Lectures on Algebraic Cycles, Second Edition (New Mathematical Monographs)

156 Pages·2010·0.803 MB·English

by  Spencer Bloch

This page intentionally left blank
LecturesonAlgebraicCycles
SecondEdition
SpencerBloch’s1979Dukelectures,amilestoneinmodernmathematics,havebeen
outofprintalmostsincetheirfirstpublicationin1980,yettheyhaveremained
influentialandarestillthebestplacetolearntheguidingphilosophyofalgebraic
cyclesandmotives.Thisedition,nowprofessionallytypeset,hasanewprefacebythe
authorgivinghisperspectiveondevelopmentsinthefieldoverthepast30years.
Thetheoryofalgebraiccyclesencompassessuchcentralproblemsinmathematics
astheHodgeconjectureandtheBloch–Katoconjectureonspecialvaluesofzeta
functions.ThebookbeginswithMumford’sexampleshowingthattheChowgroupof
zero-cyclesonanalgebraicvarietycanbeinfinitedimensional,andexplainshow
HodgetheoryandalgebraicK-theorygivenewinsightsintothisandotherphenomena.
spencer blochisR.M.HutchinsDistinguishedServiceProfessorinthe
DepartmentofMathematicsattheUniversityofChicago.
NEWMATHEMATICALMONOGRAPHS
EditorialBoard
Be´laBolloba´s
WilliamFulton
AnatoleKatok
FrancesKirwan
PeterSarnak
BarrySimon
BurtTotaro
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisit
http://www.cambridge.org/uk/series/sSeries.asp?code=NMM
1 M.CabanesandM.EnguehardRepresentationTheoryofFiniteReductiveGroups
2 J.B.GarnettandD.E.MarshallHarmonicMeasure
3 P.CohnFreeIdealRingsandLocalizationinGeneralRings
4 E.BombieriandW.GublerHeightsinDiophantineGeometry
5 Y.J.IoninandM.S.ShrikhandeCombinatoricsofSymmetricDesigns
6 S.Berhanu,P.D.CordaroandJ.HounieAnIntroductiontoInvolutiveStructures
7 A.ShlapentokhHilbert’sTenthProblem
8 G.MichlerTheoryofFiniteSimpleGroupsI
9 A.BakerandG.Wu¨stholzLogarithmicFormsandDiophantineGeometry
10 P.KronheimerandT.MrowkaMonopolesandThree-Manifolds
11 B.Bekka,P.delaHarpeandA.ValetteKazhdan’sProperty(T)
12 J.NeisendorferAlgebraicMethodsinUnstableHomotopyTheory
13 M.GrandisDirectedAlgebraicTopology
14 G.MichlerTheoryofFiniteSimpleGroupsII
15 R.SchertzComplexMultiplication
Lectures on Algebraic Cycles
Second Edition
SPENCER BLOCH
UniversityofChicago
CAMBRIDGEUNIVERSITYPRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521118422
© David Leibowitz 2010
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2010
ISBN-13 978-0-511-90031-0 eBook (EBL)
ISBN-13 978-0-521-11842-2 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
Contents
Prefacetothesecondedition pagevii
0 Introduction 3
1 Zero-cyclesonsurfaces 9
Appendix:OnanargumentofMumfordinthetheoryofalgebraic
cycles 21
2 Curvesonthreefoldsandintermediatejacobians 25
3 Curvesonthreefolds–therelativecase 37
4 K-theoreticandcohomologicalmethods 45
5 TorsionintheChowgroup 59
6 ComplementsonH2(K ) 69
2
7 Diophantinequestions 79
8 Relativecyclesandzetafunctions 97
9 Relativecyclesandzetafunctions–continued 115
Bibliography 123
Index 129

Preface to the second edition
30Yearslater...
Looking back over these lectures, given at Duke University in 1979, I can
say with some pride that they contain early hints of a number of important
themes in modern arithmetic geometry. Of course, the flip side of that coin
is that they are now, thirty years later, seriously out of date. To bring them
up to date would involve writing several more monographs, a task best left
tomathematiciansthirtyyearsyoungerthanme.WhatIproposeinsteadisto
commentfairlybrieflyonseveralofthelecturesinanattempttoputthereader
in touch with what I believe are the most important modern ideas in these
areas. The section on motives just below is intended as a brief introduction
tothemodernviewpointonthatsubject.Theremainingsectionsuntilthelast
followroughlythecontentoftheoriginalbook,thoughthetitleshavechanged
slightly to reflect my current emphasis. The last section, motives in physics,
representsmyrecentresearch.
IntheoriginalvolumeIincludedaquotefromCharlieChan,thegreatChi-
nese detective, who told his bumbling number one son “answer simple, but
questionveryveryhard.”Itseemedtomeanappropriatecommentonthesub-
jectofalgebraiccycles.Giventheamazingdeepnewideasintroducedintothe
subject in recent years, however, I think now that the question remains very
veryhard,buttheanswerisperhapsnolongersosimple...
AttheendofthisessayIincludeabriefbibliography,whichisbynomeans
complete. It is only intended to illustrate the various ideas mentioned in the
text.
viii Prefacetothesecondedition
Motives
Muchoftherecentworkinthisareaiscenteredaroundmotivesandthecon-
struction – in fact various constructions, due to Hanamura (1995), Levine
(1998),andVoevodsky(Mazzaetal.2006;Voevodskyetal.2000)–ofatrian-
gulatedcategoryofmixedmotives.IwillsketchVoevodsky’sconstructionas
it also plays a central role in his proof of the Bloch–Kato–Milnor conjecture
discussed in Lecture 5. Then I will discuss various lectures from the original
monograph.
Let k be a field. The category Cor is an additive category with objects
k
smooth k-varieties: Ob(Cor ) = Ob(Sm ). Morphisms Z: X → Y are finite
k k (cid:1)
linearcombinationsofcorrespondencesZ = nZ whereZ ⊂ X×Y isclosed
i i i
andtheprojectionπ : Z → Xisfiniteandsurjective.Intuitively,wemaythink
i i
ofZ asamap X → SymnY associatingto x ∈ X thefibre f−1(x)viewedasa
i i
zero-cycleonY.Thereisanevidentfunctor
Sm →Cor
k k
whichistheidentityonobjects.
ApresheafonacategoryCwithvaluesinanabeliancategoryAissimplya
contravariantfunctorF: Cop → A.AnA-valuedpresheaf F onCor induces
k
apresheafF| onSm .Intuitively,toliftapresheafGfromSm toCor one
Smk k k k
needs astructureoftracemapsortransfers f∗: G(Z) → G(X)forZ/X finite.
PresheavesonCor arereferredtoaspresheaveswithtransfers.
k
ForX ∈Ob(Sm )onehastherepresentablesheafZ (X)definedby
k tr
Z (X)(U)=Hom (U,X).
tr Cork
Animportantelaborationonthisideayieldsforpointedobjects x ∈ X
i i
Z ((X ,x )∧···∧(X ,x))
tr 1 1 n i (cid:2)(cid:3) (cid:5) (cid:6)
:=Coker Z (X ×···X(cid:4) ×···X )→Z ( X) .
tr 1 i n tr i
(cid:7)
Inparticular,onedefinesZ ( nG )bytakingX =A1−{0}andx =1.
tr m i i
A presheaf with transfers F is called homotopy invariant if, with obvious
notation, i∗ = i∗: F(U × A1) → F(U). The complex of chains C∗(F) on a
0 1
presheafwithtransfersF isthepresheafofcomplexes(placedincohomologi-
caldegrees[−∞,0])
C∗(F):=U (cid:7)→···→ F(U×∆n)→···→ F(U×∆0)
Here
(cid:1)
(0.1) ∆n :=Speck[t ,...,t ]/( t −1)
0 n i