestern), Musings about algebraic cycles modulo algebraic equivalence Henri Gillet (Univ. Illinois, Chicago), Arithmetic intersection theory on stacks Alexander Goncharov (Brown), Motivic fundamental groups of curves and Feynman integrals Luc Illusie (Paris-Sud), On Gabber’s recent work in ´etale cohomology Uwe Jannsen (Regensburg), Finiteness of motivic cohomology and resolution of singularities Bruno Kahn (Paris 7), Questions on weights and Albanese varieties Kazuya Kato (Kyoto), Non-commutative Iwasawa theory and Hilbert modular forms Marc Levine (Northeastern), Motivic Postnikov towers Stephen Lichtenbaum (Brown), The conjecture of Birch and Swinnerton- Dyer is misleading Arthur Ogus (Berkeley), Functoriality of the Cartier transform Wayne Raskind (USC), p-adic intermediate Jacobians Takashi Saito (Tokyo), Wild ramification and the characteristic cycle of an -adic ´etale sheaf Chad Schoen (Duke), Some surfaces of general type in abelian varieties Tony Scholl (Cambridge), Some Eisenstein series Vasudevan Srinivas (TIFR), Oriented intersection multiplicities

xi

Titles in This Series

56 Rob de Jeu and James D. Lewis, Editors, Motives and algebraic cycles: A celebration in honour of Spencer J. Bloch, 2009 55 Panos M. Pardalos and Thomas F. Coleman, Editors, Lectures on global optimization, 2009 54 Noriko Yui, Helena Verrill, and Charles F. Doran, Editors, Modular forms and string duality, 2008 53 Mikhail Lyubich and Michael Yampolsky, Editors, Holomorphic dynamics and renormalization: A volume in honour of John Milnor’s 75th birthday, 2008 52 Luigi Rodino, Bert-Wolfgang Schulze, and M. W. Wong, Editors, Pseudo-differential operators: Partial differential equations and time-frequency analysis, 2007 51 Giovanni Forni, Mikhail Lyubich, Charles Pugh, and Michael Shub, Editors, Partial hyperbolic dynamics, laminations, and Teichm¨uller flow, 2007 50 Ilia Binder and Dirk Kreimer, Editors, Universality and renormalization, 2007 49 Wayne Nagata and N. Sri Namachchivaya, Editors, Bifurcation theory and spatio-temporal pattern formation, 2006 48 Hermann Brunner, Xiao-Qiang Zhao, and Xingfu Zou, Editors, Nonlinear dynamics and evolution equations, 2006 47 Hans U. Boden, Ian Hambleton, Andrew J. Nicas, and B. Doug Park, Editors, Geometry and topology of manifolds, 2005 46 Thierry Passot, Catherine Sulem, and Pierre-Louis Sulem, Editors, Topics in kinetic theory, 2005 45 Ragnar-Olaf Buchweitz and Helmut Lenzing, Editors, Representations of algebras and related topics, 2005 44 Lajos Horv´ath and Barbara Szyszkowicz, Editors, Asymptotic methods in stochastics, 2004 43 George Janelidze, Bodo Pareigis, and Walter Tholen, Editors, Galois theory, Hopf algebras, and semiabelian categories, 2004 42 Saber Elaydi, Gerry Ladas, Jianhong Wu, and Xingfu Zou, Editors, Difference and differential equations, 2004 41 Alf van der Poorten and Andreas Stein, Editors, High primes and misdemeanours: Lectures in honour of the 60th birthday of Hugh Cowie Williams, 2004 40 Vlastimil Dlab and Claus Michael Ringel, Editors, Representations of finite dimensional algebras and related topics in Lie theory and geometry, 2004 39 Stephen Berman, Yuly Billig, Yi-Zhi Huang, and James Lepowsky, Editors, Vector operator algebras in mathematics and physics, 2003 38 Noriko Yui and James D. Lewis, Editors, Calabi-Yau varieties and mirror symmetry, 2003 37 Panos Pardalos and Henry Wolkowicz, Editors, Novel approaches to hard discrete optimization, 2003 36 Shigui Ruan, Gail S. K. Wolkowicz, and Jianhong Wu, Editors, Dynamical systems and their applications in biology, 2003 35 Yakov Eliashberg, Boris Khesin, and Fran¸cois Lalonde, Editors, Symplectic and contact topology: Interactions and perspectives, 2003 34 T. J. Lyons and T. S. Salisbury, Editors, Numerical methods and stochastics, 2002 33 Franz-Viktor Kuhlmann, Salma Kuhlmann, and Murray Marshall, Editors, Valuation theory and its applications, Volume II, 2003 32 Franz-Viktor Kuhlmann, Salma Kuhlmann, and Murray Marshall, Editors, Valuation theory and its applications, Volume I, 2002 31 A.Galves,J.K.Hale,andC.Rocha,Editors, Differential equations and dynamical systems, 2002 TITLES IN THIS SERIES

30 Roberto Longo, Editor, Mathematical physics in mathematics and physics: Quantum and operator algebraic aspects, 2001 29 Teresa Faria and Pedro Freitas, Editors, Topics in functional differential and difference equations, 2001 28 David R. McDonald and Stephen R. E. Turner, Editors, Analysis of communication networks: Call centres, traffic and performance, 2000 27 Shui Feng and Anna T. Lawniczak, Editors, Hydrodynamic limits and related topics, 2000 26 Neal Madras, Editor, Monte Carlo methods, 2000 25 A. G. Ramm, P. N. Shivakumar, and A. V. Strauss, Editors, Operator theory and its applications, 2000 24 Edward Bierstone, Boris Khesin, Askold Khovanskii, and Jerrold E. Marsden, Editors, The Arnoldfest: Proceedings of a conference in honour of V. I. Arnold for his sixtieth birthday, 1999 23 Myrna H. Wooders, Editor, Topics in mathematical economics and game theory. Essays in honor of Robert J. Aumann, 1999 22 Graciela Chichilnisky, Editor, Topology and markets, 1999 21 Shigui Ruan, Gail S. K. Wolkowicz, and Jianhong Wu, Editors, Differential equations with applications to biology, 1998 20 Peter A. Fillmore and James A. Mingo, Editors, Operator algebras and their applications II, 1998 19 William G. Dwyer, Stephen Halperin, Richard Kane, Stanley O. Kochman, Mark E. Mahowald, and Paul S. Selick (Editor-in-Chief), Editors, Stable and unstable homotopy, 1998 18 Panos M. Pardalos and Henry Wolkowicz, Editors, Topics in semidefinite and interior-point methods, 1998 17 Joachim J. R. Cuntz and Masoud Khalkhali, Editors, Cyclic cohomology and noncommutative geometry, 1997 16 Victor P. Snaith, Editor, Algebraic K-theory, 1997 15 Stephen P. Braham, Jack D. Gegenberg, and Robert J. McKellar, Editors, Sixth Canadian conference on general relativity and relativistic astrophysics, 1997 14 Mourad E. H. Ismail, David R. Masson, and Mizan Rahman, Editors, Special functions, q-series and related topics, 1997 13 Peter A. Fillmore and James A. Mingo, Editors, Operator algebras and their applications, 1997 12 Dan-Virgil Voiculescu, Editor, Free probability theory, 1997 11 Colleen D. Cutler and Daniel T. Kaplan, Editors, Nonlinear dynamics and time series: Building a bridge between the natural and statistical sciences, 1997 10 Jerrold E. Marsden, George W. Patrick, and William F. Shadwick, Editors, Integration algorithms and classical mechanics, 1996 9 W. H. Kliemann, W. F. Langford, and N. S. Namachchivaya, Editors, Nonlinear dynamics and stochastic mechanics, 1996 8 Larry M. Bates and David L. Rod, Editors, Conservative systems and quantum chaos, 1996 7 William F. Shadwick, Perinkulam Sambamurthy Krishnaprasad, and Tudor Stefan Ratiu, Editors, Mechanics day, 1996

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/.

Spencer J. Bloch has, and continues to have, a profound influence on the subject of Algebraic K-Theory, Cycles and Motives. This book, which is comprised of a number of independent research articles written by leading experts in the field, is dedicated in his honour, and gives a snapshot of the current and evolving nature of the subject. Some of the articles are written in an expository style, providing a perspective on the current state of the subject to those wishing to learn more about it. Others are more technical, representing new developments and making them especially interesting to researchers for keeping abreast of recent progress.

AMS on the Web FIC/56 www.ams.org