A Course in Hodge Theory (With Emphasis on Multiple Integrals)

380 Pages·2020·3.266 MB·English

by  Hossein Movasati

Hossein Movasati
A Course in Hodge Theory
With Emphasis on Multiple Integrals
September7,2020
Publisher

v
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Wheatwilleventuallygrowallaround;
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Itsvaluesoarsnowwithitmenarefed;
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Life,wisdom,andintelligencetheygain,
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Jalalal-DinMuhammadBalkhi,Masnavi,Book1,Section148.Translation:JawidMojaddedi,OxfordWorld’sClassic.Calligraphy:nastaliqonline.ir.

Thisbookisdedicatedtothememoryofmy
parentsRogayehandAli,andtomyfamily
SaraandOmid

Preface
ThemainobjectiveofthepresentbookistogiveanintroductiontoHodgetheory
anditsmainconjecture,theso-calledHodgeconjecture.Weaimtoexploretheori-
ginsofHodgetheorymuchbeforetheintroductionofHodgedecompositionofthe
de Rham cohomology of smooth projective varieties. This is namely the study of
elliptic,abelianandmultipleintegralsoriginatedfromtheworksofCauchy,Abel,
Jacobi, Riemann, Poincare´, Picard and Lefschetz among many others. Therefore,
the reader is warned that he or she will find in this book a partial presentation of
themodernHodgetheory.Thepresentbookisintendedtobeanincompleteresus-
citation of Picard and Simart’s treatise The´orie des fonctions alge´briques de deux
variables inde´pendantes after almosta century,keeping inmind that themain ob-
ject of study is the multiple integral itself and not other by-products. A complete
analysis of this treatise and other contributions need a historian in mathematics.
This is beyond the scope of this book. Another main emphasis of this book is on
thecomputationalaspectsofthetheorysuchascomputinghomologiesbymeansof
vanishingcycles,deRhamcohomologies,Gauss-Maninconnections,Hodgecycles,
etc. The development of Hodge theory during the last decades has put it far from
itsoriginandtheintroductionofmirrorsymmetrybystringtheoristsandtheperiod
manipulations of the B-model Calabi-Yau varieties, have risen the need for a text
inHodgetheorywithmoreemphasisonperiodsandmultipleintegrals.Weaimto
present materials which are not covered in J. Lewis’s book A survey of the Hodge
conjectureandC.Voisin’sbooksHodgetheoryandcomplexalgebraicgeometry,I,
II,therefore,thereaderwillnotfindinthisbooksomeofthefundamentaltheorems
in modern Hodge theory. We have tried to keep the text self-sufficient, however, a
basicundergraduateknowledgeofComplexAnalysis,DifferentialEquations,Alge-
braicTopologyandAlgebraicGeometrywillmakethereadingofthetextsmoother.
Thetextismainlywrittenfortwoprimarytargetaudiences:graduatestudentswho
want to learn Hodge theory and get a flavor of why the Hodge conjecture is hard
to deal with, and mathematicians who use periods and multiple integrals in their
researchandwouldliketoputtheminaHodgetheoreticframework.Wehopethat
ourtext,togetherwiththosementionedabove,makesHodgetheorymoreaccessible
toabroaderpublic.
ix
x Preface
HosseinMovasati
RiodeJaneiro,RJ,Brazil
Acknowledgements
ThepresenttextiswrittenduringmanyyearsthatItaughtHodgetheoryatIMPA,
RiodeJaneiro.Here,Iwouldliketothankthestudentsandcolleagueswhosatin
my courses and made many questions and comments. This includes, but not lim-
itedto,KhosroM.Shokri,YounesNikdelan,YadollahZare,RobertoVillaflor,Rau´l
Cha´vez,MarcusTorres,JorgeDuque,DanielLopezandWalterGaviria.Mysincere
thanksgotoSampeiUsuiandFouadElZeinwhoattheearlystagesofmyworkin
Hodgetheorysupportedmyresearchandhelpedmetounderstandwhatisdoneand
what is not done. I would also like to thank many mathematics institutes for pro-
vidingexcellentresearchambienceduringthepreparationofthepresenttext.This
includes: Instituto de Matema´tica Pura e Aplicada (IMPA) in Rio de Janeiro (my
home institute), Max-Planck Institute for Mathematics (MPIM) in Bonn, Institute
forPhysicsandMathematics(IPM)inTehranandCenterofMathematicalSciences
andApplications(CMSA)atHarvarduniversity(shortvisits).Duringtheprepara-
tion of the present book I had email contacts with many mathematicians. I would
liketothankthemforthepermissiontoreproducesomeoftheircommentsinthis
book.MysincerethanksgotoPierreDeligneforallhiscorrectionsandcomments
concerningbothhistoricalandtechnicalaspectsofHodgetheory.Inparticular,his
commentsonChapter18havebeenofagreatvaluefortheauthor.Iwouldliketo
thankDavidMumfordforhiscommentsonChapter3andsomeclarificationsonits
historical content.Thanks go toDuco van Straten,Stefan Reiter, EmreSerto¨z and
AnanyoDanfortheircommentsonthisbook.MyheartfeltthanksgotoShing-Tung
Yauforhisinterest,supportandmanyinvitationstovisitCMSAatHarvard.Iwould
liketothankmymentors,Ce´sarCamacho,PauloSadandAlcidesLinsNeto,with
whom I started my mathematical carrier. I would like to thank my wife Sara and
mysonOmidtowhomthisbookisdedicated,forprovidingalovelyatmosphereat
homewheremostofthebookiswritten.Theelaborationofthefiguresinthisbook
wouldnotbepossiblewithoutthecarefulworkofMs.DalilaOchoa,whoprepared
digitalized version from my original hand drawings, and the financial support of
FAPERJ.
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