The geometry of algebraic cycles : proceedings of the Conference on Algebraic Cycles, Columbus, Ohio, March 25-29, 2008

202 Pages·2010·1.43 MB·English

by  Reza Akhtar

The Geometry of
Algebraic Cycles

Clay Mathematics Proceedings
Volume 9
The Geometry of
Algebraic Cycles
Proceedings of the Conference
on Algebraic Cycles,
Columbus, Ohio
March 25–29, 2008
Reza Akhtar
Patrick Brosnan
Roy Joshua
Editors
American Mathematical Society
Clay Mathematics Institute
2010 Mathematics Subject Classification. Primary 14C15, 14C25, 14C30, 14C35, 14F20,
14F42, 19E15.
Library of Congress Cataloging-in-Publication Data
ConferenceonAlgebraicCycles(2008: Columbus,Ohio)
Thegeometryofalgebraiccycles: proceedingsoftheConferenceonAlgebraicCycles,Colum-
bus,Ohio,March25–29,2008/RezaAkhtar,PatrickBrosnan,RoyJoshua,editors.
p.cm. —(Claymathematicsproceedings;v.9)
Includesbibliographicalreferences.
ISBN978-0-8218-5191-3(alk.paper)
1.Algebraiccycles—Congresses. 2.Geometry,Algebraic—Congresses. I.Akhtar,Reza,1973–
II.Brosnan,Patrick,1968– III.Joshua,Roy,1956– IV.Title.
QA564.C65685 2008
516.3(cid:2)5—dc22
2010010765
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Contents
Preface vii
Transcendental Aspects 1
The Hodge Theoretic Fundamental Group and its Cohomology 3
Donu Arapura
The Real Regulator for a Self-product of a General Surface 23
Xi Chen and James D. Lewis
Lipschitz Cocycles and Poincar´e Duality 33
Eric Friedlander and Christian Haesemeyer
On the Motive of a K3 Surface 53
Claudio Pedrini
Two Observations about Normal Functions 75
Christian Schnell
Positive Characteristics and Arithmetic 81
Autour de la conjecture de Tate `a coefficients Z pour les vari´et´es sur les corps
(cid:2)
finis 83
Jean-Louis Colliot-Th´el`ene et Tama´s Szamuely
Regulators via Iterated Integrals (Numerical Computations) 99
Herbert Gangl
Zero-Cycles on Algebraic Tori 119
Alexander S. Merkurjev
Chow-Ku¨nneth Projectors and (cid:2)-adic Cohomology 123
Andrea Miller
Connections with Mathematical Physics 135
Motives Associated to Sums of Graphs 137
Spencer Bloch
Double Shuffle Relations and Renormalization of Multiple Zeta Values 145
Li Guo, Sylvie Paycha, Bingyong Xie, and Bin Zhang
v

Preface
The subject of algebraic cycles has its roots in the study of divisors, extending
as far back as the nineteenth century; however, the field truly began to blossom
inthe mid-twentieth century afterGrothendieck’s formulation of a seriesof conjec-
tureswhichnowbearhisname. Sincethen,algebraiccycleshavemadeasignificant
impact on many fields of mathematics, among them number theory, algebraic ge-
ometry, and mathematical physics, to name only a few. Spencer Bloch introduced
thehigherChowgroupsintheearly1980sextendingtheclassicalChow-groupsand
roughlyhavingthesame relationshiptoalgebraicK-theoryassingular cohomology
has to topological K-theory. The subject has risen to prominence in recent years
in light of the work of Suslin, Voevodsky, and Friedlander on motivic cohomology,
which also identified Bloch’s higher Chow groups with the latter. In particular
Voevodsky’s solution of the Milnor conjecture, and work on its extension to other
primes (the so-called Bloch-Kato conjecture) has stimulated plenty of interest and
work in this area.
Algebraic Cycles II was conceived a sequel to the Conference on Algebraic Cy-
cles, held at the Ohio State University in December 2000. The goal of both these
conferences was to stimulate further activity in this area by gathering together ex-
perts alongside younger mathematicians beginning work in the field. The scientific
program of Algebraic Cycles II focused on the study of cycles in the contexts of
arithmetic geometry, motivic cohomology, and mathematical physics. The confer-
encewasalsoheldatTheOhioStateUniversity,fromMarch25toMarch29,2008,
and was organized by Reza Akhtar (Miami University), Patrick Brosnan (Univer-
sityofBritishColumbia),RoyJoshua(OhioState)alongwithDavidEllwood(The
Clay Mathematics Institute). The conference featured eighteen 40- or 50- minute
talks and was attended by about 80 participants from all over the world.
It was felt that a volume devoted to the conference proceedings would better
servethemathematicalcommunity. WeareverythankfultotheClayMathematics
Institute for agreeing to publish this volume as part of the Clay Mathematical
Proceedings(jointlypublishedbytheClayMathematicsInstituteandtheAmerican
Mathematical Society). Several of the articles in this volume contain research
presented at this conference, while others represent separate contributions. The
editorsarehappytoacknowledgetheenthusiasticsupporttheyreceivedfrommany
mathematicians working in this general area, either by contributing a paper to the
volume, by serving as referees or by providing other technical assistance. It is our
hope that this volume will be of value both to established researchers working in
the field and to graduate students who have interest in it.
vii
viii PREFACE
TheeditorswishtoextendtheirgratitudetotheNationalSecurityAgency,the
National Science Foundation, the Clay Mathematics Institute, and The Ohio State
University for their financial support of this conference. We recognize in particular
David Ellwood of the Clay Mathematics Institute for his enthusiasm and support
in agreeing to publish these proceedings. We also wish to thank the excellent tech-
nical support we received from Vida Salahi (Clay Mathematics Institute), Marilyn
Radcliff (Ohio State University), Roshini Joshua (for the Web-page and Poster)
and finally everyone else, who helped make this conference a success.
Reza Akhtar, Patrick Brosnan and Roy Joshua
November, 2009
Transcendental Aspects